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Manifestations of One-Dimensional Electronic Correlations in Higher-Dimensional Systems

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Manifestations of One-Dimensional Electronic Correlations in Higher-Dimensional Systems
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2008

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Backscattering ( jstor )
Conductivity ( jstor )
Electrons ( jstor )
Fermions ( jstor )
Impurities ( jstor )
Magnetic fields ( jstor )
Natural logarithms ( jstor )
Specific heat ( jstor )
Temperature dependence ( jstor )
Thermodynamics ( jstor )

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MANIFESTATIONS OF ONE-DIMENSIONAL ELECTRONIC CORRELATIONS
IN HIGHER-DIMENSIONAL SYSTF:\ IS
















By

RONOJOY SAHA


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Ronojoy Saha



































To my family















ACKNOWLEDGMENTS

First and formeost I would like to thank my research supervisor, Professor

Dmitrii Maslov, for his constant encouragement and guidance throughout the

entire course of my research. His enthusiasm, dedication, and optimism towards

physics research have been extremely infectious. The countless hours I have spent

discussing physics with him were highly productive and intellectually stimulating.

I would like to thank Professor Jim Dufty, Professor Arthur Hebard and

Professor Pradeep Kumar, who were ahv-- willing and open to discuss any physics

related questions. I am honored and grateful to Professor Russell Bowers, Professor

Adrian Roitberg, Professor Khandker Muttalib, Professor Sergei Obukhov and

Professor Arthur Hebard for serving on my supervisory committee.

My thanks go to the Physics Department secretaries, Ms. Balkcom, Ms.

Latimer, Ms. Nichola and Mr. Williams; and to my friends Partho, Vidya, Suhas,

Aditi, Aparna and Karthik for their help and support.

I would like to thank my wife and best friend Sreya for being my source of

strength and inspiration through all these years.

I would like to thank my family for the unconditional love, support and

encouragement they provided through the years.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iv

LIST OF FIGURES ................... ......... vii

ABSTRACT ... .. .. .. ... .. .. .. .. ... .. .. .. ... .. .. x

CHAPTER

1 INTRODUCTION .................... ....... 1

1.1 Transport in Ultra Strong Magnetic Fields ...... ........ 2
1.1.1 Weak Localization QCC ................. .. .. 5
1.1.2 Interaction Correction to the Conductivity-Altshuler Aronov
Corrections (class I) ..... . . .... 10
1.1.3 Corrections to WL QCC due to Electron-Electron Interactions:
Dephasing (class II) ....... . . .... 17
1.2 Non-Fermi Liquid Features of Fermi Liquids: 1D Physics in Higher
Dimensions ................... . . 19
1.3 Spin Susceptibility near a Ferromagnetic Quantum Critical Point
in Itinerant Two and Three Dimensional Systems. . ... 34
1.3.1 Hertz's LGW Functional ............ .. .. .. 36

2 CORRELATED ELECTRONS IN ULTRA-HIGH MAGNETIC FIELD:
TRANSPORT PROPERTIES .................. ...... 43

2.1 Localization in the Ultra Quantum Limit . . ...... 45
2.1.1 Diagrammatic Calculation for the Conductivity ...... ..46
2.1.2 Quantum Interference Correction to the Conductivity . 50
2.2 Conductivity of Interacting Electrons in the Ultra-Quantum Limit:
Diagrammatic Approach .................. .... .. 58
2.2.1 Self-Energy Diagrams .................. ..... 61
2.2.1.1 Diagram Fig. 2-10(a) ............... .. 64
2.2.1.2 Diagram Fig. 2-11(a). .............. 67
2.2.2 Vertex Corrections .................. .. 69
2.2.2.1 Diagram 2-10(b) .................. .. 69
2.2.2.2 Diagram Fig. 2-11(b) ............... .. 71
2.2.3 Sub-Leading Diagrams .................. .. 72
2.2.4 Correction to the Conductivity ................ 73
2.2.5 Effective Impurity Potential ................ 73









2.3 Impurity Scattering Cross-Section for Interacting Electrons . 75
2.3.1 Non-Interacting Case ..... ........... ...... 76
2.3.2 Interacting Case ................ ... ... .. 78
2.4 Experiments ............... ......... .. 84
2.5 Conclusions ............... .......... .. 93

3 SINGULAR CORRECTIONS TO THERMODYNAMICS FOR A ONE
DIMENSIONAL INTERACTING SYSTEM: EVOLUTION OF THE
NONANALYTIC CORRECTIONS TO THE FERMI LIQUID BEHAVIOR 95

3.1 One-Dimensional Model ............... .. .. .. 99
3.2 Specific Heat ..... . . ............. 105
3.2.1 Specific Heat from the Second Order Self Energy ....... 107
3.2.2 Specific Heat from the Thermodynamic Potential at Second
Order . . . . .... 112
3.2.3 Specific Heat from Third Order Self Energy . ... 116
3.2.4 Specific Heat from the Sine-Gordon Model . .... 127
3.3 Spin Susceptibility .................. ........ 130
3.4 Experiments .................. ............ 136
3.5 Conclusion. .................. ........... 137

4 SPIN SUSCEPTIBILITY NEAR A FERROMAGNETIC QUANTUM
CRITICAL POINT IN ITINERANT TWO AND THREE DIMENSIONAL
SYSTEMS ................... .............. 138

4.1 Spin Susceptibility X,(H), in 2D .............. .. 141
4.2 Spin Susceptibility Xs(H), in 3D .............. .. 145
4.3 Spin Susceptibility for a Fermi Liquid in 2D . . 147
4.4 Spin Susceptibility near the Quantum Critical Point ......... 156
4.4.1 2D ........... ... ....... ........ 158
4.4.2 3D ........... ... ....... ........ 162
4.5 Conclusions . . . . . . . 165

5 CONCLUSIONS .................. .......... 166

REFERENCES .............. ........ . .. 168

BIOGRAPHICAL SKETCH ............. . . .... 174















LIST OF FIGURES
Figure page

1-1 Weak localization corrections .............. ... 5

1-2 Ladder diagram for M (diffuson) and C (Cooperon). .. . 7

1-3 Quantum corrections to conductivity for noninteracting electrons . 7

1-4 Scattering by Friedel oscillations. ................ ..... 12

1-5 Self-energy at first order in interaction with a bosonic field . ... 23

1-6 Kinematics of scattering. (a) "Any-angle" scattering leading to regular
FL terms in self-energy; (b) Dynamical forward scattering; (c) Dynamical
backscattering. Processes (b) and (c) are responsible for nonanalytic terms
in the self-energy .................. ............ .. 25

1-7 Non trivial second order diagrams for the self-energy .......... ..26

1-8 Scattering processes responsible for divergent and/or nonanalytic corrections
to the self-energy in 2D. (a) "Forward scattering -an analog of the g4
process in 1D (b) "Forward scattering with anti-parallel momenta-an
analog of the g2 process in 1D (c) "backscattering; with antiparallel momenta-
an analog of the gl process in 1D .............. ...... 29

1-9 Typical trajectories of two interacting fermions . ..... 31

2-1 Diagram (a) is the leading contribution to the self energy at fourth order 48

2-2 Dyson's series ............... ............. .. 49

2-3 Drude conductivity ............... ........... .. 49

2-4 Third and second order fan diagram. ................ .... 50

2-5 Cooperon sequence for 3D electrons in the UQL. Unlike in 1D, each term
in the series comes with a different coefficient c.. . . 54

2-6 First and second order diffuson ............... .... 55

2-7 Interference correction to conductivity .............. .. .. 56









2-8 Crossed diffuson diagrams. Left, a double-diffuson diagram, which also
acquires a mass. Right, a third-order non-cooperon diagram which, up
to a number, gives the same contribution as the third order fan diagram. 58

2-9 First order interaction corrections to the conductivity where effects of
impurities appear only in the disorder-averaged Green's functions. 61

2-10 Exchange diagrams that are first order in the interaction and with a single
extra impurity line. The Green's functions are disorder-averaged. Diagrams
(a) and (b) give In T correction to the conductivity and exchange diagrams
(c), (d) and (e) give sub-leading corrections to the conductivity. . 62

2-11 Hartree diagrams that are first order in the interaction and with a single
extra impurity line. The Green's functions are disorder-averaged. Both
diagrams give In T correction to the conductivity. . ..... 63

2-12 The self-energy correction contained in diagram 2-10(a), denoted in the
y(2--12)
text as 12) . . . . .. .. . . 64

2-13 The self-energy correction contained in diagram 2-11(a), denoted in the
text as 213) ................... ............ ..67

2-14 Diagram 2-10(b) vs diagram 2-11(b). .................. 71

2-15 Effective impurity potential .................. ..... .. 74

2-16 The handle diagram corresponds to diagrams 2-10(a) and 2-11(a) and
the crossing diagram corresponds to 2-10(b) and 2-11(b). . ... 75

2-17 Profile of the Friedel oscillations around a point impurity in a 3D metal
in the UQL. The oscillations decay as 1/z along the magnetic field direction
and have a Gaussian envelope in the transverse direction. . ... 79

2-18 Renormalized conductivities parallel (az) and perpendicular (a,,) to
the direction of the applied magnetic field. Power-law behavior is expected
in the temperature region 1/7 < T < W. ....... . ... 82

2-19 Temperature dependence of the ab-plane resistivity pxx for a graphite
crystal at the c-axis magnetic fields indicated in the legend . ... 86

2-20 Temperature dependence of the c-axis conductivity az for a graphite
crystal in a magnetic field parallel to the c axis. The magnetic field values
are indicated on the plot, with the field increasing downwards, the lowest
plot corresponds to the highest field .................. .. 87

2-21 Temperature dependence (log-log scale) of the ab-plane resistivity scaled
with the field p,,/B2 for a graphite crystal at the c-axis magnetic fields
indicated in the legend ............ . .. 88









2-22 Temperature dependence (on a log-log scale) of the ab-plane resistivity
px/B2 at the highest attained c-axis magnetic field of 17.5T for the same
graphite crystal .................. ............. .. 89

2-23 Phase breaking rate vs T due to electron-phonon scattering ...... ..92

3-1 Interaction vertices ............. . . ... 101

3-2 Non-trivial second order self energy diagrams for right moving fermions 107

3-3 Second order diagrams for the thermodynamic potential with maximum
number of explicit particle-hole bubbles .................. .. 112

3-4 The different choices for the 3rd order diagram. ............. .117

3-5 All 3rd order se diagrams for right movers which have two II2kp ... 122

3-6 All third order self energy diagrams containing two Cooper bubbles 123

3-7 Effective third order self-energy diagrams (the double line is a vertex). 124

3-8 All g2 and gi vertices at 2nd order. ............. .. 125

3-9 All third order self-energy diagrams with two Cooper bubbles or two II2ka
bubbles .................. ................. .. 126

3-10 Second order diagrams for the thermodynamic potential. . ... 132

4-1 Particle-hole type second order diagram for the thermodynamic potential. 143

4-2 Particle-hole type third order diagram for the thermodynamic potential. 144

4-3 The skeleton diagram for the thermodynamic potential. . ... 148

4-4 Fermion self-energy (a) and Bosonic self-energy (b) . . ... 158















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MANIFESTATIONS OF ONE-DIMENSIONAL ELECTRONIC CORRELATIONS
IN HIGHER-DIMENSIONAL SYSTF:\ IS

By

Ronojoy Saha

August 2006

C('! ,i: Dmitrii L. Maslov
Major Department: Physics

In this work we have studied the fundamental aspects of transport and

thermodynamic properties of a one-dimensional (ID) electron system, and

have shown that these 1D correlations pl iv an important role in understanding

the physics of higher-dimensional systems. The first system we studied is a

three-dimensional (3D) metal subjected to a strong magnetic field that confines the

electrons to the lowest Landau level. We investigated the effect of dilute impurities

in the transport properties of this system. We showed that the nature of electron

transport is one dimensional due to the reduced effective dimensionality induced

by the magnetic field. The localization behavior in this system was shown to be

intermediate, between a 1D and a 3D system. The interaction corrections to the

conductivity exhibit power law 1, 1iwi;- a oc T" with a field dependent exponent.

Next we studied the thermodynamic properties of a one-dimensional

interacting system, where we showed that the next-to-leading terms in the specific

heat and spin susceptibility are nonanalytic, in the same way as they are for

higher-dimensional (D = 2, 3) systems. We obtained the nonanalytic, TlnT term

in the specific heat in 1D and showed that although the nonanalytic corrections









in all dimensions arise from the same source, there are subtle differences in the

magnitude of the effect in different dimensions.

In the final part of this work we analyzed the nonanalytic corrections to

the spin susceptibility (Xs(H)) in higher dimensional systems. We showed that,

although there were contributions from non-1D scattering in these nonanalytic

terms, the dominant contribution came from 1D scattering. We also showed that

the second order ferromagnetic quantum phase transition is unstable both in 2D

and 3D, with a tendency towards a first order transition.















CHAPTER 1
INTRODUCTION

One-dimensional interacting systems (Luttinger-liquids) exhibit many features

which appear distinct from their higher-dimensional counterparts (Fermi-liquids).

Our goal in this thesis is to highlight the similarities between higher-D and 1D

systems. The progress in understanding of ID systems has been greatly facilitated

by the availability of exact or .,-vmptotically exact methods (Bethe Ansatz,

bosonization, conformal field theory), which typically do not work very well above

1D. The downside of this progress is that 1D effects, being studied by specifically

1D methods, look somewhat special and not really related to higher dimensions.

We are going to argue that this is not true. 1t ,ii: effects which are viewed as

the hallmarks of 1D physics, e.g., the suppression of tunneling conductance

by the electron-electron interaction, do have higher dimensional counterparts

and stem from essentially the same physics. In particular, scattering at Friedel

oscillations caused by tunneling barriers and impurities is responsible for zero-bias

tunneling anomalies in all dimensions. The difference lies in the magnitude of the

effect, not in its qualitative nature. We illustrate this similarity by showing that

1D correlations p1 iv an important role in understanding the physics of higher

dimensional systems. We studied three seemingly different problems, but as we will

show, all three of them are connected by the common feature of 1D correlations.

Our goal in the introduction is to provide a background for the physics discussed

in the three chapters of this dissertation. We have set h and kB equal to unity

everywhere.









1.1 Transport in Ultra Strong Magnetic Fields

The behavior of an isotropic three-dimensional (3D) metal in a high magnetic

field has been a focus of attention of the condensed matter community for many

decades. Due to Landau quantization of orbits, the energy of an electron in this

system depends only on the momentum along the magnetic field,


E= Pz 2 c
j+(2m 1) (11)

where wc = eH/mc is the cyclotron frequency and n is the Landau level. Thus

the system exhibits effects characteristic of one-dimensional (1D) metals, while

being intrinsically a 3D system. This reduction of effective dimensionality of

charge carriers from 3D to 1D is most pronounced in the ultra-quantum limit

(n = 0, when only the lowest Landau level remains populated) and is expected

to result in a number of unusual phases. It is well known that the ground state

of repulsively interacting electrons in the UQL is unstable with respect to the

formation of a charge density wave [1-3], which has been observed experimentally

in magneto-resistance measurements on graphite in high magnetic fields [4].

The most complete a i i ,-i; of the CDW instability for the case of short range

interactions was performed in Ref. [3], by solving the renormalization-group (RG)

equations for the interaction vertex. On the other hand, it has recently been

shown that for the case of long-range (Coulomb) interactions between electrons, a

3D metal in UQL exhibits Luttinger-liquid like (1D) behavior at energies higher

than the CDW gap [5, 6]. Biagini et al. [5] and Tsai et al. [6] showed that in the

UQL, the tunneling conductance has a power law anomaly (nonlinearities in I-V

characteristics at small biases), which is typical for a one dimensional interacting

system (Luttinger liquid). The magnetic-field-induced Luttinger liquid phase

can be anticipated from the following simplified picture. In a strong magnetic

field, electron trajectories are helices spiraling around the field lines. A bundle of









such trajectories with a common center of orbit can be viewed as a 1D conductor

( V.--i '). In the presence of electron-electron interactions, each vi..-.," considered

separately, is in the LL state. Interactions with small momentum transfers among

electrons on different i.--i -' do not change the LL nature of a single wire [7]. In

chapter 2 of this dissertation we study the transport properties of a disordered

3D metal in the UQL, both with and without electron-electron interactions. Both

the localization and interaction corrections to the conductivity show signatures

typical for one-dimensional systems. Before we get into the details of our study, we

will briefly review the physics of the interplay between the interaction effects and

disorder induced localization in diffusive systems of low dimensionality.

At low temperatures, the conductivity of disordered conductors (normal metals

and semiconductors) is determined by scattering of electrons off quenched disorder

(e.g., impurities and defects). The residual conductivity is given by the Drude
formula,

o- = (12)


where n is the electron concentration, e is the electron charge, 'r is the transport

mean free time, and m is the effective mass. The Drude formula neglects

interference between electron waves scattered by different impurities, which

occur as corrections to Eq.1-2, in the parameter (kFI)- 1 < 1 (where kF is the

Fermi momentum and f is mean free path). In low dimensions (d < 2), these

(interference) quantum corrections to the conductivity (QCC) diverge when the

temperature T decreases and eventually, drive the system to the insulating regime.

The quantum corrections to the conductivity are of substantial importance even

for conductors that are far from the strong localization regime: in a wide range

of parameters QCC, though smaller than the conductivity, determine all the

temperature and field dependence of the conductivity. The systematic study of









QCC started almost three decades ago. A comprehensive review of the status of

the problem from both theoretical and experimental viewpoints can be found in

several papers [8-11].

According to their physical origin, QCC can be divided into two distinct

groups. The correction of the first type, known as the weak localization (WL)

correction, is caused by the quantum interference effect on the diffusive motion of

a single electron. For low-dimensional (d = 1,2) infinite systems the WL QCC

diverge at T 0; this divergence is regularized either by a magnetic field or by

some other dephasing (inelastic scattering) mechanism. We will elaborate on this

type of QCC in section 1.1.1 below and also see how it changes for a 3D metal in

UQL in chapter 2.

The second type of QCC, usually referred to as the interaction effects, is

absent in the one-particle approximation; they are entirely due to interaction

between electrons. These corrections can be interpreted as the elastic scattering

of an electron off the inhomogeneous distribution of the density of the rest

of the electrons. One can attribute this inhomogeneous distribution to the

Friedel oscillations produced by each impurity. The role of the electron-electron

interactions in this type of QCC is to produce a static self-consistent (and

temperature dependent) potential which renormalizes the single particle density of

states and the conductivity. Such a potential does not lead to any real transitions

between single-electron quantum states (those require real inelastic scattering).

Therefore, it does not break the time reversal invariance of the system and neither

does it affect nor regularize the WL corrections. We will elaborate on this type of

QCC in section 1.1.2, and also study it for our case of 3D metal in UQL in chapter

2.

However the interaction between electrons is by no means irrelevant to the

WL QCC. Indeed, these interactions cause phase relaxation of the single electron






5


states, and thus result in the cut-off of the divergences in the WL corrections.

This dephasing (described by by the phase breaking time 7r(T)) requires real

inelastic collisions between the electrons and can be obtained experimentally from

the temperature dependence of magneto-resistance measurements. We will discuss

the phase breaking time due to electron-electron interactions in section 1.1.3, of

this introduction. Therefore there are two classes of interaction contribution to

the conductivity: the genuine interaction corrections (elastic scattering of Friedel

oscillation: Altshuler-Aronov corrections)-Class I, and corrections to WL QCC due

to interactions (inelastic scattering-dephasing)-Class II.

1.1.1 Weak Localization QCC

These corrections are caused by the quantum interference effect in the diffusive

motion of a single electron. In going from point A to point B a particle can travel














Figure 1-1. Weak localization corrections.


along different trajectories (Fig.1 1). The total probability W for a transfer from

point A to point B is


w I= A 1 A |-2 + Y AA. (1-3)
i i iyj

The first term in Eq.1-3 describes the sum of the probabilities for each path and

the second term corresponds to interference of various amplitudes. The interference









term drops out when averaging over many paths because of its oscillatory nature.

However, there exists special type of trajectories, i.e., the self- intersecting ones,

for which interference cannot be neglected (see Fig.1 1). If A1 is the amplitude for

the clockwise motion around the loop and A2 is the amplitude for the anticlockwise

motion, then the probability to reach point O is


W = |Ai 2 + A2 + 2+ReATA2 41A12. (1 4)


i.e., twice the value we would have obtained by neglecting interference. Enhanced

probability to find the particle at a point of origin means reduced probability

to find it at final point (B). Therefore this effect leads to a decrease in the

conductivity (increase in resistivity) induced by interference.

The relative magnitude of weak localization QCC, 6a/a, is proportional to the

probability to form a loop trajectory

6 dP dt V (1-5)
a (Dt)d/2

leading to a ~ -T 4(2-d)/2 (ln r for d=2), which diverges as T is lowered

for d < 2, leading to strong Anderson localization. Here v is the electron

velocity, D is the diffusion coefficient, A is the electron wavelength and 7r(T) is

the phase breaking time. Phase coherence is destroyed by inelastic scattering

(electron-electron, electron-phonon) or by magnetic and a.c electric fields. The

temperature dependence of the WL correction is determined by 'rT (T). Typically,

rT ~ T-P, where the exponent p depends on the inelastic scattering mechanism

(electron-electron, electron-phonon) and dimensionality. Interference effects occur

for 7- < r- (T) i.e., at low temperatures.

In the language of Feynman diagrams, the WL QCC [12] is obtained by

including the maximally crossed ladder diagram, the Cooperon (see Fig.1-2), in

the conductivity diagram. The other type of ladder (vertex ) diagram, the Diffuson










(Fig. -2), when included in the conductivity diagram changes the elastic scattering

time to the transport time. In the field-theoretic language, the Weak Localization

p+q;E+w


+
i i


C(Q;w)


Il



'K


+ Y


+


+ 'I


+ ...


+ ...


Figure 1-2. Ladder diagram for M (diffuson) and C (Cooperon).


correction, which arises due to interference of time reversed paths is determined by

the "Cooperon" mode C (Q; w), i.e., the particle-particle diffusion propagator,


c(Q;w)


27rvT (


1rDQ2
iwr + DQ27r'


(1-6)


to first order in (kft)-1. Calculation of the singular contributions to conductivity

(interference effect) at small w, Q should include diagrams containing as an internal

block the graphs which yield after summation C(Q;wG) (1-3). The WL QCC is

P q-P

< : = < + < : :> +

P q-p






Figure 1-3. Quantum corrections to conductivity for noninteracting electrons.


obtained from


De2 /
D2 J j (dQ) C (Q; w),
6~WL (127


(1-7)


So-WL









which gives

6--WL (1D) (1 8)
a \- aPT
SIn( 1 n(( 1 ) (2D), (1-9)
kpf r hpf UJT
1 1T

~ ( )2 +( )2 (3D) (1 10)

in different dimensions. Perturbation theory breaks down in one- and two-dimensions

(for w < 1, in the diffusive limit or at low temperatures, T7 ~ T-P) which -,-..-. -1i

strong localization in reduced dimensions. Anderson [13] had first shown that

at sufficiently high impurity concentration, electronic states become localized

and the system becomes an insulator. Mott and Twose [14] had predicted that

the conductivity for a one dimensional system should vanish in the limit of low

frequencies (Mott's law) which was later rigourously proved by Berezinskii [15],

who showed that electron states in 1D are strongly localized and there is no

diffusive regime in 1D. The localization length in 1D is of the order of the mean

free path (f), therefore in ID for length scales shorter than the electron motion is

ballistic and for lengths longer than i then electron motion is localized. 2D systems

are also strongly localized but the localization length is very large (Lloc ~ fekf) as

compared to 1D (Loc r~ ). Thus in 2D, the ballistic regime (L < f) crosses over to

the diffusive regime ( for i < L < Loc) and then to the localized regime (L,,o < L).

Scaling theory of localization proposed by Abrahams et al. [16] describes

Localization in higher dimensions. This theory is based on the assumption that the

only parameter that determines the behavior of the system under renormalization

is the dimensionless conductance, g (in units of e2/h). The variation of g with the

system size obeys the Gell-Mann Low equation

ding
dlnL 3g).









In the metallic regime g > 1, the conductance shows ohmic behavior for which

(g) = d 2. Corrections to 3 (g) in the metallic regime are obtained by
perturbation theory in -. For g < 1 (insulating regime), f (g) is obtained from

the simple argument that g must decrease exponentially with the system size in

this regime. In 1D and 2D, g decreases with increasing system size, which means

that the electron states are ah--iv- localized. In 3D there is a continuous phase

transition between metallic and insulating phases. This transition happens when

kff 1 (Anderson transition).

Effect of magnetic field on WL QCC. If the system is placed in a

magnetic field H, the amplitude for a particle to pass the loop clockwise and

anticlockwise (Fig.1 1) acquire additional phase factors,

(i I itHS
A1 -+ Alexp i dlA Ale o

A2 A2zexp e dlA) A2e 0o



where Oo = hc/2e is the flux quantum. The phase difference between waves passing

the loop clockwise and anticlockwise is 6 = 27r/0o, = HS is the flux through

the loop with cross-section S. Thus the magnetic field destroys interference,

reducing the probability for a particle to return to a given point, and hence reduces

the resistivity. This mechanism is responsible for negative magneto-resistance [17].

The characteristic time scale for phase breaking is tH lH2/D where IH c/eH

is the magnetic length. The typical magnetic field involved is H ~ c/eD7r. At this

field, the product wu,- satisfies wu-T ~ (EFT)-1 < 1 where EF is the Fermi energy

and w~ is the cyclotron frequency. Thus at the phase breaking field the classical

magneto-resistance, determined by the value of w,- is still small.

A weak magnetic field destroys phase coherence and increases the conductivity.

If the field is increased further, we reach the classical magneto-resistance regime,









where the conductivity decreases with the field. What happens at even higher fields

when Landau quantization becomes important? We address this issue in chapter 2

of this dissertation. We show that a three-dimensional disordered conductor in the

Ultra quantum limit, where only the lowest landau level is populated, exhibits a

new phenomenon: intermediate localization. The quantum interference correction

6a is of the order of the Drude conductivity UD (as in 1D) which indicates a

breakdown of perturbation theory. However, the conductivity remains finite at T

- 0 (as in 3D). It is demonstrated that the particle-particle correlator (Cooperon)

is massive. Its mass (in units of the scattering rate) is of the order of the impurity

scattering rate.

1.1.2 Interaction Correction to the Conductivity-Altshuler Aronov
Corrections (class I)

The effect of electron-electron interaction in disordered systems makes

it drastically different from that of pure metals, where the interaction at low

temperatures manifests itself only in the renormalization of the electron spectral

parameters [18] (the wave function renormalization Z, effective mass m*, etc.).

First we note that within the transport equation, electron-electron collisions can

in no way affect the conductivity in the case of a simple band structure and in

the absence of Umklapp processes, since electron-electron collisions conserve the

total momentum of the electron system. Inclusion of the Fermi liquid corrections

renormalizes the residual resistivity while not resulting in any dependence of

the conductivity on the temperature and frequency. However one frequently

encounters the situation that the resistivity scales as T2. This dependence is often

interpreted as the ." i i"-liquid" effect, arising from electron-electron scattering

with characteristic time T', oc T-2. In fact, the resistivity is due to Umklapp

scattering. In good metals, normal processes (which conserve the total electron

moment) and Umklapp processes (which conserve the moment up to a reciprocal









lattice vector) are equally probable and the Umklapp scattering rate entering

the resistivity also scales as T2. Note that at low temperatures this resistivity

due to electron-electron scattering (Umklapp) gives the dominant contribution

because the electron-phonon contribution to the resistivity scales as T5 (Bloch's

law, Te-ph 1/T5).

As was mentioned previously, taking into account the interference of elastic

scattering by impurities with the electron-electron interaction produces non trivial

temperature and frequency dependence of the conductivity. This correction

arises from coherent scattering of an electron from an impurity and the Friedel

oscillation it creates [19]. We will first study this correction to the conductivity in

the ballistic limit, (TT- > 1, where 7r is the elastic scattering lifetime) and then in

the diffusive regime (TT
collisions with impurities before it scatters from another electron, whereas in the

ballistic limit the electron-electron collision rate is faster than electron-impurity

rate, thus single impurity effects are important in the ballistic limit. In chapter

2 of this dissertation we will evaluate this interaction QCC in the ballistic limit

in a 3D metal in the UQL. There has been a recent renewal of interest in the

interaction QCC, (class I) due to the metal to insulator transition observed in

two-dimensional (high mobility) Si-MOSFET samples [20]. The qualitative features

of this transition was understood by Zala, Narozhny and Aleiner [19] who showed

that the insulating (logarithmic upturn in the resistivity) behavior in the diffusive

regime and metallic (linear rise in temperature) behavior of the resistivity in the

ballistic limit (2D), are due to coherent scattering at Friedel oscillations. Below,

we first outline their simple quantum mechanical scattering theory approach to

show how temperature dependent corrections to conductivity arise for scattering at

Friedel oscillations, and then extend their analysis to obtain the interaction QCC in









3D ballistic limit. In chapter 2 we evaluate this correction for a 3D system in the

UQL.





__^__--------^ --- ---





A


Figure 1-4. Scattering by Friedel oscillations.


Scattering at Friedel Oscillations Friedel oscillations in the electron

density are created due to standing waves formed as a result of interference

between incoming and backscattered electron waves (Fig.1-4). Consider an

impurity at the origin; its potential Uimp(r induces a modulation of electron

density around the impurity. In the Born approximation one can find the

oscillating correction, 6n(r) = n(r) no to the electron density n(r = Yk "' 2 ()12:

n(r) v sin(2kFr)
n(r) ~ -gv (1-11)

Here r is the distance from the impurity, kF is the Fermi momentum, g =

f Uimp (Idrj is the matrix element for impurity scattering and no is the electron

density in the absence of impurities and d is dimensionality. Taking into account

electron-electron interactions Vo(r5 r7) one finds additional scattering potential

due to the Friedel oscillations Eq.1-11. This potential can be presented as a sum of

the direct (Hartree) and exchange (Fock) terms [21]


6V(7, r2) VH(t1)6( r-) V(rl r),


(1-12)










VH(i) dWVo( )6p(7), (1-13)

1 _
VF(, 2) V Vo(f -2)Jn(T, 2), (1-14)
2

where by p(r) we denote diagonal elements of the one electron density matrix,


n(Tr, r-) W*k^ 1r) k(T2). (1-15)
k

As a function of the distance from the impurity, the Hartree-Fock energy 6V

oscillates similarly to Eq.1-11. The leading correction to conductivity is a result

of interference between two semi-classical paths shown in fig. 4. If an electron

follows path "A," it scatters off the Friedel oscillation created by the impurity

and path "B" corresponds to scattering by the impurity itself. Interference is

most important for scattering angles close to 7 (or for backscattering), since the

extra phase factor accumulated by the electron on path "A" (ei2kR) relative to

path "B" is canceled by the phase of the Friedel oscillation e-i2kFR, so that the

amplitude corresponding to the two paths are coherent. As a result, the probability

of backscattering is greater than the classical expectation (taken into account

the Drude conductivity). Therefore, accounting for interference effects lead to

a correction to the conductivity. We note that the interference persists to large

distances, limited by temperature R ~ Ik kF|-1 < vF/T. Thus there is a

possibility for the correction to have nontrivial temperature dependence. The

sign of the correction depends on the sign of the effective coupling constant that

describes electron-electron interaction. First, we will study the contribution arising

from the Hartree potential. Consider a scattering problem in the potential given in

Eq.1-13. The particle's wave function is a sum of the incoming plane wave and the

outgoing spherical wave (3D),

ikr
'I' e"' + f(O)
r









where f(0) is the scattering amplitude, which we will determine in the Born

approximation. For the impurity potential itself the amplitude f(0) weakly depends

on the angle. At zero temperature it determines the Drude conductivity oD,

while the leading temperature correction is T2 (when the scattering time energy

dependent), as is usual for Fermi systems. We now show that this is not the

case for the potential in Eq.1-12. In fact, taking into account Eq.1-12 leads to

enhanced backscattering and thus to the conductivity correction which depends

on temperature as 6a oc T2 lnT (in 3D), 6a ~ T (in 2D) and, as we will see later

6 T2" (in 3D UQL, a is the interaction parameter) all for the ballistic limit.

Far from the scatterer the wave function of a particle can be found in the first

order of perturbation theory as = eikd. + 6q(rl, where the correction is given by

[22]

(r =1 /drVH 1i) r (116)

Substituting the form of the Hartree potential from Eq: -13, and introducing

the Fourier transform of the electron-electron interaction Vo(q), we obtain for the

scattering amplitude (at large distances from the impurity)


f(0) = Vo() di (r)e. (1-17)
27 j

where q = kF/r and | 2k sin(0/2). We see that the scattering amplitude

depends on the scattering angle (0), as well as the electron's energy (e = k2/2m).

The density oscillation in 3D, with hard wall boundary condition at the origin

(impenetrable impurity), is

6n(r) 1d(k)[Tk 2 -_ 0k 2]

-2kF sin(2kF(r a)) sin(2kFr)
2 2k(r a) 2kpr '









where a is the size of the impurity and f(k) is the Fermi distribution function. We

make the s wave scattering approximation (slow particles, kFa < 1) to obtain

(2kp)2a cos(2kpr) sin (2kr) ( 8)
r2 2kFr (2kFr)2 "

Substituting the density from Eq.1-18 in Eq.1-17, we obtain for the scattering

amplitude

-2mVo(2kF sin())2kaF 1 01 sin( ) 2 0
f(0) = sin(-) + In I |Cos2 19()
sin() L 2 4 1 + sin() 2

In the limit 0 wr + x where x < 1, the scattering amplitude behaves as

f (x) Vo(2kF)[ Iln x]. The transport scattering cross section is now

At, dO sin(0) dQ(1 -cos(0)) fo + f(0) 2, (1-20)
0 Jo

where fo is the amplitude for scattering at the impurity itself (which does not

depend on 0 in the Born limit and gives a constant (T independent) value for the

Drude conductivity). The leading energy dependence comes from the interference

(cross term), which is proportional to f(0). The main contribution to the integral
comes from 0 7 backscatteringg). Expanding near 7, i.e., 0 = 7 + 01 where 01 is

small [19], 01 ~ \k k/k ~ c/Ep, we obtain for the scattering cross section

and transport rate ((rt,)- oc niVpAt, ~ 6p)

oc vVo(2k)(e)21 n(e). (1-21)
7tr, W

Then one obtains the interaction QCC from the Hartree channel [23] in 3D (using

6a/aD = -Sp/pD)
-vVo(2kF) )n( ). (1-22)
JD EF T
One obtains a similar contribution from the exchange (Fock) potential, except

now the coupling constant in front of the T2 In T term is Vo(0). The Hartree and

exchange contribution come with opposite signs. In 2D the interaction QCC is









linear in temperature [19]


-- v-[2Vo(2kF) Vo(0) (1-23)
JD EF

In 1D Yue, Glazman and Matveev [21] used the same approach and calculated the

correction to the transmission coefficient due to scattering at the Friedel oscillation

and obtained a logarithmic temperature correction at the lowest order


t -to In T |, (1-24)


where a [Vo 2V2kFvpF. Using a poor man renormalization group procedure,

they showed that the first order logarithmic correction is in fact a weak coupling

expansion of the more general power law scaling form of the transmission

coefficient,


t to )


where W is the band width. The transmission coefficient is related to the

conductance using the Landauer formula G ~ |t|2, which gives in 1D


G G ) (1-25)


This result was also obtained independently (via bosnization) by Kane and Fisher

[24]. Eq.1-22, 1-23, and Eq.1-25 give the interaction QCC in the ballistic limit in

3D, 2D and ID systems respectively. In chapter 2 of this dissertation we show that

in 3D UQL, this interaction correction to the conductivity behaves similar to that

of a true 1D system.

The interaction correction to QCC in the diffusive limit also arises from the

same physics (namely scattering at friedel oscillations) but now one has average

over many impurities diffusivee motion). This correction to the conductivity was

evaluated by Altshuler and Aronov in 3D [8] and by Altshuler, Aronov and Lee in









2D [25].

6 (2 2F) ln(TT), (2D) (1-26)

where F is the depends on the strength of the interaction, and

4 3F T
S-( ( T(3D). (1-27)
3 2 D

Scattering at the Friedel oscillations also results in a singular energy (temperature)

dependence of the local density of states which can be observed as a zero bias

anomaly in tunneling. The local DOS can be obtained from the electrons Green's

function using 6v(c) -Im f dfiGR((p), e). The correction to the Green's

function can be evaluated the same way as we evaluated the correction to the wave

function due to Friedel oscillation or it can also be evaluated diagrammatically by

calculating the electron's self energy in the presence of disorder and interaction [8].

1.1.3 Corrections to WL QCC due to Electron-Electron Interactions:
Dephasing (class II)

The basic feature underlying the quasi-particle description of electrons in

metals and semiconductors is the small width of the one electron states. The

minimum width of a wave packet and, hence, the minimum decay of a state are

determined by the wave function phase relaxation time 7r. For strongly inelastic

processes this time coincides with the out-relaxation time. In degenerate Fermi

systems where the energy transferred in each collision is of the order c, i.e., of

the order of the excitation energy measured from the Fermi level, the inverse

excitation decay time is of the order of c2/Ep and thus, is smaller than the

excitation energy, which is c. These considerations does not depend on the specific

details of the electron interaction and originate from the fact that scattering of

quasiparticles by one another is governed by large momentum transfers. Therefore

the decay is determined only by the phase volume of final states. It was believed

by analogy with the Fermi liquid, that in the case of weak disorder, kgt > 1, the









excitation decay should likewise be proportional to e2. It turns out, however, that

excitation in disordered systems decays faster, which raises the question of validity

of quasiparticle description of disordered conductors in low dimensional systems.

Apart from being important in the development of the theory, the decay time

for one electron excitations (the phase relaxation time), governs the temperature

dependence of the WL QCC.

It was shown by Altshuler and Aronov that for 3D disordered systems, the

phase relaxation time Tr is governed by large energy transfer processes and in

this regime T'r Tee (where Tee is the out relaxation time). The out relaxation

time can be calculated from the Bolztmann equation (with diffusive dynamics for

the electrons). This gives- ~ (c)3/2 in 3D, [8]. However in lower dimensions

(d = 1, 2) electron-electron collisions with small energy transfers is the dominant

mechanism for dephasing. The Bolzmann approach (which is good for large energy

transfers) fails in 2D and 1D case. Technically, there would be divergences for

small energy transfers [8] both in 2D (logarithmic) and quasi-1D (power law)

in the Bolztmann-equation result for the out relaxation rate. These divergences

must be regularized in a self-consistent manner. The phase breaking time in

lower dimensions can also be obtained by solving the equation of motion for

the particle-particle (Cooperon) propagator in the presence of space and time

dependent fluctuating electromagnetic fields which model the small energy transfer

processes [26]. This gives (rT)-'1 ~ T (in 2D) and (rT)-'1 ~ T2/3 (in quasi-ID).

In true one-dimensional systems, this subject is controversial as true 1D

systems do not have a diffusive regime (the ballistic limit crosses over to the

localized regime) and the quasiparticle description breaks down for an interacting

1D system which is in the Luttinger liquid state. As a result one cannot define ,ee.

In a recent work on this subject [27], it was shown that even for a 1D disordered

Luttinger liquid, there exists a weak localization correction to the conductivity









whose temperature dependence is governed by the phase relaxation rate, (,r)-1 oc

VT (for spinless electrons in 1D) and (r0)-1 oc T (for electrons with spin) and the

WL QCC behaves as,

( )2
WL ND ( nj), (1t28)


where aD 2= e2vvF2T is the Drude conductivity in 1D, which depends on T through

a renormalization of static disorder, 7-0/ = (Ep/T)2". Here To is non-interacting

scattering time and 7 is the renormalized (by Friedel oscillation) scattering time

and a characterizes the interaction.

At present there are no theoretical predictions for Tr- in 3D UQL. The Fermi

liquid approaches for calculating the phase breaking time are not expected to

work here because the Cooperon is not a singular diagram (it acquires a mass in

3D UQL as shown in chapter 2) and, once again, there are no single particle like

excitations as the ground state is a charge-density-wave and excitations above the

ground state are Luttinger liquid like. However in chapter 2 we will show that some

recent magneto-resistance measurements on graphite in UQL qualitatively agree

with predictions of 7T- due to electron-phonon interactions in 1D.

1.2 Non-Fermi Liquid Features of Fermi Liquids: 1D Physics in Higher
Dimensions

The universal features of Fermi liquids and their physical consequences

continue to attract the attention of the condensed-matter community for almost

50 years after the Fermi-liquid theory was developed by Landau [28]. A search

for stability conditions of a Fermi liquid and deviations from a Fermi liquid

behavior, [29-32] particularly near quantum critical points, intensified in recent

years mostly due to the non-Fermi-liquid features of the normal state of high T,

superconductors[33] and heavy fermion mat(i i-[;4].

The similarity between the Fermi-liquid and a Fermi gas holds only for the

leading terms in the expansion of the thermodynamic quantities (specific heat









C(T), spin susceptibility Xs) in the energy (temperature) or spatial (momentum)

scales. Next-to-leading terms are singular (nonanalytic) and, upon a deeper look,

reveal a rich physics of essentially ID scattering processes, embedded into higher

dimensional phase space.

In this introduction, we will discuss the difference between the i 5,il ,

processes which lead to the leading Fermi-liquid forms of thermodynamic quantities

and i ,i. ID processes which are responsible for the nonanalytic (non-Fermi

liquid) behavior. We will see that the role of these rare processes increases as the

dimensionality is reduced and, eventually, the rare processes become normal in ID,

where the Fermi-liquid description breaks down.

In a Fermi gas, thermodynamic quantities form regular, analytic series as a

function of either temperature T, or the inverse spatial scale q of an inhomogeneous

magnetic field. For T < EF and q < kF,


C(T)/T = +aT2+bT4 +..., (1-29)

X,(T,q 0) = s(0) + cT2 + dT +..., (30)

X(T 0,q) = o(0)+eq2 +f4+..., (1 31)

where 7 = 7r2 F/3, X,0 = gB2 F and vF ~ mkpD-2 is the density of states

(DOS) on the Fermi surface, g is the Lande factor and pB is the Bohr magneton

and a... f are some constants. Even powers of T occur because of the approximate

particle-hole symmetry of the Fermi function around the Fermi energy. The above

expressions are valid in all dimensions, except D = 2. This is because the DOS is

constant in 2D, the leading correction to the 7T term in C(T) is exponential in

Ep/T and X, does not depend on q for q < 2kg. However this anomaly is removed

as soon as we take into account a finite bandwidth of the electron spectrum, upon

which the universal (T2" and q2") behavior is restored.









An interacting Fermi system is described by Landau's Fermi-liquid theory,

according to which the leading terms in C(T) and X, are same as that of the Fermi

gas with renormalized parameters (replace bare mass by effective mass m*, bare g

factor by effective g-factor g* in the above Fermi gas results),


C(T)/T = 7* 7o( + (cos0F,)), (1-32)
1 +(cos OFe)
Xs(T,q) = Xs*(O) = s( + (cos (1-33)
1 + (F8,)

where Fe, F, are charge and spin harmonics of the Landau interaction function:

F(Jf,l) = F(O)I + F,(0)a.a', where 5, are the Pauli matrices. The Fermi-liquid

theory is an ..i-mptotically low-energy theory by construction, and it is really

suitable only for extracting the leading terms, corresponding to the first terms in

the Fermi gas expression. Indeed, the free energy of the Fermi-liquid of an ensemble

of quasiparticles interacting in a pairwise manner can be written as [35]


F Fo k + fk,kl'' 'nk' + (0(3k),
k k,k'

where F0 is the ground state energy, 6nk is the deviation of the fermion occupation

number from its ground state value, and fk,k' is the Landau interaction function.

As 6nk is of the order of T/EF, the free-energy is at most quadratic in T, and

so the corresponding C(T) is at most linear in T. Consequently the Fermi-liquid

(FL) theory (within the conventional formulation) does not -v- anything about the

higher order terms.

Strictly -I'" i1:ii a nonanalytic dependence of fk,k' on the deviations from

the Fermi surface k kF, accounts for the non-analytic T dependence of C(T)

[36]. Higher order terms in T or q can be obtained within microscopic models

which specify particular interaction and employ perturbation theory. Such an

approach is complimentary to the FL: the former works for weak interactions but

at arbitrary temperatures whereas FL works both for weak and strong interactions,









but only in the limit of lowest temperatures. Microscopic models (Fermi gas with

weak repulsion, electron-phonon interaction, paramagnon model, etc.) show that

the higher order terms in the specific heat and spin susceptibility are nonanalytic

functions of T and q [37-48]. For example,

C(T)/T = 7- a3T21n(Ep/T)(3D), (1-34)

C(T)/T = 72- 2T(2D), (1-35)

Xs(q) = Xs(O) + 3q2 1n(k/ql|)(3D), (1-36)

Xs(q) = Xs(0) + 2|q(2D), (1-37)

where all coefficients are positive for the case of electron-electron interaction.

As seen from the above equations the nonanalyticity becomes stronger as the

dimensionality is reduced. The strongest nonanalyticity occurs is 1D, where-at least

as long as single particle properties are concerned-the FL breaks down [49, 50]:

C(T)/T = i- ailn(EF/T)(ID), (1-38)

Xs(q) = Xs(0) 3iln(k F/ql)(1D). (1-39)

These nonanalytic corrections to the specific heat and spin susceptibility in 1D are

obtained in chapter 3. It turns out that the evolution of the non-analytic behavior

with the dimensionality reflects an increasing role of special, almost 1D scattering

processes in higher dimensions. Thus non-analyticities in higher dimensions can be

viewed as precursors of 1D physics for D > 1.

We will first study the necessary condition to obtain a FL description and then

see how relaxing these conditions lead to the nonanalytic form for the self-energy

and thermodynamic properties. Within the Fermi liquid

ReZE(R, k) -A + Bk + ... (1-40)

ImZE(R k) C(2 + 2T2) + ... (1 41)









Landau's argument for the E2 (or T2) behavior of ImER requires two conditions: (1)

quasiparticles must obey Fermi statistics, i.e., the temperature is smaller than the

degeneracy temperature TF = kFVF*, where vF* is the renormalized Fermi velocity,

(2) inter-particle scattering is dominated by processes with large (generally, of order

kF) momentum transfers. Once these two conditions were satisfied, the self-energy

assumes a universal form, Eq.1-40 and Eq.1-41, regardless of a specific type of

interaction (electron-electron, electron-phonon) and dimensionality. Consider the

self-energy of an electron (1st order) as it interacts with some boson (see Fig. 1-5

). The wavy line can be, e.g., a dynamic Coulomb interaction, phonon propagator,

etc. On the mass shell (E = k; where k = k2/2m- kF2/2m) at T = 0 and for E > 0


a) q,






k,e k-q, e-m k,e

Figure 1-5. Self-energy at first order in interaction with a bosonic field




ImER(E) w~ du dDqmGR(E u, k qImVR(w, q) (1-42)

The constraint on energy transfers (0 < u < E) is a direct manifestation of the

Pauli principle. The potential term V(r, t) is a propagator of some field which has

a classical limit, so V(r, t) is real, thus ImV(q, w) is an odd function of w and we

write it explicitly as


ImVR(w, q) = wF( w, q).


(1-43)









As a function of q, F has at least two characteristic scales. One is provided by
the internal structure of the interaction (screening wave vector for the Coulomb
potential) or by kF whichever is smaller. This scale, Q, does not depend on w and
provides the ultra-violet cutoff in the theory. In addition there is a second scale

I w/vF, and, since w is bounded from above by E and for low energies (E 0), one
can assume Q > IWu/vF. Thus in a dimensionless form

ImV"(w, q) =- U (( Q). (1

The angular integration over ImGR yields on the mass shell

S/'liG -T dOR(dO (- vF. + q2/2m) = 1 AD( 2/2), (1 45)
J J vFq vpq

where the subscript D stands for the dimensionality, and

A3(x) 0(1 Ix),
(I IXl)
A2(x) 0(1x
1 X2

The function AD primarily serves to impose a lower cutoff q > |w|/vF and we can
ignore the specific functional form. Using Eq.1 45 and Eq.1 44 into Eq.1 42, one
obtains

ImZR() j dW dq d U/2U( WQ (1-46)
JO q>II/VF Q VFQ

Now if the momentum integral is dominated by large moment of the order of
Q, then the function U to leading order can be considered to be independent of
frequency (since Q > IwI/vF), and one can set w = 0 in U, and also replace the
lower limit of the q integral by zero. The momentum and frequency integrals then
decouple, (the momentum integral gives a pre-factor and the frequency integral
gives E2), and one obtains an analytic E2 dependence for ImE. Then the linear
in E term in ReE can be obtained by using the Kramers-Kronig relation. Thus








we see that large momentum (and energy independent) transfers and decoupling
of the momentum and frequency integral are essential to obtain a FL behavior.
The E2 result seems to be quite general under the assumptions made. When and
why are these assumptions violated? Long-range interaction, associated with

Q-A

Im Occ W: ( I Q- co



Non-analytic part of Imi

Q-w /vF IQ-2kF, aI/vF


/ /4 Q-^ rQo




Figure 1-6. Kinematics of scattering. (a) "Any-angle" scattering leading to
regular FL terms in self-energy; (b) Dynamical forward ., i. li i-.
(c) Dynamical backscattering. Processes (b) and (c) are responsible for
nonanalytic terms in the self-energy

small-angle scattering, is known to destroy the FL. For example, transverse long
range (current-current [51] or gauge [52]) interactions which, unlike the Coulomb
interaction are not screened, lead to the breakdown of the Fermi-liquid. But these
interactions occur under special circumstances (e.g., near half-filling for gauge
interactions). For a more generic case, it turns out that even if the bare interaction
is of the most benign form, e.g., a delta-function in real space, there are deviations
from a FL behavior. These deviations get amplified as the dimensionality is
reduced, and, eventually, lead to a complete breakdown of the FL in ID. Already
for the simplest case of a point-like interaction, the second order self-energy shows









p

a) p-q b)


k k+q k k p p+q k+q k

Figure 1-7. Non trivial second order diagrams for the self-energy


a nontrivial frequency dependence. For a contact interaction the two self-energy

diagrams of Fig. can be lumped together (the second diagram is -1/2 the first

one). Two given fermions interact via polarizing the medium consisting of other

fermions. Hence the effective interaction at the second order is proportional to the

polarization bubble, which just shows how polarizable the medium is,


ImVR(w, q)= -U2Im R(w, q). (1 47)


For small angle scattering q < 2kg, w < EF, the particle-hole polarization bubble

has the same scaling form in all three dimensions [53],


ImHR (q; u vD BD (RD (1-48)
vpq vFq

where D = aDrmkFD-2 is the density of states in D dimensions (a3 r-2, a2

-1, al = 2/7) and BD is a dimensionless function whose main role is to impose

a constraint w < vpq in 2D and 3D, and u = vFq in 1D. The above form of the

polarization operator indicates Landau damping: Collective excitations (spin and

charge density waves) decay into particle-hole pairs, this decay occurs only within

the particle-hole continuum whose boundary for D > 1 is at u = vFq for small U, q,

therefore, decay occurs for w < vFq. Using the polarization operator in Eq.1-42 one









gets in 3D,

ImER(E) U2 d dqq 2 duu ],
0 Jiwl/VF VFq Vq JO wo VF
FL
beyond FL
~ a2- bl3, (149)

where the first term originates from the large momentum transfer regime and is

the Fermi-liquid result whereas the sub-leading second term originates from the

small-momentum-transfer regime and is nonanalytic. The fraction of phase space

for small angle scattering is small: most of the self-energy comes from large-angle

scattering events (q ~ Q), but we already start to see the importance for small

angle processes. Applying Kramers-Kronig transformation to the non-analytic part

(I 13) in ImER, we get a corresponding non-analytic contribution to the real part
as (ReER)non-an OC 3 In II and, finally, using the specific heat formula (see Eq.3-14

in chapter 3) we get a nonanalytic T3 In T contribution which has been observed

experimentally both in metals [54] (mostly heavy fermion materials) and He3 [55].

Similarly in 2D

ImER() ~ U2 Edw F dqq ~ ri 2 n In (1-50)
Jo wl/J F UqVFq Vp

and ReER (E) oc xE| and this results in the T2 non-analyticity for the specific heat

which has been observed in recent experiments on .i .ii,. i rs of He3 adsorbed on

solid substrate [56].

In 1D, as we show in chapter 3, the situation is slightly different. Even though

the same power counting arguments lead to ImER oc iE and ReER oc Eln F1 for

the second order self-energy, C(T) is linear (analytic) in T at second order and

the nonanalytic TIn T shows up only at third order in interaction and only for

fermions with spin. This difference is due to the fact that in 1D, small momentum

transfers (here particle-hole continuum shrinks to a single line w = vFq, so decay of









collective excitations is possible only on this line) do not lead to the specific heat

nonanalyticity which occurs solely from the nonanalyticity of the backscattering

(at q ~ 2kF) particle-hole bubble or the Kohn anomaly. Thus, we have the

same singular behavior of the bubble (response functions) and the results for the

self-energy differs because the phase volume qD is more effective in suppressing the

singularity in higher dimensions than in lower ones.

In addition to the forward scattering nonanalyticity, there is also a nonanalytic

contribution to the self-energy and thermodynamics arising from q w 2kF, part

of the response function, i.e., the Kohn anomaly. Usually, the Kohn anomaly is

associated with the 2kF nonanalyticity of the static particle-hole bubble and its

most familiar manifestation is the Friedel oscillation in electron density produced

by a static impurity (see section 1.1.2, of this thesis). Here the static Kohn

anomaly is of no interest as we are dealing with dynamical processes. However, the

dynamical bubble is also singular near 2kF, e.g., in 2D


ImHR(q ~ 2kF, ) ox O(2kF q). (1 51)
IkF(2kF q)

Due to the one-sided singularity in ImHR as a function of q, the 2kF effective

interaction oscillates and falls off as a power law in real space. By power counting,

since the static Friedel oscillation falls off as SlD then the dynamical one

behaves as:


Ssin(2kFr) (1 52)
U oc (1-52)
TD-1

Dynamical Kohn anomaly results in the same kind of non-analyticity in the

self-energy (and thermodynamics) as the forward scattering. The singularity

now comes from \q 2kF ~ wU/vF, i.e., dynamic backscattering. Therefore

the nonanalytic term in the self-energy is sensitive only to strictly forward or









backscattering events, whereas processes with intermediate momentum transfers

contribute to the analytic part of the self-energy.

(a) (b) (c)
k, ca ki k l CC ki k ik

k2" U(O)P k k B U(2k) p k2 ki U(2kFp k2

Figure 1-8. Scattering processes responsible for divergent and/or nonanalytic
corrections to the self-energy in 2D. (a) 1. i v ,ird scattering -an analog
of the g4 process in 1D (b) "Forward scattering with anti-parallel
momenta-an analog of the g2 process in 1D (c) ., I.:-, ii I, ui, with
antiparallel momenta- an analog of the gi process in 1D


We will now perform a kinematic a iJll~i ; and show that the nonanalytic terms

in the self-energy and specific heat in 2D comes from only 1D scattering processes.

Consider the self-energy diagram of Fig. 1-7.(a). The nonanalytic E2 In E term in the

self-energy came from two q-1 singularities: one from the angular average of ImGR

and the other one from the dynamic, U/vFq part of the particle-hole bubble. This

form of the bubble arises only in the limit u < vpq,


ImIR (, q) ~ Im dDidEG(F u,- q0G(E ,pq G dOS(cos 0 ),

~ -(foraw vpq). (1-53)
Vpq 1 Vpq
VFq IF 2 VFq
U2 V q2

From the delta function, cos = wo/vpq < 1, which means that the angle between f

and q is 0 7r/2 or j and q are perpendicular to each other. Similarly the angular

averaging of ImGR(k a,) also pins the angle between k and q to 90 degrees.


(ImGR(k ,E)),j ~ d016(e qvp cos0i)

S cos 01 .- < 1 81 ~ 7/2
vUq Vpq

Thus f; and k (the two incoming moment of the fermions) are almost perpendicular

to the same vector j. In 2D, this means that they are either almost parallel to each

other or anti-parallel to each other, and since the momentum transfer is either









small, q 0 or near 2kF, i.e., Iq 2kF I 0, we essentially have three 1D scattering

processes (see Fig.1-8 ) which are responsible for the nonanalytic corrections to

the self-energy. These three processes are (a) two fermions with almost parallel

moment (k ~ kC) collide and transfer a small momentum (q ~ 0)and leave

with outgoing momentum which are almost parallel to each other (k1' ~ k2') and

parallel to their incoming moment (k1' ~ k, k2 ~ k2I): analogous to the "94"

scattering mechanism in 1D (see Fig.1-8 (a) and chapter 3) (b) two fermions with

almost anti-parallel moment collide (ki ~ -k2) and transfer a small momentum

(q ~ 0) and leave with outgoing momentum which are almost anti-parallel to each

other (kl' ~ 2') but parallel to their incoming momenta(ki' k', kc' ~ k2):

analogous to the "g2" scattering mechanism in 1D (see Fig.1-8(b) and chapter 3),

(c) two fermions with anti-almost parallel moment collide (ki ~ -k2) and transfer

a large momentum q ~ 2kF and leave with outgoing momentum which are almost

anti-parallel to each other (kl' ~ -k2') and also anti-parallel to their incoming

moment (kI' ~ -ki, k2' ~ -kI): analogous to the "g1" scattering mechanism in

1D (see Fig.1-8(c) and chapter 3). Therefore the nonanalytic 2 InE term in the

self-energy in 2D comes from 1D scattering events, the only difference is that 2D

trajectories do have some angular spread, which is of the order of UIJ/EF. It turns

out (Ref. [44]), that out of the three 1D processes, the g2 process and gi process

are directly responsible for nonanalytic corrections (NAC) to C(T) in 2D and only

the gl process leads to NAC to C(T) in 1D. The g4 process although leads to a

mass-shell singularity in the self-energy in both 2D and 1D, but does not give any

NAC to thermodynamics.

In 3D the situation is slightly different, f I q and k _I mean that both

; and k lie in the same plane. However, it is still possible to show that for the

thermodynamic potential, j and k are either parallel or anti-parallel to each

other. Hence, the nonanalytic term in C(T) also comes from the 1D processes. In








addition, the dynamic forward scattering events (marked with a star in Fig.1-9.)
which, although not being 1D in nature, does lead to a nonanalyticity in 3D.
Thus the T3 InT anomaly in C(T) comes from both 1D and non-1D processes
[47]. The difference is that the former start already at the second order in
interaction whereas the latter occur only at third order. In 2D, the entire T2
nonanalyticity in C(T) comes from 1D processes. The nonanalytic correction to the
spin susceptibility will be the subject of discussion in chapter 4 of this thesis, where
we will show that the nonanalyticity in Xs, both in 2D and 3D comes from both 1D
and non 1D scattering processes.








"g4"






d c "any-angle" scattering event
dynamic forward scattering Regular (FL) contribution
+ I D dynamic
forward or backscattering Q r Q-2kF 0)




Figure 1-9. Typical trajectories of two interacting fermions

Our kinematic arguments can be summarized in the following pictorial way.
Suppose we follow the trajectories of two fermions, as shown in Fig. -9. There









are several types of scattering processes. First, there is a "any-angle" scattering

which, in our particular example, occurs at a third fermion whose trajectory is not

shown. This scattering contributes a analytic, FL terms both to the self-energy

and thermodynamics. Second, there are dynamic forward scattering events, when

q ~ wIUJ/vF. These are non-1D processes, as the fermions enter the interaction

region at an arbitrary angle to each other. In 3D, a third order in such a process

leads to a T3 n T term in C(T). In 2D dynamic forward scattering does not lead

to a nonanalyticity. Finally there are 1D scattering processes marked with a Sirius

star where fermions conspire to align their moment either parallel or anti-parallel

to each other. These processes determine the nonanalytic part of E and C(T) in

2D and 1D.

Therefore the nonanalytic terms in the two-dimensional self-energy and

thermodynamics are completely determined by 1D processes, 2D scattering does

not pl. i any role in the nonanalytic terms. As a result, if the bare interaction has

some q dependence, only two Fourier components matter: U(O) and U(2kF) e.g., in

2D

ImZR(E) oc [U2(O) + U2(2) U(O)U(2kF)E21 n (1-54)

ReR (E) oc [U2(0) + U2(2kF) U(O)U(2kFp)]EE|, (1-55)

C(T)/T = a[U2(0) + U2(2k) U(0)U(2kF)]T, (1-56)

Xs(Q, T) = Xs*(O) + bU2(2kF)max[vFQ, T, H], (1-57)

where a and b are some coefficients. These perturbative results can be generalized

for the Fermi-liquid case, when interaction is not weak. Then the vertices U(O)

and U(2kF), occurring in the perturbative expressions are replaced by scattering

amplitude (F) at angle 0 = 7,


F(j, ) F (0)1 + F,(0).a',


(1-58)









where c and s refer to the charge and spin sectors respectively. Thus in 2D [45],

T
C(T)/T = -a[F2() + 32,2). (1-59)

The additional (In T)2 factor in the denominator comes from the Cooper channel

renormalization of the backscattering amplitude [47, 48]. In 3D, the Ts lnT

nonanalyticity in the specific heat arises from both 1D (excitation of a single

particle-hole pair) and non-1D (excitation of three particle-hole pairs) scattering

processes [47].


C(T) 3 n T + FaF2a,0 + F3l + ...' (1 60)
(1 + g InT)2 V
ID, one p-h pair non ID, three p-h pairs

where subscript a = c, s and 0,1, 2... indicate the harmonics of the expansion.

Again, the additional (1 + g In T)2 factor in the denominator comes from the

Cooper channel renormalization of the backscattering amplitude [47, 48].

We saw that the nonanalytic corrections to the specific heat in D = 2, 3, arise

from one dimensional scattering processes, (and they show up at second order in

perturbation theory), and the degree of nonanalyticity increases with decrease in

dimensionality. This predicts that the strongest nonanalyticity in the specific heat

should occur in 1D. However, it was shown in Ref.[57], that the specific heat in

1D is linear in T, at least in second order in perturbation theory. In addition, the

bosonization solution of a one-dimensional interacting system, predicts that the

C(T) is linear in T. We resolve this paradox by showing (in chapter 3) that the

general argument for nonanalyticity in D > 1 at the second order in interaction

breaks down in 1D, due to a subtle cancelation and the nonanalytic T InT term

in the specific heat occurs at third order and only for electrons with spin. This is

verified by considering the RG flow of the marginally irrelevant operator in the

Sine-Gordon theory (which appears in the bosonization scheme for fermions with

spin). For spinless electrons we show that the nonanalyticities in particle-particle









and particle-hole channels completely cancel out and the resulting specific

heat is linear in T (the bosonized theory is gaussian). The singularity in the

particle-hole channel results in a nonanalytic behavior for the spin-susceptibility

Xs oc lnmax[|Q |HI, T], present at the second order.

The spin susceptibility both in 2D and 3D gets nonanalytic contributions from

both 1D and non-1D processes. These corrections will be described in detail in

C'i lpter 4 of this thesis where we also study the nonanalytic corrections near a

ferromagnetic quantum critical point.

1.3 Spin Susceptibility near a Ferromagnetic Quantum Critical Point in
Itinerant Two and Three Dimensional Systems.

The physics of quantum phase transitions has been a subject of great interest

lately. In contrast to the usual classical (thermal) phase transitions, quantum

phase transitions occur at zero temperature as a function of some non-thermal

control parameter (e.g., pressure or doping), and the fluctuations that drive the

transition are quantum rather than thermal. Among the transitions that have been

investigated are various metal-insulator transitions, the superconductor-insulator

transition in thin metal films, and (the first one to be studied in detail and the

subject of this thesis), the ferromagnetic transition of itinerant electrons that

occurs as a function of the exchange coupling between the electron spins. In a

pioneering paper, Hertz [58] derived a Landau-Ginzburg-Wilson (LGW) functional

for this transition by considering a simple model of itinerant electrons that interact

only via the exchange interaction. Hertz analyzed this LGW functional by means

of the renormalization group (RG) methods that generalize the Wilson's treatment

of classical phase transitions. He concluded that the ferromagnetic order in an

itinerant system sets in via a continuous (or 2nd order) quantum phase transition

and the resulting state is spatially uniform. Furthermore, he showed that the

critical behavior in the physical dimensions d = 3 and d = 2 is mean-field-like,









since the dynamical critical exponent z = 3, (which arises due to the coupling

between statics and dynamics in a quantum problem), decreases the upper critical

dimension from d+c = 4 for the classical case to d+c = 1 in the quantum case.

Hertz's theory which was later extended by Millis [59] and Moriya [60], (it is

commonly referred as the Hertz Millis Moriya (HMM) theory), is believed to

explain the quantum critical behavior in a number of materials [61]; however, there

are other systems which do not agree with the HMM predictions and show a first

order transition, (e.g., UGe2), to the ordered state. This contradiction motivated

the theorists to re-examine the assumptions made in the HMM theory.

The HMM scenario of a ferromagnetic quantum phase transition is based on

the assumption that fermions can be integrated out so that the effective action

involves only fluctuations of the order parameter. This assumption has recently

been questioned, as microscopic calculations reveal non-analytic dependence of

the spin susceptibility on the momentum (q), magnetic field (H), and for D / 3,

temperature (T) [42, 44] both away and near the quantum critical point (see the

discussion in section 1.2). For example, in 2D


xs(H,Q,T) = const. + max(HI, IQI, T), (1-61)

and in 3D


Xs(H, Q) = const. + (q2, H2) ln[max(lHI, |Q)], (1-62)

where H, q and T are measured in appropriate units. The dependence on T is

nonanalytic in the sense that the Sommerfeld expansion for the Fermi gas can only

generate even powers of T. Of particular importance is the sign of the nonanalytic

dependence: ,s(H, Q) increases both as a function of H and q (at 2nd order in

perturbation theory) for small H,q. As ,s(H, Q) must definitely decrease for H

and q exceeding the atomic scale, the natural conclusion is then it has a maximum









at finite H and q. This means that the system shows a tendency either to a first

order transition to a uniform ferromagnetic state (the metamagnetic transition as

a function of the field), or ordering at finite q, (to a spiral state). The choice of the

particular scenario is determined by an interplay of the microscopic parameters.

In C'! lpter 4 of this thesis, we will obtain the nonanalytic corrections to X,(H)

in second and third order in perturbation theory and show that these corrections

oscillate between positive at 2nd order, (which points towards a metamagnetic

transition), and negative at 3rd order (which points towards a continuous second

order phase transition) values. Thus it is impossible to predict the nature of the

phase transition by investigating the nonanalytic terms at the lowest order in

perturbation theory. Furthermore, in real systems interactions are not weak and

one cannot terminate the perturbation theory to a few low orders. To circumvent

this inherent problem with perturbative calculations and to make predictions

for realistic systems (e.g., He3), we obtain the nonanalytic field dependence for

a generic Fermi liquid by expressing our result in terms of the lowest harmonics

of the Landau interaction parameters. We also describe the nonanalytic field

dependence near the quantum critical point using the self-consistent spin-fermion

model, and show that the sign of the corrections is metamagnetic. Here, in the

introduction, we briefly review Hertz's theory of the second order phase transition.

1.3.1 Hertz's LGW Functional

Hertz considered the Hubbard model with the lagrangian L given by


L -i,,0 -It)= t1-P-t1,7/C
i,c l,l',

+ (hT + n11)2 1 1( n1)2 (1-63)
4 4

The partition function is obtained by performing a Hubbard-Stratonovich

transformation to decouple the four-fermion interaction in the charge and spin

channel. The charge channel is assumed to be non-critical and is thus discarded,









whereas the partition function for the spin channel takes the following form;


Z J= DDCtDCe- fo dL(,Ct,C)


where


L(p, Ct, C)


Y Cti, (, po)C,C t1 -Ct1,C1',
i,, 1,l',
U +U
4+ 2 ni il).
1 1


The field Q' is the conjugate field to the ni nil, which can be considered as the

magnetic field acting on the fermions. Performing the functional integration over

the fermion operators (C, Ct) he arrived at the partition function

Z = D Se-s(ff)


with the effective action;


Seff (0)


U drT 27) Trln[(O, t- t1_, + ].
4 0 2


(1-66)


The Mean-field-theory would correspond to the saddle point approximation to the

functional integration with respect to 4. To deduce an effective (LGW) functional,

one expands the interaction term (Tr n term) in Q'. The matrix (M) in the Tr In

term in Eq.1-66 in the Fourier space becomes


(M )(k, iLL, ) ; k', ium, a')


S[(-ILWn + (k))n,wm kk
crU -
+ (k k',i0 ikm).
2Q17


(1-64)


(1-65)


(1-67)









The first term on the right hand side of the above equation is the inverse Green's

function for free fermions (-G- o), the second term is the ,l'- I I I- ,i" (V). Then


Trln[M] Tr -G- G )] Trln[ ( V) n[-G-lo] + Trln[ GoV]

Trln[-G-lo] Tr(GoV).
n=

Expanding up to fourth order in V the effective action is


1

+4QV > v4Uq (q, iw j Ii2,i, Ii3)

4 4
x(4,wy iw4)J i)K K ). (1-68)
i= 1 i 1

The coefficients v, in Eq.1-68 are the irreducible bare m-point vertices in

the diagrammatic perturbation theory language. The quadratic coefficient is

v2(q iUi) = 1 Uxo(q, i1l), where xo(q, iui) is the free electron susceptibility given

by the Lindhard function (Polarization bubble), which at small q and small u/qvp

behaves as


ol, iumm) = -V- YGo'(E in)Go( + i+ iumrn),
k,iEs

F[1 t)2- _Q q...]. (1-69)
3 2kF 2 \qvF

Hertz assumed the all the higher order coefficients v, starting with v4 can be

approximated as constants as they vary on the scale of q ~ 2kF and w ~ EF. In

appropriate units Hertz's form of the effective LGW functional is


q,2Wm qiL;,1i
x (q3, I3) (-qT i -iL iw 2 iL3) (1-70)









where ro = 1 UVF -= -2 ( is the correlation length which diverges at the phase

transition), is the distance from the critical point and Uo = U4v"F/12 is a constant.

Thus, Hertz's effective action is almost of the same form as the classical LGW

functional (for the 44 theory), except for the presence of the frequency dependent

term in v2 which contains the essential information about the dynamics. The

action therefore describes a set of int( ,il ii:- weakly Landau-damped (due to the

:.,. I/qvp term) excitations: paramagnons.

Hertz then applied Wilson's momentum shell renormalization group transformation

to the above quantum functional. Here, q and w have to be re-scaled differently.

This is due to the fact that in the paramagnon propagator (v2-1), q and uWm\

appear in a non-symmetric way. Therefore, the system is anisotropic in the

time and space directions. As a result it becomes necessary to introduce a new

parameter, the dynamical critical exponent z for scaling


S~ NqZ. (1-71)


For the quantum ferromagnetic transition which we study here, z = 3. In the RG

procedure consists of the following steps (a) high energy states (with q and w) in

the "outer shell" (A > q > A/b; A > w > A/b; b > 1, A is a cut-off) are integrated

out; (b) variables q and w, are re-scaled as q' = qe and / = ..' with I being

infinitesimal. (c) fields Q are also re-scaled so that in terms of the new fields and

re-scaled q and a, the q2 and wul/q terms in the quadratic part of the action looks

like those in the original functional. Performing all these steps, Hertz obtained the

following RG equations

dr
d = 2r + 12uf2, (1-72)
du
u 18u2f4, (1 73)
dl









where e = 4 (d + z) and the expressions for f2 and f4 can be found in Ref. [58].

The second RG equation shows that the Gaussian fixed point, with u 0, is stable

if c is negative, that is, if d > 4 z. For z = 3, we should therefore expect a stable

Gaussian fixed point and Landau exponents in d = 2, 3.

The two main assumptions that Hertz made in arriving at his LGW functional

(Eq. 1-68 and 1-70) were: (1)the coefficients Vn,mn>4 are nonsingular and can

be approximated by constants and (2) the static spin susceptibility has regular

q2 momentum dependence. For the 2D ferromagnetic transition, nonanalytic

terms in v, were found by Chubukov et al., [62], however, the authors claimed

that these nonanalyticities do not give rise to an anomalous exponent in the spin

susceptibility and therefore were not dangerous. In chapter 4 of this dissertation

we examine the second assumption (2) more carefully. The reasoning behind

Hertz's second assumption was the belief that in itinerant ferromagnets the q

dependence of the 02 term comes solely from fermions with high energies, of

the order of EF or bandwidth, in which case the expansion in powers of (q/p)2

should generally hold for q < pp. This reasoning was disputed in Refs. [42, 44].

These authors considered a static spin susceptibility ,s(q) in a weakly interacting

Fermi liquid, i.e., far away from a quantum ferromagnetic transition, and argued

that for D < 3 and arbitrary small interaction, the small q expansion of Xs(q)

begins with a nonanalytic Iqd-1 term, with an extra logarithm in D = 3. This

nonanalyticity originates from the 2pF singularity in the particle-hole polarization

bubble [42-44] and comes from low energy fermions (in the vicinity of the Fermi

surface), with energies of the order of vFq < Ep. These nonanalytic terms

arise when one considers the reference action So as the one which contains the

particle-hole spin singlet channel interaction (charge channel) and the Cooper

channel interaction, which were neglected in the Hertz model (Hertz's reference

action was just the noninteracting one). Furthermore, the pre-factor of this









term turns out to be negative, which indicates the breakdown of the continuous

transition to ferromagnetism. Thus according to Ref. [42, 63] the modified effective

action near the critical point should be

q2 t i( (1D1 2 74)
Seff) 2 (o -q +D- + 2 )(; 2 +b44+... (174)
q,

with an extra logarithm in D = 3. The weak point of this argument is that

within the RPA, one assumes that fermionic excitations remain coherent at the

quantum-critical point (QCP). Meanwhile, it is known [64] that upon approaching

the QCP, the fermionic effective mass m* diverges as In in D = 3 and (3-D

in smaller dimensions. It can be shown that m/m* appears as a prefactor of the

Iq D-1 term; which would mean that the nonanalytic term vanishes at the QCP.

This still does not imply that Eq.1-70 is valid at the transition because, as we show

in chapter 4, the divergence in m* does not completely eliminate the nonanalytic

term, it just makes it weaker than away from the QCP.

Our approach will be to use the low-energy effective spin-Fermion hamiltonian,

which is obtained by integrating the fermions with energies between the fermionic

bandwidth W and a lower cut-off A (with A < W), out of the partition function

[64, 65]:


H VF(p- pF)ctp,acpa+ Xs,o-l(q)SqS-q + g Ctp+q,a7O, iSq. (1-75)
p,a q p,q

Here Sq describe the collective bosonic degrees of freedom in the spin channel,

and g is residual spin-fermion coupling. In Hertz's approach, all fermions were

integrated out, whereas in the Spin-Fermion model only the high-energy fermions

are integrated out while keeping the low-energy ones. This will turn out to be

important because the spin fluctuation propagator is renormalized by the fermions,

and the fermion self energy is renormalized by interaction with bosons. This model

has to be solved self-consistently as it takes into account the low-energy (mass)






42


renormalization of the spin fluctuation propagator. In chapter 4 of this dissertation

we use this model to obtain the magnetic field dependence of the spin susceptibility

near the quantum critical point, and analyze the stability of the second order

quantum phase transition.















CHAPTER 2
CORRELATED ELECTRONS IN ULTRA-HIGH MAGNETIC FIELD:
TRANSPORT PROPERTIES

One-dimensional systems exhibit unique physical properties which reflect the

influence of strong correlations. The effective dimensionality of charge carriers in

a bulk metal may be reduced from 3D to 1D by applying a strong magnetic field.

It has recently been shown that this reduction leads to formation of a strongly

correlated state, which belongs to the universality class of a Luttinger liquid [5].

The tunneling density of states exhibits a characteristic scaling behavior for the

case of long-range repulsive interaction [5, 6]. This effect is most pronounced in the

ultra-quantum limit (UQL), when only the lowest Landau level remains occupied.

Here, in this chapter we investigate the effect of dilute impurities on the transport

properties of the system. For good metals, the quantizing field is too high (of

the order of 104 Tesla), but semi-metals and doped semiconductors have a low

carrier density and quantizing fields of the order of 1 10 Tesla and allow for a

experimental test of the theoretical predictions made here.

In section 2.1 we discuss localization effects for non-interacting electrons in

the UQL. We find that the localization behavior is intermediate between 1D (

6a ~ CD: strong localization) and 3D (6a < CD: weak localization). We show that

the particle-particle correlator (Cooperon) is massive in the strong magnetic field

limit. It's i: i--" (in units of the scattering rate) is of the order of the impurity

scattering rate. Therefore, localization in the strong-field limit proceeds as if a

strong phase-breaking process is operating as frequently as impurity scattering.

Even at T= 0, this phase-breaking exists as it is provided by the magnetic field

and as a result complete localization never occurs in 3D UQL. On the other









hand, the particle-hole correlator (the diffuson) remains massless, which means

that normal quasi-classical diffusion takes place. Our findings are in agreement

with previous work on this subject [66, 67], where the localization problem was

analyzed for the case of long ranged disorder, whereas in our study we have

analyzed the case of short ranged disorder. Our result for conductivity in the UQL

is t5coop + mDrude = UDrude/2.

In section 2.2 we calculate the corrections to the conductivity due to

electron-electron interactions using finite-temperature diagrammatic technique

where disorder is treated in the ballistic limit. Due to this reduced effective

dimensionality, to first order in interaction, the leading corrections are logarithmic

in temperature. Another way of obtaining the conductivity is to calculate the

interaction correction to the scattering cross-section through an impurity (in a

Hartree-Fock approximation) and use a Drude relation between the cross-section

and the conductivity. We show in section 2.3 that, to first order in the interaction,

the two approaches are equivalent. This is important since, while a higher order

calculation using the diagrammatic technique would be extremely lengthy, the

interaction correction to the cross-section is obtained to all orders via an exact

mapping on to a 1D problem of tunneling conductance of interacting electrons

through a barrier [21]. We find that the Drude conductivities parallel (r = +1)

and perpendicular (r = -1) to the magnetic field exhibit the scaling laws

aJ oc T"2", where a is a function of the magnetic field. The physical reason for

such a behavior of the conductivity is a nearly 1D form of the Friedel oscillation

around an impurity in the strong magnetic field.

The ground state of repulsively interacting electrons in the UQL is known to

be unstable to the formation of a charge-density wave (CDW) [1-3]. This has been

confirmed, for example, by experiments on graphite in high magnetic fields [4].

Both the Hartree-Fock and the diagrammatic calculations presented here are done









without taking into account renormalization corrections for the interaction vertices

themselves. This is justified at energies much larger than the CDW gap but breaks

down at low enough energies. In order for our results to hold, there should exist an

intermediate energy interval in which the renormalization of the interaction vertices

due to CDW fluctuations is not yet important but the power-law renormalization

of the scattering cross-section is already significant. That such an interval exists

for the case of long-range electron-electron interaction was shown by solving the

full RG equations for the vertices and for the cross-section. We have not included

this discussion here for brevity. We discuss possible experimental verification of our

results in section 2.4 and conclude in Section 2.5.

2.1 Localization in the Ultra Quantum Limit

In this section we analyze the localization effects for electrons in the UQL.

As an external magnetic field is applied to the system, a question is whether the

reduction of the effective dimensionality leads to re-entrance of interference effects,

which were initially suppressed by a weak magnetic field. It seems plausible (and

was indeed -, .-.- -1. I by some authors in the past) that the application of a strong

magnetic field may result in a strong localization of carriers, similarly to what

happens in a truly 1D system. A physical argument rules out this possibility

[66, 68], at least for short-range impurities. In this case, while scattering at an

impurity, an electron moves transverse to the field by a distance of the order of the

magnetic length 1H = 1/eH. The flux captured by the electron's trajectory is

then of the order of the flux quantum, and thus interference is destroyed. However,

an application of the standard cross-technique to the calculation of the conductivity

in the UQL fails for the same reason that it does in 1D: all diagrams which go

beyond the Boltzmann equation level give contributions of the order of the Drude

conductivity aD itself. Therefore, perturbation theory breaks down. In 1D, a

similar breakdown is a signal but not the proof of strong localization. In the









following subsection we illustrate the different scenario which arises for 3D electrons

in the UQL.

2.1.1 Diagrammatic Calculation for the Conductivity

In this section we will use the zero temperature diagrammatic formalism to

evaluate first the conductivity and then the WL type quantum corrections. We

choose to work in the symmetric gauge, in which the eigenfunction of a three

dimensional electron gas in UQL is [22]


',m (p, 0, z .) = ezz Rm (p) ,
i27r

where m(pz) is the angular (linear) momentum quantum number in the direction of

the field. In the UQL the allowed values of m are m = 0, 1,2, 3... and


Rm (p) pmexp (-p2 /41

The single particle Green's function, in the Matsubara representation is


G (r,r'; i) (r') (r),


where p = (pz2 pF2)/2m + w/2,(wc is the cyclotron frequency). The Green's

function in the mixed representation (momentum and position representation

because the magnetic field breaks translational invariance) separates into a 1D and

a transverse part

1 e -im(4p-4')
G (pz,p, p'O ; I) 2= m-(p) R (Pp) (2-1)
P m=0
SGld(pz; i) GI (ri, r'), (2-2)

where (r = p, Q). The reason for this separability is the degeneracy of the Landau

level; the energy does not depend on the transverse quantum numbers. The 1D

part, GID, is in the momentum space and the transverse part, G, has been

kept in real space. The disorder averaged Green's function is obtained by doing









perturbation theory in the impurity potential U (for weak disorder kF >> 1, so
that the small parameter is 1/kF) and employing standard cross technique [18] for

disorder averaging. The perturbative (in U) solution of the Schrodinger equation
for the Green's function is


G (r, r'; iE)


Go (r,'; ) + i d3r1Go (r, r1; i) U (r) Go (r r'; i)


+ ff d3rd3r2Go (r, ri; i) U (rl)Go (ri, r2; iE) U (r2)Go (r2, r'; i) ...

where G(Go) is the full (bare) Green's function, U(r) is the random impurity
potential which is chosen to have a 6 correlated gaussian distribution with

(U (r)) = 0 and (U (r) U (r')) = r' ,,,23 (r r'). The number density of impurities
is ni, and uo is the impurity strength (for short ranged point like impurities).
The leading contribution to the self-energy comes from the second order

diagram. The first and third order corrections are zero as (U (r)) = 0. We work in
the Born limit, neglecting processes where an electron scatters from more than two

(same) impurities. The second order correction is


G2 (r, r'; i))

which gives


S 3 ri d3r2Go (r r; ;i) Go (ri, r2;i) Go (r2, r';i) {U


(r) U(r2)),


(G2 (p, p p'O, P iE))


[-ii (UO)2 VHHisgn (E)] IF x_


2x Rm (p) R (p') ,
27m
rn0


and the fourth order correction gives


(G4 Pz, p, p ; iE))


[-n (UO)2 VH7jSgn (E)] 2 IF-w


S-- Rm2 (p) Rm (p')
27e^









where the transverse part in each of these expressions is simply Gi(ri, r'). At

fourth order, there are three diagrams, the rainbow diagram (Fig.2-l,(b)) and

the intersecting diagram (Fig.2-1, (c)) are small by a factor l/kFY compared to

the leading one ((G4)) (Fig.2-1,(a)) for short-range weak disorder. In the Born





b



Figure 2-1. Diagram (a) is the leading contribution to the self energy at fourth
order

approximation, the scattering rate in a magnetic field, is 1/7 = 2,' 1,,,2vH, where

VH = 1/(27r2VF1H2) is the 3D density of states in the presence of a magnetic field,

and the self-energy is E = -isgn (E) /27. The full Dyson's series (Fig. 2-2) can be

summed to give:


(p,,rr, ri) 1 )G i(rr) 1 + + -- 2

S ) Gi(ri, r') (2-3)


Therefore, the effect of impurity scattering enters only in the 1D part of the

Green's function. Using the above form of the Green's function and the Kubo

formula we now evaluate the Drude conductivity. The Kubo formula for the

longitudinal d.c. conductivity (E || H I| z) in the kinetic equation approximation is

1 e2 d dp
azz lim e 2dt i (G i (pz, rI, r; IE
wo a m2 2 j 2w

(G (p, r i vit i si s2

The diagram for the Drude conductivity is shown in (Fig. 2-3). Due to the






49


/ -\ I \ / x
+ -<--- + +



G = Go + dG



dG = + +


Figure 2-2. Dyson's series






Figure 2-3. Drude conductivity


factorizability of the Green's function, the conductivity is also separable into a 1D

and a transverse part: uzz = -1-d x al. The 1D part is in the standard form and

gives the famous Einstein's relation for the d.c conductivity a1d = e2vD = e2/r,

where D = vFr is the diffusion coefficient and vl = l/(rvF) is the density of

states in ID. Using the orthonormality: f dpp [Rm, (p)] 2 1 and completeness:

E, [R. (p)]2 = /(1H2) properties of the wave function, the transverse part can be
shown to be equal to 1 with the degeneracy factor of the lowest landau level.

SC-m(p c4>) in(p'-) R
Sdp d 21F Rm (p) R () 2(p R, (p') R (p) ,
m,n= O
21


Thus az e2Dv1/(27rl) = e2DvH is the Drude result in UQL. The Diffuson

(or particle-hole correlator) corrections to the conductivity can be shown to be

zero for a delta function impurity potential. In the next sub-section we show

that the particle-particle correlator, or the Cooperon acquires a mass in a strong










magnetic field and evaluate the Cooperon correction to the conductivity. The

Diffuson remains massless which means normal quasiclassical diffusion occurs in the

particle-hole channel. Implications of these results on electron localization will be

discussed.

2.1.2 Quantum Interference Correction to the Conductivity

Every diagram for the conductivity, even higher order ones, can be split

into a 1D and a transverse part. The transverse part is ahv-- a number, cn,

multiplied by the Landau level degeneracy factor (1/2r12)"'+1, where n denotes

the order of the diagram (the number of dashed lines in the vertex diagrams), so

that 6Jz = 6(1D x or and ar = n x (1/27w1l)'+1 We show that for the cooperon

type diagrams, = 1/(2'-1) and for the diffuson type C, = 1. In 1D all quantum

corrections to conductivity (QCC) are of the same order as the Drude value,

16aiD JD. This indicates the breakdown of perturbation theory in 1D. Similarly,

for 3D electrons in the UQL, 16ao-z ~ D, because the transverse numbers Cn are

of order one. Therefore perturbation theory also fails in the UQL. However, we will

see below that these transverse numbers Cn make strong localization impossible in a

strong field. In technical terms, they are responsible for generating a finite mass for

the Cooperon.

p, r 2 r P
Pq qp r p qp ,
r qs p rr


Sp2 q p P q
r qp q +P



r q 1 q


Figure 2-4. Third and second order fan diagram.


We calculate these coefficients for the lowest order Cooperon diagrams (2nd

and 3rd order fan diagram shown in Fig. 2-4) explicitly and then state the general

argument by which these numbers can be obtained for all higher order diagrams.









For the second order fan diagram,

S1 2 e2 fdE fdr idpz f dq, f dpi
6jzz = hm (r I,, ) Im dr' dri dr2
w- v2 727 27 27i 27
Pz (qz pz) G (pz, i-, rl_; iE) G (plz, ril, r72; iE+) G (qz pr2, r i; iE+)

x G (q\ pY, ,, rl; iE_) G (qZ pl, r1_, r2, ; iE_)

xG (p', r2, r7; iF_) (2-4)
iw--w+id /
The one-dimensional part of Eq. (2-4) is given by

6ID lim Im 2 R (q) X (q, ) (2-5)
w-o0 m2 [ 2 27J

where R (q) is the one dimensional rung in the particle-particle channel and

X (q, w) is the part containing the vertices:

R (q) = (r ,,2)2 GID(p1; ipi+) GID(q-pi; iF-), (2-6)

X(q, ) J J 2d pp (q p) G1D (p; i+) G1D (q p; i+)

x G1D(q p; iF_) G1D (p; i_). (2-7)

We use the linear spectrum approximation, f d (...) = J d1 (...) where
v = 1/27rv = vi/2 and vl is the 1D density of states, and assume small (total)

momentum, q < pF to obtain for the rung (w > 0):




(1 + wr)2+ q
R(q) (nuo2)22-rvlT- (1 +J j)q +2q2 (2 8)

For the vertex, linearizing and using q < pF, (we cannot set q = 0 in the vertex a

priori because our cooperon will acquire a mass) followed by the pole integration in
(, and the E-integration, we obtain for w > 0,

X (q, ) -2 1PFW (2-9)
(1 + wr) ((1 + wT)2 + q2f2)









and finally for the 1D part of the conductivity


61D -C(2rlH2)2. (2-10)
47r

Note that 6ald ~ OD, so perturbation theory breaks down in the UQL. The

transverse part of the conductivity is,


L J = dr'J dril dr2 Gi (ri,ri) GL (ri, r2) G (r2, r)

xGi (r ri) Gi (rl ,r2) G (r2,r), (2-11)

where r = (p, 0) and G_ is defined in Eq.(2-2). After performing the azimuthal

integration, we obtain
o lm 14rn r 2
Sn () dpp'R2 (/) dPm (P1) Rn (P1) 1 (P1) 1)
,mOn0 n=(20)3
oo l+rn T2(
>n o 2m 3 [Aimn ]2. (2-12)
i,m=0 n=0 (2)

where n' = 1 + m n. Notice that the second order fan diagram has two radial

integration, likewise third order fan diagrams will have three radial integration,

and so on. Using the integral representation of the Gamma function, Aimn,, is

1 1 1 12
A*Tnn' t 2t r2 (n + )! (2-13)
212 2m+' m!nln'!

and the transverse part of the conductivity becomes:
2
1 0o l+m rn 2m 1
4 (27rl/H2) 3 H 2m22(T+1)m!
H11n 1 (,r1 0 n =0(214)

x .! t* .[(' + 1)!]2 (2-14)
mWnI! (I + m n)!









The sum over n is done using the binomial property and the sum over I is a

tabulated sum [69],


S = 1 2H (2 2- 2(m+1), (2-15)
4(27lH2)3 0 2H 2mm!

1 (2-16)
2 (2 lH2)3

The coefficient of the transverse part of the second order fan diagram for the

conductivity is c2 = 1/2. Combining Eq. (2-10) and (2-16), the QCC from the fan

diagram at second order is 6Jao = -e2DvH/8.

Similarly, the higher order diagrams can be evaluated. The third order

fan diagram has the same vertex as the second order one but has one extra

factor of the rung R (q). This gives for the one dimensional part: 6 lD

-(3e2k/167) (27rl)3. The transverse part now has three radial integration (3

factors of A's) is given as:

To m+q 2
mL = Rt, 4 () [Amnqs] [As,'] [A m,] (2 17)
q,m=0n,n'=0 (

where s' m + q n and s = m + q n'. The radial integration can be performed

as before to obtain the A's. Performing the sums, we find
S o R2 m+q m+q
L (2x7)4 (21H 2)3 m!q!23(-+q) (m + q )!
Sm,q=0 n=0 n'=0
S) (2-18)
4 (2lH2)4

Thus for the third order fan diagram, c3 = 1/4, and the QCC is az

-3e2DvH/64. The nth-order fan diagram has n radial integration, each of which

gives a factor of 1/2 so that one has a coefficient (1/2)". The summation over

angular momentum indices gives a factor 2 regardless of the diagram's order. So,

the overall coefficient in the nth-order fan diagram due to the transverse integration

is c = 1/2-1.









n-1>
S 1 + 1/2 + 1/4 +.. (1/2) +*


Figure 2-5. Cooperon sequence for 3D electrons in the UQL. Unlike in 1D, each
term in the series comes with a different coefficient c,.

We construct the Cooperon sequence in UQL as shown in Fig. (2-5), with

the prefactors indicating c,, the numbers obtained after transverse integration at

each order. These numbers are responsible for the mass of the Cooperon. The DOS

factor at nth order is (1/27lH2) +1. The dashed line in the figure denotes g(gi), the

correlator in 3D (ID), where g = 1/27uvHT niuo2 = 2rlH2/(27vl') g127lH2. R

is the one dimensional rung in the particle-particle channel (small total momentum)

evaluated in the diffusive limit.

R = G D(P; i +)GID(q p; ) -1 (1 Drq2 +...), (2-19)

For 3D electrons in the UQL, the cooperon sequence gives:

g g2 R g3 R2
C(q; iw) + x-+ x-+...,
(2H1 2)2 (2 lH2)3 2 (21 1H2)4 4
S gR g12R2 (gR) (220)-1
7T'2 (t + + 4 + ...+ + ...) (220)
2lH 2 4 2"-1

and using Eq. (2-19), this becomes,

91/ (2xlH2)
C(q; iw) g (2 ) (221)
1/2 + D q2/2 |11|/2

In the limit q, u 0, C becomes a constant. There are no infrared divergence,

because we have a massive Cooperon. The mass in units of the scattering rate is

a pure number (1/2). It can be interpreted as 7-/-rH so that 7-TH is of the order

of the impurity scattering time 7r. This indicates that localization in a strong field

proceeds as if a strong phase-breaking process is operating simultaneously with

impurity scattering. This is dephasing by the field and it persists even at T -+ 0.










We now contrast this situation which arises in the UQL with that of any other

dimensions (1D,2D,3D) without the magnetic field. In the absence of the field the

c,'s are all one (for all dimensions) and the cooperon sequence is singular (there is

a diffusion like pole for real frequencies, for q, w ~ 0).


CHO (q;W) (I + gR + g2R2 +9 9 (2 22)
1- gR Drq2 Iw1

This gives the weak-localization correction to the conductivity (WL QCC discussed

in chapter 1) in 2D and 3D [12]. In 1D although the cooperon diagram has a pole,

all non-cooperon diagrams are also of the same order, and one needs to sum over

all the diagrams to get strong localization [15].

In the ultra quantum limit the transverse numbers for the particle-hole

diffusion propagator (the diffuson) are all equal to unity (c, = 1). Therefore

diffuson remains massless in a strong field and normal quasi-classical diffusion

occurs in the particle-hole channel. We will now evaluate the transverse part

k k

r' p-k r P k
P k


k
rl P,

Figure 2-6. First and second order diffuson


of the first and second order diffuson correction of Fig. (2-6), assuming a long

ranged impurity potential such that cld / 0. We do not attempt to calculate the

longitudinal part of the conductivity, (ald) as this will be more complicated due

to the long range disorder potential. We just assume that the longitudinal part is

finite. In the short ranged impurity case the diffuson correction to conductivity

is zero (because the ad = 0). The transverse part for the first order Diffuson









correction is,


aj= Jdri Jdr'G (r ,rl )G (r ,r') GI (r',rl )G (r, ,r1), (2-23)


where GI is defined in Eq. (2-2). Performing the azimuthal integration,


L (27)2 [Rm (P)]2 J dp1 [Rm (p )]2 [R (p1)]2 dt'' [ ( 2. (2-24)
(2'r)2" m,n=o o

Using orthonormality and completeness, we get or1 1/ (27lH2)2 X 1, so that

cl 1. For the second order diffuson we perform the azimuthal integration and

obtain,
OO
l (27 -)3 n [0' (P/)]2 dpp l[ k (p)]2l1 [R (p)]2 X
l,k,n 0

Jd2P2 [Rk (p2)]2 [Rn (P2)]2 J dp'p' [R (p')]2 (2-25)

and using orthonormality and completeness, we obtain aor 1/ (27rlH2)3 x 1, and

c2 = 1. Any nth-order diagram can be calculated in the same way, giving c, 1.

Therefore the longitudinal diffusion is free and the diffuson remains massless.

C(q;w)


P q--P


Figure 2-7. Interference correction to conductivity


Next we calculate the quantum interference correction to the conductivity in

the ultra quantum limit (see Fig. (2-7)):

1 e e2 ( dq
6cp = -lim C (q;u ) X (q; ) ), (2-26)
W->0 a) m 2 27ir+i









where the transverse integration have been performed. The vertex part of this

diagram has already been evaluated in Eq. (2-9). Using Eq. (2-21),


C (; )X(q;







The localization correct


coop -
bcoop =


[dq gl
J 27 (27lH2) (1/2 + wr/2 + Dq2r/2)

--2viT3pF2w ]
S(1 + wr) ((1t + r)2 + 2f2)
-2vr3ppF22g1 7T3 (v/ + 1)
(27w/H2) (1 + w;T) 27 (DTr2) (1 + wT)2 W

*tion (in the diffusive limit wur < 1,q < 1) is

lim -Im 2 T)2 ( -)
L-oa m2 [27f (27iH2) (+ ;T)3 i i6
- (1 ) D (2
27 27lH2 2


The above result indicates that perturbation theory fails in the UQL (6acoop/a =

-1/2) in the same manner as it does in a one dimensional system. However,

contrary to what happens in 1D, there is no strong localization in the UQL. The

crossed diffuson diagram (next order in l/kF, see Fig. 2-8), is also non singular

and massive. The transverse coefficient for the lowest order crossed diffuson

diagram is c = 1/3/2. It is different from the transverse coefficients of the 2nd and

3rd order fan and diffuson diagrams. Therefore one cannot sum the perturbation

series in the crossed diffuson diagram and obtain a simple geometric series. We

propose that intermediate localization (as opposed to weak or strong) occurs in a

3D metal in the UQL, by which we mean that although all interference corrections

are of the order of the Drude conductivity itself, the zero temperature value

remains finite (finite suppression of the Drude conductivity):

H 1
JT- a D < a < 1. (2-28)
2-


-27)









The lower bound for a is based on the fact that 6acoop + acD x aJH. The

non-cooperon type diagrams, at least in the lowest order, have the opposite sign as

that of the cooperon. They are also of the same order as the cooperon, so it is not

clear what they will add up to. It may happen that all the non-Cooperon diagrams

modify our prediction for a and may make a anywhere from 0 -i 1. To obtain a

better estimate for a one needs to generalize Berezinskii's [15] diagram technique

(developed for the 1D localization problem) to 3D UQL. Our results also agree with

those obtained by the authors of Ref. [66, 67]. These authors considered long range

disorder, < 1H, and obtained all (-)2cD. If one uses the author's formula

for the short range impurity case, where ~ 1H then one obtains all ~ JD, which

means that the number a ~ 1.

In the next section we use the finite temperature diagrammatic technique to

calculate the corrections to the conductivity due to electron-electron interactions

(interaction QCC). We will show that these corrections are logarithmic in

temperature and thus they confirm that the system behavior is one-dimensional.






Figure 2-8. Crossed diffuson diagrams. Left, a double-diffuson diagram, which also
acquires a mass. Right, a third-order non-cooperon diagram which,
up to a number, gives the same contribution as the third order fan
diagram.

2.2 Conductivity of Interacting Electrons in the Ultra-Quantum Limit:
Diagrammatic Approach

In this section we study the electron interaction corrections (Altshuler-Aronov

corrections) to the conductivity in UQL in the ballistic limit. We work in the

Landau gauge and use the basis,

ei(pz+pzz)
', ,,n(x, y, ) L= L (y + pl ) (2-29)
VL\LZ H









for the single-electron wave function, where

1 2 12
(u) 1 -_U/21H (U/1H) (2-30)
(2"n !i /1H 1) /2

Here 1H = 1/eH is the magnetic length and H, is a Hermite polynomial. The

Green's function in Matsubara representation can be written as


G(E, pz,p, y, y') H H (2-3 1)
i WnPz)

where the sum is over all Landau levels, ,(pz) = (p2 p2)/2m + nw,, and

c = eH/m is the electron cyclotron frequency. We will need only excitations near

the Fermi level for our calculation, so in the UQL (EF < c;,) contributions to the

Green's function coming from n > 0 terms in the sum in Eq. (2-31) are negligible

due to the large mass term (of order w,) in the denominator. Neglecting these

terms, the total Green's function is written as the product of a longitudinal and a

perpendicular part


G(, z,p,p, y, y') = G1D( pz)Gi(p,, (t') (2-32)

with Gi(p, ,'/1') =o(Y +Pxl),) o(y' + pl). As shown in the previous section the

disorder-averaged longitudinal Green's function corresponds to G1D(,Pz) = 1/(i -

jpz + isgn(E)/2r) where 1/7 = -2ImE. Calculating the conductivity using this

Green's function gives the Drude formula, with density of states vH = VlD/27rFl.

The (dynamically) screened Coulomb potential in the ultra-quantum limit is

given by [70]
47re2
VR (, q) q2 q 2q2 l R(q /D' (2-33)
q + q K2-q I R (H qz) V1D
where the screening wavevector is related to the density of states via the usual

relation K2 = 47e2H and HR (w, q,) is the polarization bubble of 1D electrons.

In what follows, we will need only some limiting forms of the potential. For









1/T < L < p E and l/f < q < kF,


n ((w, )= D (2-34)
g (a, + iO)

independent of the temperature (up to (T/Ep)2-terms). In the static limit, when

the transverse moment are small (ql2 2 < 1), the potential reduces to an isotropic

form
47re2
VR (0, q) = 2, (2-35)
2 2+2(2 35)

which differs from a corresponding quantity in the zero magnetic field only in that

K scales with H as K ~ 1H H. As it will be shown below, the leading correction

to the conductivity (as well as to the tunneling density of states [5, 6]) comes with

processes with q_ ~ K < 1H1. Therefore, the Gaussian factor in the denominator of

Eq. (2-33) can be replaced by unity for all cases of interest.

The polarization bubble exhibits a 1D Kohn anomaly for q, near 2kF. Such

large momentum transfers are important only in Hartree diagrams, where the

potential is to be taken at w = 0 in the ballistic limit. Near the Kohn anomaly, the

static polarization bubble can be written as

1 EF
II2k (0, q) IlD In EF, (2-36)
2 maxIIqz 2pF\, T/VFI

to logarithmic accuracy.

Finally, the pole of the potential in Eq. (2-33) corresponds to a collective

mode -magnetoplasmon. For w, qvF < EF and qlH < 1, the dispersion relation

of the magnetoplasmon mode is given by

2
p2 cO2 q, 22 2 22
U 2+vpO s+ Vz (2-37)


where upo = /4rne2/m is the plasmon frequency of a 3D metal in zero magnetic

field and s vpFV1 + 2/q2 is the magnetoplasmon velocity. In all situations of

interest for this problem, typical longitudinal moment is much smaller than the









transverse ones, |qz, < q so that one can write

I 2
S VF 1+ N. (2-38)










(a) (b)
Figure 2-9. First order interaction corrections to the conductivity where effects of
impurities appear only in the disorder-averaged Green's functions.

We now proceed to compute the first order interaction correction to the

conductivity in the ballistic limit (TTr > 1). This includes contributions from

diagrams shown in Fig. 2-9, where effects of impurities appear only in the

disorder-averaged Green's functions. It also includes diagrams with one interaction

line and one extra impurity line. These can be separated further into exchange

(Fig. 2-10) and Hartree (Fig. 2-11) diagrams. In this section we show that

diagrams 2-10(a), 2-10(b), 2-11(a) and 2-11(b) give a leading In (T/Ep)

-correction to the conductivity, whereas all other diagrams give sub-leading

contributions.

2.2.1 Self-Energy Diagrams

Diagrams Fig. 2-9(a), 2-10(a), 2-10(c), and 2-11(a) involve corrections to the

self-energy due to electron-electron interaction. Diagram 2-9(a) describes inelastic

scattering of an electron on a collective mode (plasmon), which would have existed

even for a system without disorder. As the electron-electron interaction cannot

lead to a finite conductivity in the translationally invariant case, this diagram

is canceled by the counter-correction of the vertex type [Fig. 2-9(b)]. Diagrams

Fig. 2-10(a), 2-10(c), and 2-11(a) describe correction to the self-energy due to















(a) (b)





-__




(c) (d) (e)



Figure 2-10. Exchange diagrams that are first order in the interaction and
with a single extra impurity line. The Green's functions are
disorder-averaged. Diagrams (a) and (b) give InT correction to the
conductivity and exchange diagrams (c), (d) and (e) give sub-leading
corrections to the conductivity.

interference between electron-electron and electron-impurity scattering. A general

form of the correction to the conductivity for all diagrams of the self-energy type

can be written as

y 2 lim -1 TpzT GI (En, P) GiD (En ,, Pz) 6ID (En, P,)

(2-39)

where 6E1D (,,pz,) is the correction to the (1 ii -ubara) self-energy of the effective

1D problem, to which the original problem is reduced upon integrating out

transverse coordinates. This is possible due to the fact that the Green's functions

are factorized into a 1D and a transverse part, as shown in Eq. (2-32), and the

integration over transverse variables can be carried out and simply give the

















(a) (b)



Figure 2-11. Hartree diagrams that are first order in the interaction and
with a single extra impurity line. The Green's functions are
disorder-averaged. Both diagrams give InT correction to the
conductivity.

degeneracy factor 1/27l In this effective 1D problem, electrons interact via an

effective potential

V (D z, q,) (22 V (, q) e-I (2-40)
S(24)
whereas each impurity line carries a factor nu^r /2rl = VF/2r, where ni is the

concentration of impurities and uo is the impurity potential. The overall factor of

2 in Eq.(2-39) is the combinatorial coefficient associated with each diagram of the

self-energy type.

Substituting (2-33) into (2-40) and using the condition KIH < 1, we obtain

1
V1D (w, q,) e2 In 1 2 (. (2-41)
1H [qz K2n (, q,) Iz D\

Performing the analytic continuation in Eq.(2-39), we obtain

C VF= i O 0 dp1 GR C 2 [-ImGR ImJZ + ReG DReJ6El]

(2-42)

where GD = 1/(E--p+i/2-) and GEZD is the interaction correction to the retarded

self-energy of the non-interacting electrons which is Yo = -i/2r. (For brevity, we

suppressed the arguments of GR1 and E1ZD, which are E,p).









2.2.1.1 Diagram Fig. 2-10(a)

We start with the exchange diagram Fig. 2-10(a) which corresponds to a

correction in the self-energy as shown in Figure 2-12. As u and q, are expected

to be large compared to 1/7 and 1/f, respectively, it suffices to replace the

Green's functions in the self-energy by those in the absence of disorder. In

the rest of the diagram for the conductivity, the Green's functions are taken

in the presence of disorder. In 1D, it is convenient to separate the electrons

into left- and right-movers described by the Green's functions G (,,p)

1/(in T vpp + isgnFn/2-), where p = pz T PF. Accordingly, there are also

two self-energies E, for left- and right- moving electrons. The contribution for

E+ is shown in Fig. 2-12. The Green's functions of right/left electrons are labeled

by in the diagram. Processes in which an electron is forward-scattered twice

at the same impurity do not contribute to the conductivity and are therefore

not considered in this calculation. The diagram with backscattering both at an

impurity and other electrons involves states far away from the Fermi surface and

is thus neglected. The only important diagram is the one shown in Fig. 2-12

where the electron is backscattered by an impurity and forward scattered by other

electrons.



+ + +
P \p's p'-q / p-q pe



Figure 2-12. The self-energy correction contained in diagram 2-10(a), denoted in
y(2--12)
the text as E 12


At first, we neglect the frequency dependence of the potential. The momentum

carried by the interaction line is small, q, E/vF T/vF, and at low temperatures,

such that T/vF < K, one can neglect q, compared to K in V1D and replace VID by a









constant, VID 2./,,* where


go= (e2vF) in 1 (2-43)
KiH

is a dimensionless coupling constant. The perturbation theory is valid for go < 1.

Having in mind that the retarded self-energy is obtained from the Matsubara

one by the analytic continuation ie -iE + iO for En > 0, we choose En to be

positive. Performing an elementary integration over p', we arrive at

S(2--12) 1 dq, 1
ST Wm>En 2r [L + I ][* ( o;) F (p q,)]'

where 1/T = niu~v1D/l1 Although the integral over q, is convergent at the upper

limit, it is instructive to calculate it with an ultraviolet cut-off qmax ~ PF. Doing so,

we obtain


y 2--12) ) = 290T 1x
7 i (2u; ,E) + VF

-1 a m -1 a;"m En
1 tan1 --- + tan- n (2-44)
7T VFqmax VFqmax

Now we see that to logarithmic accuracy it is safe to cut the sum at WUM -

VFqmax ~ EF and omit the factor in the curly brackets in Eq. (2-44):

12--12) ,) g 2T (1 (2-45)
+ p) 2T W i (2-; E, ) + UFP
rm >En
go InEF _(1 +n VFP
2T i[ 2-T 2 4+ T

Performing analytic continuation and separating real and imaginary parts, we

obtain


Re (2--12 (,) go tanh + V (246)
"4+ 4rT

Im (2--12) ( i) r In- t + Re i + VF
27T 27T (2 47T
go Ei
Sgo n EF (2-47)
27,T max {|lE +VUF, T}









To obtain the real part in a form given in Eq. (2-46) we used an identity

S( ix) 2 = r/cosh 7x, whereas the last line in Eq. (2-47) is valid to

logarithmic accuracy. The self-energy of left-moving electrons is obtained from

Eqs. (2-46), (2-47) by making a replacement E + vFp E vFp.

We pause here to discuss the physical meaning of the results contained in

Eqs. (2-46) and (2-47). Eq. (2-46) shows that the correction to the effective mass

is T-dependent: for IE + vFpp < T, 6m oc T-1. In principle, such a correction

might result in an additional T-dependence of the conductivity. However, this

T-dependence occurs only in the next-to-leading order in the parameter (T-r)- <1

1 of the ballistic approximation. The leading correction to the conductivity comes

from the imaginary part of the self-energy, Eq. (2-47). This correction exhibits a

characteristic 1D logarithmic singularity, which signals the breakdown of the Fermi

liquid (to the lowest order in the interaction).

The main contribution to the conductivity comes from the correction to the

imaginary part of the self-energy [Eq. (2-47)]. Substituting Eq. (2-47) into Eq.

(2-42) and adding a similar contribution from the left-moving electrons, we obtain

6,(2--10a) go EF e2 1 EF
--2In -I ln I n (2-48)
a 7r T TTVF IH T

We note that the above result was obtained using the static form of the

interaction potential. We now return to the full dynamic potential and show that

the frequency dependence of the potential does not change the results given by

Eqs. (2-46) and (2-47), to logarithmic accuracy. For a dynamic potential it is more

convenient to perform the integration over qj at the very end so that the potential

entering the calculation is of the 3D form

4xre2
V( W q) 472 (2-49)


g 2 + K2 qv z + 22
q,2+q +q2
42e 2vq2 + w2
q {_+ K2 2 q2 2 2 (2 50)
I F z T n d.









where we used that q, < q_ and introduced a2 (q2) = q/ (qj + 02). The integral

over p' gives the same result as for the static potential. Integrating over q,, we

obtain for the effective ID self-energy instead of (2-45),

(2--12) e2 d2q1 e-qlzH 1 EF (1 n iFp
+ ( ) J (27[)2 ( + 2 a (q2) 27T 2 27T [1 + a (q)]

To log-accuracy, the integral over q_ is solved by taking the limits q_/K -- oo and

qlH -- 0 in the integrand and cutting the integral at q_ = r and q_ = 1H as the
lower and upper limits, respectively. In this approximation, a (q ) is replaced by

1, and the result for (2-12b) coincides with that obtained for the static potential,

Eqs. (2-46) and (2-47). Coming back to Eqs. (2-49) and (2-50), we can interpret

this result in the following way. The difference between the dynamic potential and

the static one is in the presence of the dynamic polarization bubble multiplying

K2 in the denominator of Eq. (2-49). If the potential is taken in the static form,

typical frequencies are related to typical moment as u ~ vFpq, which means that

this factor is of order of unity and K must be replaced by K* ~ K. But because the

final result for E depends on K only via a (large) logarithmic term, log( ln KfB ),

such a renormalization of K is beyond the logarithmic accuracy of the calculation.

2.2.1.2 Diagram Fig. 2-11(a).

+ k+p-p'



k e'

+ I +



Figure 2-13. The self-energy correction contained in diagram 2-11(a), denoted in
the text a 2--13)
the text as +









Diagram Fig. 2-11(a) is a Hartree counter-part of the exchange diagram of

Fig. 2-10(a). Separating the contributions of left- and right movers, the diagram

corresponding to backscattering at the static impurity potential is shown in

Fig. 2-13. Again, diagrams corresponding to forward scattering at the impurity

potential do not contribute to the conductivity and do not need to be considered

here. The diagram of Fig. 2-13 also includes backscattering at a Friedel oscillation.

Although this diagram contains a particle-hole bubble, it is more convenient to

label the moment as shown in Fig. 2-13, integrate over p' first, and then over k.

For ', 1l i:-I I. lii,- the 1D potential of Eq. (2-41) becomes

V2 (m, q') V= (V q'+ 2kg)
= 2 In 1 (2kF)2 +K2 ln 2kF / ,T/ 12-51)
H .max {\q', UWm /UF,TI/VF})

where the last line is valid to logarithmic accuracy. As a first approximation, we

neglect the q-dependence of the interaction potential, replacing VT2 in Eq. (2-51)

by a constant V --2 2g2kpVF. This results in
EF
R-1) ( ) -2 92k T V (2-52)
7 2 (2 m Fn) + VFp

-2I [n E i- iF + (2-53)
27r7- 27T 2 47T )

which, up to a sign and overall factor of the coupling constant, is the same as the
R}{(2--13) in Eq. (245).
exchange contribution 213) in Eq. (2 45).

When the dependence of V'F on q' is restored, the result in Eq. (2-53)

changes only in that the coupling constant acquires a weak T- dependence


g2kF g92k (T) e2/2v1) ln 1 [(2kp)2 + 2Inn E/T] I.


(2-54)









Calculating the contribution of Eq. (2-53) with Eq. (2-54) to the conductivity,

we find the correction to the conductivity from diagram Fig. 2-11(a) to be:

6c(2--lla) e2 24k+ 2 In E/Tr EF
a 27VF Kk2 T

Notice that in the limit of very low T and/or very strong fields, the screening

wavevector drops out of the result and the net T-dependence of the conductivity

becomes In x In (In x) where x Ep/T.

2.2.2 Vertex Corrections

Two other important diagrams are the ones in Fig. 2-10(b) and Fig. 2-11(b).

These are the vertex corrections counterparts of the self-energy diagrams in Fig.

2-10(a) and Fig.2-ll(b), correspondingly.

2.2.2.1 Diagram 2-10(b)

Diagram 2-10(b) can be shown to give the same contribution as 2-10(a). In

this Section we show this by reducing diagram 2-10(b) to 2-10(a) without doing

explicit integration over q, and Matsubara summations.

Decomposing diagram 2-10(b) into contributions from left and right fermions,

we obtain

6 (2--0b) 2 lm T 2 M M_ V.D (, q,) im.Q+i
r 2w/ Q-O i72 2wQ 1q
wl En
(2-56)

where M are the vector vertices


M J G(En, p)G(E Qm,p)G(E 1, p q,).
27T

It is obvious that M_ = -M. When evaluating M1, one needs to consider all

choices of signs for Matsubara frequencies. For the cases


En > 0, n F- < 0, and F l < 0; (2-57)

En > 0, En m, < 0, and Fn ,l > 0, (2-58)









we have


M+1 / (2-59)
1
VF(Q + 1/T)[1*L T I + 1/T] ,

M( +/)[( ) /' (2-60)

respectively. For all other cases, the results can be shown either to vanish because

of the locations of the poles or to cancel each other. In the ballistic limit, the

product M+M_ can be simplified in both cases to

1
M+ M_ = (2-61)
M ( + 1/')2[w + 2, (2

The subsequent integration of this expression gives a Icu l--singularity and it is this

singularity which gives the In T-dependence of the correction to the conductivity.

Now we go back to diagram 2-10(a). In Sec. 2.2.1.1, we found the contribution

of this diagram by evaluating the self-energy first and then substituting the result

into the Kubo formula. To prove the equivalence of diagrams 2-10(a) and (b) it is

convenient to consider the full diagram 2-10(a) without singling out the self-energy

part. Summing up the contribution of left and right fermions, we obtain

(2__10a) 2v 1 (e22v2 2
i(2-"-10a) _- T t)2 PVD C1F ) im-(O,+i,,
-T 27 H U L 27- 7
(2-62)

where


P = G (,, p)G (E Q, p)G (E W,,p q,) x
27T

S G (En, p') GT (E z)

A non-zero result for Eq. (2-62) is obtained only for the case given in Eq. (2-57),

when













P++- P_ = + 2+ Q, (2-63)
vF (m + 1/L)2 [ + I i *+ i/ T[iu I i *i + 1Tr]

with


Q = (q -q,) *
V 2(i i, + i/r)2 (i( n) i )(i + i, + i/T) (q

Neglecting the q,-dependence of VID, we see that f 1. 0 0. An expansion in q

results in non-divergent terms which do not bring any non-trivial T-dependence.

Making a ballistic approximation in the rest of Eq. (2-63), we see that it coincides

with Eq. (2-61). The Matsubara summation goes over a twice smaller interval of

frequencies compared to that in Eq. (2-56). We see that Eqs. (2-62) and (2-56)

give the same result and thus


bo(2--10b) 6(2--10a). (264)


2.2.2.2 Diagram Fig. 2-11(b)

p-q
E-0o


C-0

E p-q p'q P P I~ P C



P e- --- p' e-n P C---- p' E-Q

(a) (b)

Figure 2-14. Diagram 2-10(b) vs diagram 2-11(b).


The diagram in Fig. 2-11(b) is a vertex correction counterpart of the Hartree

self-energy diagram Fig. 2-11(a), and it gives the same contribution as Fig.

2-11(a). To see this, we compare the diagrams in Figs. 2-10 (b) and 2-11(b)










labeling them as shown in Fig. 2-14. For a q- and cu-independent interaction,

diagram Fig. 2-14(b) is of the same magnitude but opposite sign as diagram Fig.

2-14(a). For a q-dependent interaction, the T-dependent parts of these diagrams

differ also in the overall factor of the coupling constant: diagram 2-14(a) contains

go whereas diagram 2-14(b) contains g2k,. Electron-electron backscattering in

diagram (a) and electron-electron forward scattering in diagram (b) give either

sub-leading or T-independent contributions. Thus


ba(2--11b) g 2F2kF (2--10b)
go


g2kF j(2--10a) 6(2--11)
go


2.2.3 Sub-Leading Diagrams

All other diagrams give sub-leading contributions. Diagram Fig. 2-10(c) gives

a self-energy type contribution to the conductivity so we use Eq. (2-42). If the

interaction potential is taken to be static, the contribution from this diagram is

zero. Using the dynamical potential, the leading contribution from this diagram is

a In T-correction to the conductivity

ba(2--1c) 1 C2 E.
IjC 2-6


-- .1T"
a 27 vF T

This contribution is smaller than that from diagrams Fig. 2-10(a) [Eq.(2-48)]

and Fig. 2-11(a) [Eq.(2-55)] (and diagrams Fig. 2-10(b) and 2-11(b)) by a

T-independent log-factor.

Diagrams 2-10(d) and 2-10(e) give mutually canceling contributions of the


)uu


form:


r(2--10d) e2 1
----- (2-67)
a 24vF TTr
6b(2--10c) e2 1
2-4 go-" (2-68)
a 24 1T

Each of these contributions is small since we are in the ballistic limit (Tr > 1).


(2-65)


*\









All the calculations shown here are done considering the dynamic interaction

potential at small frequencies. At high frequencies, i.e., at frequencies close to the

magnetoplasmon frequency, the contributions from all the conductivity diagrams

cancel out. That this has to be the case was pointed out recently in Ref. [19].

This is a very useful result because each individual diagram, taken separately,

may give singular corrections. In our case we have also explicitly checked that

this cancellation indeed occurs. Contributions from diagrams (2-9a), (2-10a) and

(2-10c) cancel each other. Contribution from (2-9b) cancels that of (210b), and

finally (2-10d) and (2-10e) cancel each other.

2.2.4 Correction to the Conductivity

Adding up the results from Eqs.(2-48), (2-55), (2-64), and (2-65), we find the

leading correction to the conductivity

c6 6,(2--10a) 7(2--11a) a(2--10b) a(2--11b)
+ + +
2 4p +2 In EF/T EF
In In (2 69)
TWV, K2 T

Eq. (2 -69) is the main result of this Section.

2.2.5 Effective Impurity Potential

The fact that only four diagrams, Figs. 2-10(a,b) and Figs. 2-11(a,b),

determine the leading correction to the conductivity -i.-.-. -I- that there must be

some simple reason for these diagrams to be the dominant ones. Indeed, only these

diagrams arise if one considers scattering of electrons by "effectil impurities that

consist of a combination of bare impurities and the Coulomb fields of electrons

surrounding the bare impurities. For weak delta-function bare impurities, the

effective impurity potential corresponds to "d' --:i, the impurity with the mean

field of Hartree and exchange interactions (see Fig. 3-1).


Vo(, p, p') = Vo + VH(p p') + V,(, p,p').









The first term in this equation is the strength of a bare impurity, the second one is

the Coulomb potential of electrons whose density is modulated due to the presence

of the bare impurity, and the third one is an exchange potential for electrons

interacting and scattering through a weak impurity.




X X x +



Figure 2-15. Effective impurity potential


Due to the exchange contribution, the effective impurity potential is non-local,

and it may depend on the energy, if the interaction is dynamical. Performing the

impurity averaging, we obtain the correlation function of the effective impurity

potential


C = nVo(, p,p') 2 = nV2 + 2nVo[VH(p p') + V(E, p,p')] + O(g2), (2-70)


where g = e2/vF is the interaction strength. Diagrammatically, C corresponds

to a dashed line of the cross-technique [18]. The first term (bare impurities) is

taken into account in the leading order in 1/EFr < 1 by summing infinite series

for the single-particle Green's function and then using the Kubo formula for

the conductivity. Because the bare impurities are short-range, there is only one

diagram for the conductivity-the usual !i iil!." diagram; the vertex correction

to this diagram vanishes. Corrections to the conductivity result from the Hartree

and exchange terms in Eq. (2-70). To first order there are two diagrams, shown

in Fig. 2-16. Although the bare impurity is point-like, the Hartree and exchange

potentials it generates have slowly decaying tails and also oscillate in space. Thus

the vertex correction, Fig. 2-16(b), is not zero. The self-energy diagram, Fig.









2-16(a), corresponds to two diagrams: Fig. 2-10(a) and Fig. 2-11(a). Diagram

Fig. 2-16(b) corresponds to the diagrams in Fig. 2-10(b) and Fig. 2-11(b). For

an arbitrary impurity potential, it can be shown that contributions of 2-16(a) and

2-16(b) coming from forward scattering cancels each other. For backscattering, the

contribution from 2-16(a) and 2-16(b) are the same.









(a) (b)



Figure 2-16. The handle diagram corresponds to diagrams 2-10(a) and 2-11(a) and
the crossing diagram corresponds to 2-10(b) and 2-11(b).


2.3 Impurity Scattering Cross-Section for Interacting Electrons

In this Section we apply a different approach to the conductivity of interacting

electrons in the UQL. In Section 2.2.5 we demonstrated that, to first order in

the interaction, the only diagrams which are important for 6a correspond to

scattering at an effective impurity potential. This -i-i.:- -I that the result for

6a can be obtained by calculating the interaction correction to the impurity

scattering cross-section and then substituting the corrected cross-section into the

Drude formula. In this section we show that to first order this procedure gives

a result identical to that of the diagrammatic approach of Sec. 2.2. Unlike the

diagrammatic series in the interaction for the conductivity, the perturbation

theory for the scattering cross-section can be summed up to all orders via

a renormalization group procedure. This will lead to a Luttinger-liquid-like

power-law scaling of the conductivity, discussed at the end of this section.









2.3.1 Non-Interacting Case

For electron scattering off an impurity potential Vimp(r) that is axially

symmetric about the direction of the magnetic field, the component of the

electron's angular momentum is conserved. In particular, any spherically symmetric

impurity satisfies this condition. For this reason, it is convenient to work in the

symmetric gauge, where the basis of single-electron states is labeled by pz, the

momentum in the direction of the magnetic field, and mz, the projection of the

angular momentum in the magnetic field direction:

CipzZ
'Pp ,m (r) Xm (C),
VILi

where = (x + iy)/1H and


Xm.() 1= 1m ('z exp(- (2/4) (2-71)
1HV 2m z+1xmz

with m = 0, 1, 2....

Electrons are restricted to the lowest Landau band and therefore there are only

two types of scattering events: forward and backward. Only backscattering events

contribute to the scattering cross-section, which can be written as A where

N is the number of electrons backscattered per unit time and J is the total flux

of incoming electrons. Using a Landauer-type scheme, the scattering cross-section

in each channel of conserved m, can be related to a reflection coefficient in this

channel via A, = 271i l r, 2 The total cross-section is obtained from the sum of

the cross-sections in each channel [71]:
00
A 2z |rFm2 (2-72)
m,=0









The coefficients rF are the reflection amplitudes of ID scattering problems, given

by a set of 1D Schrodinger equations

t 82
m 2 + vmz(z) () (+E ( /2) (z)

with effective 1D impurity potentials Vm,(z) = (ml Vip(r) m,) obtained by

projecting the impurity potential on the angular momentum channel mz. The

kinetic energy of the electron is denoted by p/2m. The cross-section A is

related to the backscattering time via the usual relation, 1/RH = ivFA, where ni

is the density of impurity scattering centers. When the electric field is along the

magnetic field and for T = 0, the corresponding component of the conductivity is

related to TH via

az = e21DVFTH/2. (2-73)

An impurity of radius a < 1H can be modeled by a delta-function: Vmp(r)

Vob(r). For a delta-function potential, only the m, = 0 component of Vmz (z) is
non-zero, Vm (z) = (Vo/2712l)6m ,o0(z). In this case, the scattering cross-section for

non-interacting electrons is simply
A= (V/~12 )2
A Am 0 2/02 1 2 122 (Vo/2711)2
A=A,0-o=2lro -27 1 2) 2 + (2-74)
H (Vo/21)2 + vz

where v, = pz/m. Consequently, at T = 0 the conductivity is given by

n e2 1 2i NiVF
2ol+'. (2-75)
z ni mv2 27lf Vo1

In the Born limit (when Vo < 27v pF) we recover the result for the conductivity as

found by using the Kubo formula for weak, delta-correlated disorder [Eq. (2-73)].

In the opposite (unitary) limit A = 27ln and


z, = e2/4 2/ 4 / .


(2-76)









2.3.2 Interacting Case

Now we turn to the calculation of interaction correction to the scattering

cross-section A. For an effective delta-function impurity potential Vmp(r) Vo6(r),

only the m, = 0 component is scattered. The free-electron wave function in this

channel is given by:
Pm(z)' 0Xm, O()
VzL
where far away from the impurity site, the .,i-i:,'l 1 ic form of the z-component of

the unperturbed wave-function is:


Sto ep z > 0
eiPz + r0 e-ipz Z < 0
(2-77)
e-ipzz + ro eip z > 0
to e-iz z < 0

By calculating the electron-electron interaction correction to the wave function,

one obtains the correction to amplitudes to and ro, and therefore to the scattering

cross-section via Eq. (2-72). Since now the problem has been mapped onto a 1D

scattering problem [21, 24], one can anticipate that this interaction correction has

an infrared logarithmic singularity, as it does in the pure 1D case.

The 1D nature of the system in the UQL is also clearly manifested by the

behavior of the Friedel oscillations around the impurity. The profile of the electron

density around the impurity site is given by Sn(r) = f dr'I(r, r')Vtp(r'), where

II(r, r') is the polarization operator. For a weak delta impurity potential, we obtain


n(r) no 2 ) sin( z exp(-r2/21 ), (278)

which shows only a slow, 1/z decay (see Fig. 2-17), characteristic of one-dimensional

systems (in contrast to the 1/r3 decay in 3D systems). Correspondingly, the

Hartree VH(r) and exchange Ve(r, r') potentials, that an incoming electrons feels









when being scattered from an impurity, also exhibit 2pF-oscillations and decay

as 1/z away from the impurity and along the magnetic field direction. In the

transverse plane, the density, and thus the potentials, have Gaussian envelopes

which fall off on the scale of the magnetic length (see Fig. 2-17).



















Figure 2-17. Profile of the Friedel oscillations around a point impurity in a 3D
metal in the UQL. The oscillations decay as 1/z along the magnetic
field direction and have a Gaussian envelope in the transverse
direction.

The interaction correction to the wave function due to the Hartree and

exchange potentials is


6p =o(r)= dr'G(r, r'; E) dr" [VH(r"(r' r") + Ve(r', r")] ,, o(r")
(2-79)

As discussed in the previous section, for the UQL the Green's function is the

product of a longitudinal (1D) and a transverse part, G(r, r'; E) = GlD(, z'p)Gi(rr, r'),

where the .*i-mptotic form of the longitudinal part as z oo is

S'' <
GD(Z, Z')= -+- <0 (2-80)
1pz ipz(z-z') + (z+z') z' > 0








and, in the symmetric gauge, the transverse part is

1 e (1(12 + (' 2(*(')4 (281)
Gi(ri, r) Z- Y X (r )x (r) 21 exp (2-81)
mr
For z > 0, Eq. (2-79) directly gives the correction for the transmission
amplitude t. We first consider the exchange potential,

V(r, r') -V(r- r') > j p [z,4 (r')]* z,(r) (2-82)

which can be factored as

V(r, r') -V(r r')f(z, z')Gi(ri, r) (2-83)

where

f zz') j P { [d ()] 0 (z') + [ O_ (z)]* v' (z')} (2 84)

From the form of f(z, z') one can see that the exchange potential also has terms
with 1/(z z') and 1/(z + z') decay. For example, for z, z' > 0,

sin pp( z') (eiPF(-) 1) (e- ipp(-z) 1)
f (z,z') + ro 2_- r*o (2-85)
7(z z') 27i(z + z') 2i(z + z')

The 1/(z + z') decay leads to a log-divergent correction to Itl. Decomposing the
screened Coulomb potential V(r r') into Fourier components, all the dependence
of Eq. (2-79) on the transverse coordinates rI can be collected into the factor

Tm o(ri) dr' i drGi(rir')Gi(r',r")e--iq .(r r)x(rrr') (2 -86)

Performing the integrals which appear in Eq. (2-86) for the exchange contribution,
we find that the part containing perpendicular coordinates simply enters the
interaction correction as a form factor:

Tm o(ri) Xm o(ri) exp (-q2H (2-87)









Therefore, the transverse part of the free wave function Xm,=o(r) simply remains

unchanged in the rhs of Eq. (2-79). The remaining exponential term appears in the

definition of the effective 1D potential, as in Eq. (2-40)

VlD (q) Jd V(q', q1) exp q .2H (2-88)

The same result is obtained for the transverse part of the Hartree contribution in

Eq. (2-79). Once the transverse part is solved and the effective 1D potential is

defined, the rest of the calculation is exactly equivalent to the calculation of Yue et

al. [21] for tunneling of weakly-interacting 1D electrons through a single barrier.

The interaction correction to the transmission amplitude t is directly obtained from

the correction to the wave function, Eq. (2-79). Just as in 1D, a logarithmically

divergent correction for t is obtained from the longitudinal part of this equation,

after integrating over z and z'.

It is straightforward to see why there is a log-divergent term. The Hartree

term of Eq. (2-79), after integration of the transverse coordinates, is


Z', (z) = dz'GID(z,z')VH(z', (z') (2-89)

where


VH(z) = d VWD(q )e-iqz(z-Z' ) (2-90)
Jo J-oo 27

Let's consider for simplicity |ro0 1 and Ito
gives GlD(z, z') = (2m/pz) exp(ipzz) sinpzz'. The Hartree potential behaves as

VH(z) V1D(2PF) sin(2pFZ)/z so that Eq. (2-89) gives

t to [0 sin(2ppz') / (2 91)
-o sin(pzz')V1D(2pF) s (PF) z (2-91)
to o Z/

The 1/z term gives a logarithmic singularity only in the limit pz pF, so

that Jt/to oc VD(2pF) ln[1/(p PF)]. The Hartree contribution corresponds










to enhancement of to. The exchange contribution has opposite sign and is

proportional to V1D(O). The general answer, can be written as [72]

S-a |r0o2 In (2-92)
to 1HIPz PFI

where a = (go g2k)/2r, and go and g2kF are defined in Eqs. (2-43) and (2-54),

respectively.

0


3 T

0 \0 /

(CO \ /P
a T


0 0 )
0"^- -- _____- ^
1/T W
temperature T


Figure 2-18.


Renormalized conductivities parallel (a,,) and perpendicular (axx)
to the direction of the applied magnetic field. Power-law behavior is
expected in the temperature region 1/7 < T < W.


The second-order contribution to the transmission amplitude was calculated

explicitly in Ref. [21]. The higher-order contributions can be summed up by using

a renormalization group (RG) procedure. Without repeating all the steps of Ref.

[21], we simply state here that in our case the transmission amplitude satisfies the

same RG equation, as in the purely 1D case. i.e.,

dt
at (1- t12), (2-93)









where In (1/ Ip pI 1H) and t( 0) to. The solution of Eq. (2-93) is

tto ) (p pF)H a"
;(0)2 + t2 (p pF)H 2a

The renormalized cross-section is given by Eq. (2-72), but now written in terms

of the renormalized reflection coefficient r|12 1 t12 The final result for the

conductivity can be cast in a convenient form by expressing the bare reflection and

transmission coefficients via bare conductivities in the Born and unitary limits, aoz

and a zu, given by Eqs. (2-75) and (2-76), respectively:

0T2
u0 + (0 u0 ) (2-94)


where W is an ultraviolet cut-off of the problem and a = (go g2kF)/2r, and go

and g2kF are defined in Eqs. (2-43) and (2-54), respectively which shows that a

scales with the magnetic field as a ~ H In H. We are interested in temperature

dependence of the conductivities due to electron-electron interaction corrections

and we assume here that the bare conductivities ao- and ao, have only weak

T-dependence which can be neglected.

Eq. (2-94) is the main result of this Section and is shown in Fig. 2-18. It has

a simple physical meaning: At T = W, the conductivity is equal to its value for

non-interacting electrons. At temperatures T < W, the conductivity approaches

its unitary-value limit, which means any weak impurity is eventually renormalized

by the interaction to the strong-coupling regime. However, if the impurity is

already at the unitary limit at T = W, it is not renormalized further by the

interactions. We emphasize that Eq. (2-94) is applicable only for high-enough

temperatures, i. e., T > max [1/7-, A]. The first conditions is necessary to

remain in the ballistic (single-impurity) regime, the second one allows one to

consider only the renormalization of the impurity's scattering cross-sections by the

interaction without renormalizing the interaction vertex. The latter process leads









eventually for a charge-density-wave instability at a temperature T ~ A, where

A is the charge-density-wave gap [1-3]. For the power-law behavior [Eq. (2-94)]

to have a region of validity, there should be an interval of intermediate energies

in which the renormalization of the interaction coupling constants due to CDW

fluctuations is not yet important but the corrections to the cross-section leading

to the formation of power-law is already significant. Such an interval exists for a

long-range Coulomb interaction (I KIH < 1) both for the conductivity and the

renormalization of the tunneling density of states [6].

The dissipative conductivity in a geometry when the current is parallel to

the electric field but both are perpendicular to the magnetic field, axx, occurs

via jumps between .,-li i,:ent cyclotron trajectories. In the absence of impurities,

electrons are localized by the magnetic field and a,, = 0. In the presence of

impurities, a,, is dir. /hl; rather than inversely, proportional to the scattering rate.

In particular, for short-range impurities, Ua, oc 1/T oc U- 1 and the temperature

dependence of Ua, is opposite to that of azz In the scaling regime, az oc T2" and

a,, oc T-2,. This situation is illustrated in Fig. 2-18, where a = (go g2k)/27,

and go and g2kf are defined in Eqs. (2-43) and (2-54), respectively which shows

that a(H) ~ H In H. In the next section we discuss possible experimental studies

for observing the localization and correlation effects mentioned in the first three

sections.

2.4 Experiments

For experimental observation of the effects described here, the right choice of

material is crucial. Firstly, a low-density material is needed so that the UQL may

be achieved at feasible magnetic fields. For a good metal, the quantizing field is too

high (of the order of 104 Tesla). Semi-metals, such as bismuth and graphite, and

doped semiconductors have low carrier density and quantizing fields of the order

of 1 10 Tesla. Another important condition for observing the Luttinger-liquid









like behavior is that the systems be relatively clean, so that there is a sizable range

of temperatures in which the system is in the ballistic regime (1/7 < T < EF).

This rules out doped semiconductors [73-75] since the charge carriers come from

dopants which act as impurity centers in the system. An additional condition for

occurrence of the power-law scaling behavior and formation of charge-density-wave

or Wigner crystal, is that the electron-electron interaction is strong enough.

Bismuth i --i I- can be made extremely pure; however, the charge carriers in

bismuth are extremely weakly interacting due to a large dielectric constant (~ 100)

of the ionic background. Therefore, the log-corrections calculated here can be

estimated to be very small and would be difficult to be observed experimentally.

C'i ge-density wave instability have been observed in graphite [4] -1 --. i ii-; that

interaction of charge carriers in this system is important in strong magnetic fields

and at very low temperatures. Thus graphite would be an ideal material to observe

the correlation and localization effects mentioned here. Below we present some

recent experimental results of transport measurements in graphite first in weak

magnetic fields [76] and then in ultra quantum regime and try to interpret them in

view of our findings.

Graphite has a low carrier density, high purity, relatively low Fermi-energy

(~ 220K), small effective mass (along the c-axis) and an equal number of electrons

and holes (compensated semi-metal). The metallic T dependence of the in-plane

resistivity in zero field turns into an insulating like one when a magnetic field of the

order of 10 mT is applied normal to the basal (ab) plane. Using magnetotransport

and Hall measurements, the details of this unusual behavior were shown [76], to

be captured within a conventional multiband model. The unusual temperature

dependence di-,1 i.' 1 in (Fig. 2-19) can be understood for a simple two-band case











20E-6
E 15


1E-6 o "
E 0 0 o--
T (K)
C Temeraur deedec of the ai

P 1 E-7 P 0 mT
20 mT
.P 2)2H40 mT
v 60 mT
o 80 mT
A 100 mT
1E-80 v 200 mT
0 70 140

T (K)

Figure 2-19. Temperature dependence of the ab-plane resistivity p,, for a graphite
crystal at the c-axis magnetic fields indicated in the legend


where p,, is given by [77],

SPP2(P + 2) + (P22 + P212)H2
p (p1 + p2 + ^) 2 + 92H2

with pi, and Ri = 1/qin (i = 1, 2) being the resistivity and Hall coefficient of the

two ii P ii fly electron and hole bands, respectively. At not too low temperatures

(where the measurements were performed) electron-phonon scattering is the

main mechanism for the resisitivity in the band. Assuming that p1,2 oc T" with

a > 0, we find that for perfect compensation, (R = -R2 = IRI), Eq. 2-95

can be decomposed into two contributions: a field-independent term oc T' and

a field-dependent term oc R2(T)H2/T. At high T, the first term dominates and

metallic behavior ensues. At low T, R(T) oc 1/n(T) saturates and the second

term dominates, giving insulating behavior oc T-". Although this interpretation

explains the qualitative features of the field induced metal-insulator behavior shown

in Fig.2-19, the actual situation is somewhat more complicated due to the presence









of a third (minority) band, T dependence of the carrier concentration and imperfect

compensation between the i, Pi i ily bands. For more details see Ref. [76].

Let us now direct our attention on transport measurements in the ultra

quantum regime in which we expect to see the power law conductivity behavior

similar to what is shown in Fig. 2-18. Below we present some recent data on the

same graphite samples on which the weak-field measurements were performed.


4.8E-2
8T
10T
12 T
14T
16T
17.5 T
c 4.0
N
N




3.2
0 5 10
T (K)

Figure 2-20. Temperature dependence of the c-axis conductivity a,, for a graphite
crystal in a magnetic field parallel to the c axis. The magnetic field
values are indicated on the plot, with the field increasing downwards,
the lowest plot corresponds to the highest field


Within the experimentally studied temperature range (5K < T < 10K), the

c-axis conductivity exhibits an '-:, ,l.i.:, linear temperature dependence, a, oN T

as shown in Fig. 2-20, whereas the transverse resistivity exhibits a metallic power

law temperature dependence, px oc T1/3, as shown in Fig. 2-21 and Fig. 2-22.

Although the insulating sign of the temperature dependence (for za, see Fig.2-18)

is consistent with the model of a field-induced Luttinger liquid, the independence

of the exponent of the field is not. The slope (exponent) of both a,, and px, (for

various magnetic fields), are independent of the magnetic field whereas in the










25
20
15

N 10 T
D 4T
S6T
8T
.5 io. o
14T
16T
17.5T

1 10 100
T(K)

Figure 2-21. Temperature dependence (log-log scale) of the ab-plane resistivity
scaled with the field p,,/B2 for a graphite crystal at the c-axis
magnetic fields indicated in the legend


field-induces Luttinger liquid model the exponent of the power law should depend

on the field (see Eq. 2.94, a ~ H In H). Also, the exponents of the T scaling in a,,

(which is 1), and pxx (which is 1/3) are different (as seen in experiments) whereas

they were predicted to be the same in the Luttinger liquid model.

We are going to argue that the unusual temperature behavior of azz and pxx,

can be understood within a model which includes phonon-induced dephasing of

one-dimensional electrons (in the UQL) and the correlated motion in the transverse

direction due to the memory effect of scattering at long ranged disorder. Before

we get into the details of the model, let us keep in mind a few numbers for the

system we are about to describe. For our graphite samples the Fermi energy is

EF = 220K, the Bloch-Gruniesen temperature (which separates the region of T and

T5 contribution to the resistivity) is o = 2kFs ~ 10K and the Dingle temperature

(which gives the impurity scattering rate) is 3K. Also the transport relaxation

time is much longer (by a factor of 30), than the total scattering time (or life time)

Ttr > T, indicating the long range nature of the impurities.










1800
H=17.5T
1500 slope =0.312(1)

1200

S900


600

1 10 100
T (K)

Figure 2-22. Temperature dependence (on a log-log scale) of the ab-plane
resistivity p,,/B2 at the highest attained c-axis magnetic field of
17.5T for the same graphite ( i-- I 1

We first outline an argument by Murzin [73] which shows that the transverse

motion of the electron is correlated due to drift motion in a crossed magnetic and

electric field. The disorder model is assumed to be ionized impurity type and is

therefore long ranged. The transverse displacement (after a single scattering act)

is assumed to satisfy ri < 1H < rD (rD being the screening radius). Electrons

are assumed to diffuse in the z direction. An electron re-enters the region where

the impurity's electric field is -I i, i-. (rl < rD), many times as it moves in

the transverse direction. Thus electron's motion in the transverse direction is

correlated. Only after an electron has traveled a distance greater than rD in the

transverse direction, its motion becomes diffusive with the diffusion coefficient

D,, rD2/TD. We estimate TD (the time in which electron has moved a distance

rD I to H) by finding the transverse displacement AX(t) for AX(t) < rD, and
then obtain TD by setting AX(TD) rD. The probability to find an electron again




Full Text

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FirstandformeostIwouldliketothankmyresearchsupervisor,ProfessorDmitriiMaslov,forhisconstantencouragementandguidancethroughouttheentirecourseofmyresearch.Hisenthusiasm,dedication,andoptimismtowardsphysicsresearchhavebeenextremelyinfectious.ThecountlesshoursIhavespentdiscussingphysicswithhimwerehighlyproductiveandintellectuallystimulating.IwouldliketothankProfessorJimDufty,ProfessorArthurHebardandProfessorPradeepKumar,whowerealwayswillingandopentodiscussanyphysicsrelatedquestions.IamhonoredandgratefultoProfessorRussellBowers,ProfessorAdrianRoitberg,ProfessorKhandkerMuttalib,ProfessorSergeiObukhovandProfessorArthurHebardforservingonmysupervisorycommittee.MythanksgotothePhysicsDepartmentsecretaries,Ms.Balkcom,Ms.Latimer,Ms.NicholaandMr.Williams;andtomyfriendsPartho,Vidya,Suhas,Aditi,AparnaandKarthikfortheirhelpandsupport.IwouldliketothankmywifeandbestfriendSreyaforbeingmysourceofstrengthandinspirationthroughalltheseyears.Iwouldliketothankmyfamilyfortheunconditionallove,supportandencouragementtheyprovidedthroughtheyears. iv

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page ACKNOWLEDGMENTS ............................. iv LISTOFFIGURES ................................ vii ABSTRACT .................................... x CHAPTER 1INTRODUCTION .............................. 1 1.1TransportinUltraStrongMagneticFields .............. 2 1.1.1WeakLocalizationQCC .................... 5 1.1.2InteractionCorrectiontotheConductivity-AltshulerAronovCorrections(classI) ....................... 10 1.1.3CorrectionstoWLQCCduetoElectron-ElectronInteractions:Dephasing(classII) ....................... 17 1.2Non-FermiLiquidFeaturesofFermiLiquids:1DPhysicsinHigherDimensions ............................... 19 1.3SpinSusceptibilitynearaFerromagneticQuantumCriticalPointinItinerantTwoandThreeDimensionalSystems. .......... 34 1.3.1Hertz'sLGWFunctional .................... 36 2CORRELATEDELECTRONSINULTRA-HIGHMAGNETICFIELD:TRANSPORTPROPERTIES ........................ 43 2.1LocalizationintheUltraQuantumLimit ............... 45 2.1.1DiagrammaticCalculationfortheConductivity ....... 46 2.1.2QuantumInterferenceCorrectiontotheConductivity .... 50 2.2ConductivityofInteractingElectronsintheUltra-QuantumLimit:DiagrammaticApproach ........................ 58 2.2.1Self-EnergyDiagrams ...................... 61 2.2.1.1DiagramFig. 2{10 (a) ................ 64 2.2.1.2DiagramFig. 2{11 (a). ................ 67 2.2.2VertexCorrections ....................... 69 2.2.2.1Diagram 2{10 (b) ................... 69 2.2.2.2DiagramFig. 2{11 (b) ................ 71 2.2.3Sub-LeadingDiagrams ..................... 72 2.2.4CorrectiontotheConductivity ................. 73 2.2.5EectiveImpurityPotential .................. 73 v

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..... 75 2.3.1Non-InteractingCase ...................... 76 2.3.2InteractingCase ......................... 78 2.4Experiments ............................... 84 2.5Conclusions ............................... 93 3SINGULARCORRECTIONSTOTHERMODYNAMICSFORAONEDIMENSIONALINTERACTINGSYSTEM:EVOLUTIONOFTHENONANALYTICCORRECTIONSTOTHEFERMILIQUIDBEHAVIOR 95 3.1One-DimensionalModel ........................ 99 3.2SpecicHeat .............................. 105 3.2.1SpecicHeatfromtheSecondOrderSelfEnergy ....... 107 3.2.2SpecicHeatfromtheThermodynamicPotentialatSecondOrder ............................... 112 3.2.3SpecicHeatfromThirdOrderSelfEnergy .......... 116 3.2.4SpecicHeatfromtheSine-GordonModel .......... 127 3.3SpinSusceptibility ........................... 130 3.4Experiments ............................... 136 3.5Conclusion ................................ 137 4SPINSUSCEPTIBILITYNEARAFERROMAGNETICQUANTUMCRITICALPOINTINITINERANTTWOANDTHREEDIMENSIONALSYSTEMS ................................... 138 4.1SpinSusceptibilitys(H),in2D .................... 141 4.2SpinSusceptibilitys(H),in3D .................... 145 4.3SpinSusceptibilityforaFermiLiquidin2D ............. 147 4.4SpinSusceptibilityneartheQuantumCriticalPoint ......... 156 4.4.12D ................................ 158 4.4.23D ................................ 162 4.5Conclusions ............................... 165 5CONCLUSIONS ............................... 166 REFERENCES ................................... 168 BIOGRAPHICALSKETCH ............................ 174 vi

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Figure page 1{1Weaklocalizationcorrections. ........................ 5 1{2LadderdiagramforM(diuson)andC(Cooperon). ............ 7 1{3Quantumcorrectionstoconductivityfornoninteractingelectrons. 7 1{4ScatteringbyFriedeloscillations. ...................... 12 1{5Self-energyatrstorderininteractionwithabosoniceld ........ 23 1{6Kinematicsofscattering.(a)\Any-angle"scatteringleadingtoregularFLtermsinself-energy;(b)Dynamicalforwardscattering;(c)Dynamicalbackscattering.Processes(b)and(c)areresponsiblefornonanalytictermsintheself-energy ............................... 25 1{7Nontrivialsecondorderdiagramsfortheself-energy ........... 26 1{8Scatteringprocessesresponsiblefordivergentand/ornonanalyticcorrectionstotheself-energyin2D.(a)\Forwardscattering"-ananalogoftheg4processin1D(b)\Forwardscattering"withanti-parallelmomenta-ananalogoftheg2processin1D(c)\backscattering"withantiparallelmomenta-ananalogoftheg1processin1D ...................... 29 1{9Typicaltrajectoriesoftwointeractingfermions .............. 31 2{1Diagram(a)istheleadingcontributiontotheselfenergyatfourthorder 48 2{2Dyson'sseries ................................. 49 2{3Drudeconductivity .............................. 49 2{4Thirdandsecondorderfandiagram. .................... 50 2{5Cooperonsequencefor3DelectronsintheUQL.Unlikein1D,eachtermintheseriescomeswithadierentcoecientcn. ............. 54 2{6Firstandsecondorderdiuson ....................... 55 2{7Interferencecorrectiontoconductivity ................... 56 vii

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58 2{9Firstorderinteractioncorrectionstotheconductivitywhereeectsofimpuritiesappearonlyinthedisorder-averagedGreen'sfunctions. ... 61 2{10Exchangediagramsthatarerstorderintheinteractionandwithasingleextraimpurityline.TheGreen'sfunctionsaredisorder-averaged.Diagrams(a)and(b)givelnTcorrectiontotheconductivityandexchangediagrams(c),(d)and(e)givesub-leadingcorrectionstotheconductivity. ..... 62 2{11Hartreediagramsthatarerstorderintheinteractionandwithasingleextraimpurityline.TheGreen'sfunctionsaredisorder-averaged.BothdiagramsgivelnTcorrectiontotheconductivity. ............. 63 2{12Theself-energycorrectioncontainedindiagram 2{10 (a),denotedinthetextas( 212 )+. ................................ 64 2{13Theself-energycorrectioncontainedindiagram 2{11 (a),denotedinthetextas( 213 )+ 67 2{14Diagram 2{10 (b)vsdiagram 2{11 (b). .................... 71 2{15Eectiveimpuritypotential ......................... 74 2{16Thehandlediagramcorrespondstodiagrams 2{10 (a)and 2{11 (a)andthecrossingdiagramcorrespondsto 2{10 (b)and 2{11 (b). ........ 75 2{17ProleoftheFriedeloscillationsaroundapointimpurityina3DmetalintheUQL.Theoscillationsdecayas1=zalongthemagneticelddirectionandhaveaGaussianenvelopeinthetransversedirection. ......... 79 2{18Renormalizedconductivitiesparallel(zz)andperpendicular(xx)tothedirectionoftheappliedmagneticeld.Power-lawbehaviorisexpectedinthetemperatureregion1=TW. ................. 82 2{19Temperaturedependenceoftheab-planeresistivityxxforagraphitecrystalatthec-axismagneticeldsindicatedinthelegend ........ 86 2{20Temperaturedependenceofthec-axisconductivityzzforagraphitecrystalinamagneticeldparalleltothecaxis.Themagneticeldvaluesareindicatedontheplot,withtheeldincreasingdownwards,thelowestplotcorrespondstothehighesteld ..................... 87 2{21Temperaturedependence(log-logscale)oftheab-planeresistivityscaledwiththeeldxx=B2foragraphitecrystalatthec-axismagneticeldsindicatedinthelegend ............................ 88 viii

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................................ 89 2{23PhasebreakingratevsTduetoelectron-phononscattering ....... 92 3{1Interactionvertices .............................. 101 3{2Non-trivialsecondorderselfenergydiagramsforrightmovingfermions 107 3{3Secondorderdiagramsforthethermodynamicpotentialwithmaximumnumberofexplicitparticle-holebubbles ................... 112 3{4Thedierentchoicesforthe3rdorderdiagram. .............. 117 3{5All3rdordersediagramsforrightmoverswhichhavetwo2kF. ..... 122 3{6AllthirdorderselfenergydiagramscontainingtwoCooperbubbles ... 123 3{7Eectivethirdorderself-energydiagrams(thedoublelineisavertex). 124 3{8Allg2andg1verticesat2ndorder. ..................... 125 3{9Allthirdorderself-energydiagramswithtwoCooperbubblesortwo2kFbubbles. .................................... 126 3{10Secondorderdiagramsforthethermodynamicpotential. ......... 132 4{1Particle-holetypesecondorderdiagramforthethermodynamicpotential. 143 4{2Particle-holetypethirdorderdiagramforthethermodynamicpotential. 144 4{3Theskeletondiagramforthethermodynamicpotential. .......... 148 4{4Fermionself-energy(a)andBosonicself-energy(b). ........... 158 ix

PAGE 10

Inthisworkwehavestudiedthefundamentalaspectsoftransportandthermodynamicpropertiesofaone-dimensional(1D)electronsystem,andhaveshownthatthese1Dcorrelationsplayanimportantroleinunderstandingthephysicsofhigher-dimensionalsystems.Therstsystemwestudiedisathree-dimensional(3D)metalsubjectedtoastrongmagneticeldthatconnestheelectronstothelowestLandaulevel.Weinvestigatedtheeectofdiluteimpuritiesinthetransportpropertiesofthissystem.Weshowedthatthenatureofelectrontransportisonedimensionalduetothereducedeectivedimensionalityinducedbythemagneticeld.Thelocalizationbehaviorinthissystemwasshowntobeintermediate,betweena1Danda3Dsystem.Theinteractioncorrectionstotheconductivityexhibitpowerlawscaling,/Twithaelddependentexponent. Nextwestudiedthethermodynamicpropertiesofaone-dimensionalinteractingsystem,whereweshowedthatthenext-to-leadingtermsinthespecicheatandspinsusceptibilityarenonanalytic,inthesamewayastheyareforhigher-dimensional(D=2;3)systems.Weobtainedthenonanalytic,TlnTterminthespecicheatin1Dandshowedthatalthoughthenonanalyticcorrections x

PAGE 11

Inthenalpartofthisworkweanalyzedthenonanalyticcorrectionstothespinsusceptibility(s(H))inhigherdimensionalsystems.Weshowedthat,althoughtherewerecontributionsfromnon-1Dscatteringinthesenonanalyticterms,thedominantcontributioncamefrom1Dscattering.Wealsoshowedthatthesecondorderferromagneticquantumphasetransitionisunstablebothin2Dand3D,withatendencytowardsarstordertransition. xi

PAGE 12

One-dimensionalinteractingsystems(Luttinger-liquids)exhibitmanyfeatureswhichappeardistinctfromtheirhigher-dimensionalcounterparts(Fermi-liquids).Ourgoalinthisthesisistohighlightthesimilaritiesbetweenhigher-Dand1Dsystems.Theprogressinunderstandingof1Dsystemshasbeengreatlyfacilitatedbytheavailabilityofexactorasymptoticallyexactmethods(BetheAnsatz,bosonization,conformaleldtheory),whichtypicallydonotworkverywellabove1D.Thedownsideofthisprogressisthat1Deects,beingstudiedbyspecically1Dmethods,looksomewhatspecialandnotreallyrelatedtohigherdimensions.Wearegoingtoarguethatthisisnottrue.Manyeectswhichareviewedasthehallmarksof1Dphysics,e.g.,thesuppressionoftunnelingconductancebytheelectron-electroninteraction,dohavehigherdimensionalcounterpartsandstemfromessentiallythesamephysics.Inparticular,scatteringatFriedeloscillationscausedbytunnelingbarriersandimpuritiesisresponsibleforzero-biastunnelinganomaliesinalldimensions.Thedierenceliesinthemagnitudeoftheeect,notinitsqualitativenature.Weillustratethissimilaritybyshowingthat1Dcorrelationsplayanimportantroleinunderstandingthephysicsofhigherdimensionalsystems.Westudiedthreeseeminglydierentproblems,butaswewillshow,allthreeofthemareconnectedbythecommonfeatureof1Dcorrelations.Ourgoalintheintroductionistoprovideabackgroundforthephysicsdiscussedinthethreechaptersofthisdissertation.WehavesethandkBequaltounityeverywhere. 1

PAGE 13

2; where!c=eH=mcisthecyclotronfrequencyandnistheLandaulevel.Thusthesystemexhibitseectscharacteristicofone-dimensional(1D)metals,whilebeingintrinsicallya3Dsystem.Thisreductionofeectivedimensionalityofchargecarriersfrom3Dto1Dismostpronouncedintheultra-quantumlimit(n=0,whenonlythelowestLandaulevelremainspopulated)andisexpectedtoresultinanumberofunusualphases.ItiswellknownthatthegroundstateofrepulsivelyinteractingelectronsintheUQLisunstablewithrespecttotheformationofachargedensitywave[ 1 { 3 ],whichhasbeenobservedexperimentallyinmagneto-resistancemeasurementsongraphiteinhighmagneticelds[ 4 ].ThemostcompleteanalysisoftheCDWinstabilityforthecaseofshortrangeinteractionswasperformedinRef.[ 3 ],bysolvingtherenormalization-group(RG)equationsfortheinteractionvertex.Ontheotherhand,ithasrecentlybeenshownthatforthecaseoflong-range(Coulomb)interactionsbetweenelectrons,a3DmetalinUQLexhibitsLuttinger-liquidlike(1D)behavioratenergieshigherthantheCDWgap[ 5 6 ].Biaginietal.[ 5 ]andTsaietal.[ 6 ]showedthatintheUQL,thetunnelingconductancehasapowerlawanomaly(nonlinearitiesinI-Vcharacteristicsatsmallbiases),whichistypicalforaonedimensionalinteractingsystem(Luttingerliquid).Themagnetic-eld-inducedLuttingerliquidphasecanbeanticipatedfromthefollowingsimpliedpicture.Inastrongmagneticeld,electrontrajectoriesarehelicesspiralingaroundtheeldlines.Abundleof

PAGE 14

suchtrajectorieswithacommoncenteroforbitcanbeviewedasa1Dconductor(\wire").Inthepresenceofelectron-electroninteractions,each\wire,"consideredseparately,isintheLLstate.Interactionswithsmallmomentumtransfersamongelectronsondierent\wires"donotchangetheLLnatureofasinglewire[ 7 ].Inchapter2ofthisdissertationwestudythetransportpropertiesofadisordered3DmetalintheUQL,bothwithandwithoutelectron-electroninteractions.Boththelocalizationandinteractioncorrectionstotheconductivityshowsignaturestypicalforone-dimensionalsystems.Beforewegetintothedetailsofourstudy,wewillbrieyreviewthephysicsoftheinterplaybetweentheinteractioneectsanddisorderinducedlocalizationindiusivesystemsoflowdimensionality. Atlowtemperatures,theconductivityofdisorderedconductors(normalmetalsandsemiconductors)isdeterminedbyscatteringofelectronsoquencheddisorder(e.g.,impuritiesanddefects).TheresidualconductivityisgivenbytheDrudeformula, m; wherenistheelectronconcentration,eistheelectroncharge,isthetransportmeanfreetime,andmistheeectivemass.TheDrudeformulaneglectsinterferencebetweenelectronwavesscatteredbydierentimpurities,whichoccurascorrectionstoEq. 1{2 ,intheparameter(kF`)11(wherekFistheFermimomentumand`ismeanfreepath).Inlowdimensions(d2),these(interference)quantumcorrectionstotheconductivity(QCC)divergewhenthetemperatureTdecreasesandeventually,drivethesystemtotheinsulatingregime.Thequantumcorrectionstotheconductivityareofsubstantialimportanceevenforconductorsthatarefarfromthestronglocalizationregime:inawiderangeofparametersQCC,thoughsmallerthantheconductivity,determineallthetemperatureandelddependenceoftheconductivity.Thesystematicstudyof

PAGE 15

QCCstartedalmostthreedecadesago.Acomprehensivereviewofthestatusoftheproblemfromboththeoreticalandexperimentalviewpointscanbefoundinseveralpapers[ 8 { 11 ]. Accordingtotheirphysicalorigin,QCCcanbedividedintotwodistinctgroups.Thecorrectionofthersttype,knownastheweaklocalization(WL)correction,iscausedbythequantuminterferenceeectonthediusivemotionofasingleelectron.Forlow-dimensional(d=1;2)innitesystemstheWLQCCdivergeatT!0;thisdivergenceisregularizedeitherbyamagneticeldorbysomeotherdephasing(inelasticscattering)mechanism.WewillelaborateonthistypeofQCCinsection 1.1.1 belowandalsoseehowitchangesfora3DmetalinUQLinchapter2. ThesecondtypeofQCC,usuallyreferredtoastheinteractioneects,isabsentintheone-particleapproximation;theyareentirelyduetointeractionbetweenelectrons.Thesecorrectionscanbeinterpretedastheelasticscatteringofanelectronotheinhomogeneousdistributionofthedensityoftherestoftheelectrons.OnecanattributethisinhomogeneousdistributiontotheFriedeloscillationsproducedbyeachimpurity.Theroleoftheelectron-electroninteractionsinthistypeofQCCistoproduceastaticself-consistent(andtemperaturedependent)potentialwhichrenormalizesthesingleparticledensityofstatesandtheconductivity.Suchapotentialdoesnotleadtoanyrealtransitionsbetweensingle-electronquantumstates(thoserequirerealinelasticscattering).Therefore,itdoesnotbreakthetimereversalinvarianceofthesystemandneitherdoesitaectnorregularizetheWLcorrections.WewillelaborateonthistypeofQCCinsection1.1.2,andalsostudyitforourcaseof3DmetalinUQLinchapter2. HowevertheinteractionbetweenelectronsisbynomeansirrelevanttotheWLQCC.Indeed,theseinteractionscausephaserelaxationofthesingleelectron

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states,andthusresultinthecut-oofthedivergencesintheWLcorrections.Thisdephasing(describedbybythephasebreakingtime(T))requiresrealinelasticcollisionsbetweentheelectronsandcanbeobtainedexperimentallyfromthetemperaturedependenceofmagneto-resistancemeasurements.Wewilldiscussthephasebreakingtimeduetoelectron-electroninteractionsinsection1.1.3,ofthisintroduction.Thereforetherearetwoclassesofinteractioncontributiontotheconductivity:thegenuineinteractioncorrections(elasticscatteringofFriedeloscillation:Altshuler-Aronovcorrections)-ClassI,andcorrectionstoWLQCCduetointeractions(inelasticscattering-dephasing)-ClassII. Figure1{1. Weaklocalizationcorrections. alongdierenttrajectories(Fig. 1{1 ).ThetotalprobabilityWforatransferfrompointAtopointBis TherstterminEq. 1{3 describesthesumoftheprobabilitiesforeachpathandthesecondtermcorrespondstointerferenceofvariousamplitudes.Theinterference

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termdropsoutwhenaveragingovermanypathsbecauseofitsoscillatorynature.However,thereexistsspecialtypeoftrajectories,i.e.,theself-intersectingones,forwhichinterferencecannotbeneglected(seeFig. 1{1 ).IfA1istheamplitudefortheclockwisemotionaroundtheloopandA2istheamplitudefortheanticlockwisemotion,thentheprobabilitytoreachpointOis i.e.,twicethevaluewewouldhaveobtainedbyneglectinginterference.Enhancedprobabilitytondtheparticleatapointoforiginmeansreducedprobabilitytonditatnalpoint(B).Thereforethiseectleadstoadecreaseintheconductivity(increaseinresistivity)inducedbyinterference. TherelativemagnitudeofweaklocalizationQCC,=,isproportionaltotheprobabilitytoformalooptrajectory ZdPZdtv2 leadingto(2d)=2(lnford=2),whichdivergesasTisloweredford2,leadingtostrongAndersonlocalization.Herevistheelectronvelocity,Disthediusioncoecient,istheelectronwavelengthand(T)isthephasebreakingtime.Phasecoherenceisdestroyedbyinelasticscattering(electron-electron,electron-phonon)orbymagneticanda.celectricelds.ThetemperaturedependenceoftheWLcorrectionisdeterminedby(T).Typically,Tp,wheretheexponentpdependsontheinelasticscatteringmechanism(electron-electron,electron-phonon)anddimensionality.Interferenceeectsoccurfor(T)i.e.,atlowtemperatures. InthelanguageofFeynmandiagrams,theWLQCC[ 12 ]isobtainedbyincludingthemaximallycrossedladderdiagram,theCooperon(seeFig. 1{2 ),intheconductivitydiagram.Theothertypeofladder(vertex)diagram,theDiuson

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(Fig. 1{2 ),whenincludedintheconductivitydiagramchangestheelasticscatteringtimetothetransporttime.Intheeld-theoreticlanguage,theWeakLocalization Figure1{2. LadderdiagramforM(diuson)andC(Cooperon). correction,whicharisesduetointerferenceoftimereversedpathsisdeterminedbythe\Cooperon"modeC(Q;!),i.e.,theparticle-particlediusionpropagator, 21 torstorderin(kf`)1.Calculationofthesingularcontributionstoconductivity(interferenceeect)atsmall!;QshouldincludediagramscontainingasaninternalblockthegraphswhichyieldaftersummationC(Q;!)( 1{3 ).TheWLQCCis Figure1{3.

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whichgives !+1 indierentdimensions.Perturbationtheorybreaksdowninone-andtwo-dimensions(for!1 13 ]hadrstshownthatatsucientlyhighimpurityconcentration,electronicstatesbecomelocalizedandthesystembecomesaninsulator.MottandTwose[ 14 ]hadpredictedthattheconductivityforaonedimensionalsystemshouldvanishinthelimitoflowfrequencies(Mott'slaw)whichwaslaterrigourouslyprovedbyBerezinskii[ 15 ],whoshowedthatelectronstatesin1Darestronglylocalizedandthereisnodiusiveregimein1D.Thelocalizationlengthin1Disoftheorderofthemeanfreepath(`),thereforein1Dforlengthscalesshorterthan`,theelectronmotionisballisticandforlengthslongerthan`thenelectronmotionislocalized.2Dsystemsarealsostronglylocalizedbutthelocalizationlengthisverylarge(Lloc`ekf`)ascomparedto1D(Lloc`).Thusin2D,theballisticregime(L`)crossesovertothediusiveregime(for`
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Inthemetallicregimeg1,theconductanceshowsohmicbehaviorforwhich(g)=d2.Correctionsto(g)inthemetallicregimeareobtainedbyperturbationtheoryin1 1{1 )acquireadditionalphasefactors,A1!A1expie 0;A2!A2expie 0; 17 ].ThecharacteristictimescaleforphasebreakingistHlH2=DwherelH=p Aweakmagneticelddestroysphasecoherenceandincreasestheconductivity.Iftheeldisincreasedfurther,wereachtheclassicalmagneto-resistanceregime,

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wheretheconductivitydecreaseswiththeeld.WhathappensatevenhighereldswhenLandauquantizationbecomesimportant?Weaddressthisissueinchapter2ofthisdissertation.Weshowthatathree-dimensionaldisorderedconductorintheUltraquantumlimit,whereonlythelowestlandaulevelispopulated,exhibitsanewphenomenon:intermediatelocalization.ThequantuminterferencecorrectionisoftheorderoftheDrudeconductivityD(asin1D)whichindicatesabreakdownofperturbationtheory.However,theconductivityremainsniteatT!0(asin3D).Itisdemonstratedthattheparticle-particlecorrelator(Cooperon)ismassive.Itsmass(inunitsofthescatteringrate)isoftheorderoftheimpurityscatteringrate. 18 ](thewavefunctionrenormalizationZ,eectivemassm?,etc.).Firstwenotethatwithinthetransportequation,electron-electroncollisionscaninnowayaecttheconductivityinthecaseofasimplebandstructureandintheabsenceofUmklappprocesses,sinceelectron-electroncollisionsconservethetotalmomentumoftheelectronsystem.InclusionoftheFermiliquidcorrectionsrenormalizestheresidualresistivitywhilenotresultinginanydependenceoftheconductivityonthetemperatureandfrequency.HoweveronefrequentlyencountersthesituationthattheresistivityscalesasT2.Thisdependenceisofteninterpretedasthe\Fermi-liquid"eect,arisingfromelectron-electronscatteringwithcharacteristictimeee/T2.Infact,theresistivityisduetoUmklappscattering.Ingoodmetals,normalprocesses(whichconservethetotalelectronmomenta)andUmklappprocesses(whichconservethemomentauptoareciprocal

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latticevector)areequallyprobableandtheUmklappscatteringrateenteringtheresistivityalsoscalesasT2.Notethatatlowtemperaturesthisresistivityduetoelectron-electronscattering(Umklapp)givesthedominantcontributionbecausetheelectron-phononcontributiontotheresistivityscalesasT5(Bloch'slaw,eph1=T5). Aswasmentionedpreviously,takingintoaccounttheinterferenceofelasticscatteringbyimpuritieswiththeelectron-electroninteractionproducesnontrivialtemperatureandfrequencydependencesoftheconductivity.ThiscorrectionarisesfromcoherentscatteringofanelectronfromanimpurityandtheFriedeloscillationitcreates[ 19 ].Wewillrststudythiscorrectiontotheconductivityintheballisticlimit,(T1,whereistheelasticscatteringlifetime)andtheninthediusiveregime(T1).Inthediusivelimitanelectronundergoesmultiplecollisionswithimpuritiesbeforeitscattersfromanotherelectron,whereasintheballisticlimittheelectron-electroncollisionrateisfasterthanelectron-impurityrate,thussingleimpurityeectsareimportantintheballisticlimit.Inchapter2ofthisdissertationwewillevaluatethisinteractionQCCintheballisticlimitina3DmetalintheUQL.TherehasbeenarecentrenewalofinterestintheinteractionQCC,(classI)duetothemetaltoinsulatortransitionobservedintwo-dimensional(highmobility)Si-MOSFETsamples[ 20 ].ThequalitativefeaturesofthistransitionwasunderstoodbyZala,NarozhnyandAleiner[ 19 ]whoshowedthattheinsulating(logarithmicupturnintheresistivity)behaviorinthediusiveregimeandmetallic(linearriseintemperature)behavioroftheresistivityintheballisticlimit(2D),areduetocoherentscatteringatFriedeloscillations.Below,werstoutlinetheirsimplequantummechanicalscatteringtheoryapproachtoshowhowtemperaturedependentcorrectionstoconductivityariseforscatteringatFriedeloscillations,andthenextendtheiranalysistoobtaintheinteractionQCCin

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3Dballisticlimit.Inchapter2weevaluatethiscorrectionfora3DsystemintheUQL. Figure1{4. ScatteringbyFriedeloscillations. 1{4 ).Consideranimpurityattheorigin;itspotentialUimp(~r)inducesamodulationofelectrondensityaroundtheimpurity.IntheBornapproximationonecanndtheoscillatingcorrection,n(r)=n(r)n0totheelectrondensityn(~r)=Pkjk(~r)j2: Hereristhedistancefromtheimpurity,kFistheFermimomentum,g=RUimp(~r)d~risthematrixelementforimpurityscatteringandn0istheelectrondensityintheabsenceofimpuritiesanddisdimensionality.Takingintoaccountelectron-electroninteractionsV0(~r1~r2)onendsadditionalscatteringpotentialduetotheFriedeloscillationsEq. 1{11 .Thispotentialcanbepresentedasasumofthedirect(Hartree)andexchange(Fock)terms[ 21 ]

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2V0(~r1~r2)n(~r1;~r2);(1{14) whereby(~r)wedenotediagonalelementsoftheoneelectrondensitymatrix, Asafunctionofthedistancefromtheimpurity,theHartree-FockenergyVoscillatessimilarlytoEq. 1{11 .Theleadingcorrectiontoconductivityisaresultofinterferencebetweentwosemi-classicalpathsshowning.4.Ifanelectronfollowspath\A,"itscattersotheFriedeloscillationcreatedbytheimpurityandpath\B"correspondstoscatteringbytheimpurityitself.Interferenceismostimportantforscatteringanglescloseto(orforbackscattering),sincetheextraphasefactoraccumulatedbytheelectrononpath\A"(ei2kR)relativetopath\B"iscanceledbythephaseoftheFriedeloscillationei2kFR,sothattheamplitudecorrespondingtothetwopathsarecoherent.Asaresult,theprobabilityofbackscatteringisgreaterthantheclassicalexpectation(takenintoaccounttheDrudeconductivity).Therefore,accountingforinterferenceeectsleadtoacorrectiontotheconductivity.Wenotethattheinterferencepersiststolargedistances,limitedbytemperatureRjkkFj1vF=T.Thusthereisapossibilityforthecorrectiontohavenontrivialtemperaturedependence.Thesignofthecorrectiondependsonthesignoftheeectivecouplingconstantthatdescribeselectron-electroninteraction.First,wewillstudythecontributionarisingfromtheHartreepotential.ConsiderascatteringprobleminthepotentialgiveninEq. 1{13 .Theparticle'swavefunctionisasumoftheincomingplanewaveandtheoutgoingsphericalwave(3D),=ei~k:~r+f()eikr

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wheref()isthescatteringamplitude,whichwewilldetermineintheBornapproximation.Fortheimpuritypotentialitselftheamplitudef()weaklydependsontheangle.AtzerotemperatureitdeterminestheDrudeconductivityD,whiletheleadingtemperaturecorrectionisT2(whenthescatteringtimeenergydependent),asisusualforFermisystems.WenowshowthatthisisnotthecaseforthepotentialinEq. 1{12 .Infact,takingintoaccountEq. 1{12 leadstoenhancedbackscatteringandthustotheconductivitycorrectionwhichdependsontemperatureas/T2lnT(in3D),T(in2D)and,aswewillseelaterT2(in3DUQL,istheinteractionparameter)allfortheballisticlimit. Farfromthescattererthewavefunctionofaparticlecanbefoundintherstorderofperturbationtheoryas=ei~k:~r+(~r),wherethecorrectionisgivenby[ 22 ] SubstitutingtheformoftheHartreepotentialfromEq: 1{13 ,andintroducingtheFouriertransformoftheelectron-electroninteractionV0(q),weobtainforthescatteringamplitude(atlargedistancesfromtheimpurity) 2Zd~rn(r)ei~q:~r;(1{17) where~q=~kk~r=randj~qj=2ksin(=2).Weseethatthescatteringamplitudedependsonthescatteringangle(),aswellastheelectron'senergy(=k2=2m).Thedensityoscillationin3D,withhardwallboundaryconditionattheorigin(impenetrableimpurity),isn(r)=Zd~kf(k)[jk(~r)j2j0k(~r)j2];=2kF 2kF(ra)sin(2kFr) 2kFr;

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whereaisthesizeoftheimpurityandf(k)istheFermidistributionfunction.Wemaketheswavescatteringapproximation(slowparticles,kFa1)toobtain r2cos(2kFr) 2kFrsin(2kFr) (2kFr)2:(1{18) SubstitutingthedensityfromEq. 1{18 inEq. 1{17 ,weobtainforthescatteringamplitude 2sin( 4lnj1sin( 1+sin( Inthelimit+xwherex1,thescatteringamplitudebehavesasf(x)V0(2kF)[xlnx].Thetransportscatteringcrosssectionisnow wheref0istheamplitudeforscatteringattheimpurityitself(whichdoesnotdependonintheBornlimitandgivesaconstant(Tindependent)valuefortheDrudeconductivity).Theleadingenergydependencecomesfromtheinterference(crossterm),whichisproportionaltof().Themaincontributiontotheintegralcomesfrom(backscattering).Expandingnear,i.e.,=+1where1issmall[ 19 ],1p 1 ThenoneobtainstheinteractionQCCfromtheHartreechannel[ 23 ]in3D(using=D==D) DV0(2kF)T EF2ln(EF Oneobtainsasimilarcontributionfromtheexchange(Fock)potential,exceptnowthecouplingconstantinfrontoftheT2lnTtermisV0(0).TheHartreeandexchangecontributioncomewithoppositesigns.In2DtheinteractionQCCis

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linearintemperature[ 19 ] D[2V0(2kF)V0(0)]T EF:(1{23) In1DYue,GlazmanandMatveev[ 21 ]usedthesameapproachandcalculatedthecorrectiontothetransmissioncoecientduetoscatteringattheFriedeloscillationandobtainedalogarithmictemperaturecorrectionatthelowestorder where[V02V2kF]=vF.Usingapoormanrenormalizationgroupprocedure,theyshowedthattherstorderlogarithmiccorrectionisinfactaweakcouplingexpansionofthemoregeneralpowerlawscalingformofthetransmissioncoecient,t=t0T W; W2:(1{25) Thisresultwasalsoobtainedindependently(viabosnization)byKaneandFisher[ 24 ].Eq. 1{22 1{23 ,andEq. 1{25 givetheinteractionQCCintheballisticlimitin3D,2Dand1Dsystemsrespectively.Inchapter2ofthisdissertationweshowthatin3DUQL,thisinteractioncorrectiontotheconductivitybehavessimilartothatofatrue1Dsystem. TheinteractioncorrectiontoQCCinthediusivelimitalsoarisesfromthesamephysics(namelyscatteringatfriedeloscillations)butnowonehasaverageovermanyimpurities(diusivemotion).ThiscorrectiontotheconductivitywasevaluatedbyAltshulerandAronovin3D[ 8 ]andbyAltshuler,AronovandLeein

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2D[ 25 ]. whereFisthedependsonthestrengthoftheinteraction,and 33F D;(3D):(1{27) ScatteringattheFriedeloscillationsalsoresultsinasingularenergy(temperature)dependenceofthelocaldensityofstateswhichcanbeobservedasazerobiasanomalyintunneling.ThelocalDOScanbeobtainedfromtheelectronsGreen'sfunctionusing()=ImRd~pGR(~(p);).ThecorrectiontotheGreen'sfunctioncanbeevaluatedthesamewayasweevaluatedthecorrectiontothewavefunctionduetoFriedeloscillationoritcanalsobeevaluateddiagrammaticallybycalculatingtheelectron'sselfenergyinthepresenceofdisorderandinteraction[ 8 ].

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excitationdecayshouldlikewisebeproportionalto2.Itturnsout,however,thatexcitationindisorderedsystemsdecaysfaster,whichraisesthequestionofvalidityofquasiparticledescriptionofdisorderedconductorsinlowdimensionalsystems.Apartfrombeingimportantinthedevelopmentofthetheory,thedecaytimeforoneelectronexcitations(thephaserelaxationtime),governsthetemperaturedependenceoftheWLQCC. ItwasshownbyAltshulerandAronovthatfor3Ddisorderedsystems,thephaserelaxationtimeisgovernedbylargeenergytransferprocessesandinthisregimeee(whereeeistheoutrelaxationtime).TheoutrelaxationtimecanbecalculatedfromtheBolztmannequation(withdiusivedynamicsfortheelectrons).Thisgives1 D)3=2in3D,[ 8 ].Howeverinlowerdimensions(d=1;2)electron-electroncollisionswithsmallenergytransfersisthedominantmechanismfordephasing.TheBolzmannapproach(whichisgoodforlargeenergytransfers)failsin2Dand1Dcase.Technically,therewouldbedivergencesforsmallenergytransfers[ 8 ]bothin2D(logarithmic)andquasi-1D(powerlaw)intheBolztmann-equationresultfortheoutrelaxationrate.Thesedivergencesmustberegularizedinaself-consistentmanner.Thephasebreakingtimeinlowerdimensionscanalsobeobtainedbysolvingtheequationofmotionfortheparticle-particle(Cooperon)propagatorinthepresenceofspaceandtimedependentuctuatingelectromagneticeldswhichmodelthesmallenergytransferprocesses[ 26 ].Thisgives()1T(in2D)and()1T2=3(inquasi-1D). Intrueone-dimensionalsystems,thissubjectiscontroversialastrue1Dsystemsdonothaveadiusiveregime(theballisticlimitcrossesovertothelocalizedregime)andthequasiparticledescriptionbreaksdownforaninteracting1DsystemwhichisintheLuttingerliquidstate.Asaresultonecannotdeneee.Inarecentworkonthissubject[ 27 ],itwasshownthatevenfora1DdisorderedLuttingerliquid,thereexistsaweaklocalizationcorrectiontotheconductivity

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whosetemperaturedependenceisgovernedbythephaserelaxationrate,()1/p );(1{28) whereD=e2vF2istheDrudeconductivityin1D,whichdependsonTthrougharenormalizationofstaticdisorder,0==(EF=T)2.Here0isnon-interactingscatteringtimeandistherenormalized(byFriedeloscillation)scatteringtimeandcharacterizestheinteraction. Atpresenttherearenotheoreticalpredictionsforin3DUQL.TheFermiliquidapproachesforcalculatingthephasebreakingtimearenotexpectedtoworkherebecausetheCooperonisnotasingulardiagram(itacquiresamassin3DUQLasshowninchapter2)and,onceagain,therearenosingleparticlelikeexcitationsasthegroundstateisacharge-density-waveandexcitationsabovethegroundstateareLuttingerliquidlike.Howeverinchapter2wewillshowthatsomerecentmagneto-resistancemeasurementsongraphiteinUQLqualitativelyagreewithpredictionsofduetoelectron-phononinteractionsin1D. 28 ].AsearchforstabilityconditionsofaFermiliquidanddeviationsfromaFermiliquidbehavior,[ 29 { 32 ]particularlynearquantumcriticalpoints,intensiedinrecentyearsmostlyduetothenon-Fermi-liquidfeaturesofthenormalstateofhighTcsuperconductors[ 33 ]andheavyfermionmaterials[ 34 ]. ThesimilaritybetweentheFermi-liquidandaFermigasholdsonlyfortheleadingtermsintheexpansionofthethermodynamicquantities(specicheat

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Inthisintroduction,wewilldiscussthedierencebetweenthe\regular"processeswhichleadtotheleadingFermi-liquidformsofthermodynamicquantitiesand\rare"1Dprocesseswhichareresponsibleforthenonanalytic(non-Fermiliquid)behavior.Wewillseethattheroleoftheserareprocessesincreasesasthedimensionalityisreducedand,eventually,therareprocessesbecomenormalin1D,wheretheFermi-liquiddescriptionbreaksdown. InaFermigas,thermodynamicquantitiesformregular,analyticseriesasafunctionofeithertemperatureT,ortheinversespatialscaleqofaninhomogeneousmagneticeld.ForTEFandqkF, where=2F=3,s0=gB2FandFmkFD2isthedensityofstates(DOS)ontheFermisurface,gistheLandefactorandBistheBohrmagnetonanda:::faresomeconstants.EvenpowersofToccurbecauseoftheapproximateparticle-holesymmetryoftheFermifunctionaroundtheFermienergy.Theaboveexpressionsarevalidinalldimensions,exceptD=2.ThisisbecausetheDOSisconstantin2D,theleadingcorrectiontotheTterminC(T)isexponentialinEF=Tandsdoesnotdependonqforq2kF.Howeverthisanomalyisremovedassoonaswetakeintoaccountanitebandwidthoftheelectronspectrum,uponwhichtheuniversal(T2nandq2n)behaviorisrestored.

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AninteractingFermisystemisdescribedbyLandau'sFermi-liquidtheory,accordingtowhichtheleadingtermsinC(T)andsaresameasthatoftheFermigaswithrenormalizedparameters(replacebaremassbyeectivemassm?,baregfactorbyeectiveg-factorg?intheaboveFermigasresults), whereFc;FsarechargeandspinharmonicsoftheLandauinteractionfunction:^F(~p;~p0)=Fc()^I+Fs()~:~0,where~,arethePaulimatrices.TheFermi-liquidtheoryisanasymptoticallylow-energytheorybyconstruction,anditisreallysuitableonlyforextractingtheleadingterms,correspondingtothersttermsintheFermigasexpression.Indeed,thefreeenergyoftheFermi-liquidofanensembleofquasiparticlesinteractinginapairwisemannercanbewrittenas[ 35 ]FF0=Xk(k)nk+1 2Xk;k0fk;k0nknk0+O(n3k); Strictlyspeaking,anonanalyticdependenceoffk;k0onthedeviationsfromtheFermisurfacekkF,accountsforthenon-analyticTdependenceofC(T)[ 36 ].HigherordertermsinTorqcanbeobtainedwithinmicroscopicmodelswhichspecifyparticularinteractionandemployperturbationtheory.SuchanapproachiscomplimentarytotheFL:theformerworksforweakinteractionsbutatarbitrarytemperatureswhereasFLworksbothforweakandstronginteractions,

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butonlyinthelimitoflowesttemperatures.Microscopicmodels(Fermigaswithweakrepulsion,electron-phononinteraction,paramagnonmodel,etc.)showthatthehigherordertermsinthespecicheatandspinsusceptibilityarenonanalyticfunctionsofTandq[ 37 { 48 ].Forexample, whereallcoecientsarepositiveforthecaseofelectron-electroninteraction.Asseenfromtheaboveequationsthenonanalyticitybecomesstrongerasthedimensionalityisreduced.Thestrongestnonanalyticityoccursis1D,where-atleastaslongassingleparticlepropertiesareconcerned-theFLbreaksdown[ 49 50 ]: Thesenonanalyticcorrectionstothespecicheatandspinsusceptibilityin1Dareobtainedinchapter3.Itturnsoutthattheevolutionofthenon-analyticbehaviorwiththedimensionalityreectsanincreasingroleofspecial,almost1Dscatteringprocessesinhigherdimensions.Thusnon-analyticitiesinhigherdimensionscanbeviewedasprecursorsof1DphysicsforD>1. WewillrststudythenecessaryconditiontoobtainaFLdescriptionandthenseehowrelaxingtheseconditionsleadtothenonanalyticformfortheself-energyandthermodynamicproperties.WithintheFermiliquid

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Landau'sargumentforthe"2(orT2)behaviorofImRrequirestwoconditions:(1)quasiparticlesmustobeyFermistatistics,i.e.,thetemperatureissmallerthanthedegeneracytemperatureTF=kFvF?,wherevF?istherenormalizedFermivelocity,(2)inter-particlescatteringisdominatedbyprocesseswithlarge(generally,oforderkF)momentumtransfers.Oncethesetwoconditionsweresatised,theself-energyassumesauniversalform,Eq. 1{40 andEq. 1{41 ,regardlessofaspecictypeofinteraction(electron-electron,electron-phonon)anddimensionality.Considertheself-energyofanelectron(1storder)asitinteractswithsomeboson(seeFig. 1{5 ).Thewavylinecanbe,e.g.,adynamicCoulombinteraction,phononpropagator,etc.Onthemassshell("=k;wherek=k2=2mkF2=2m)atT=0andfor">0 Figure1{5. Self-energyatrstorderininteractionwithabosoniceld ImR(")Z"0d!ZdD~qImGR("!;~k~q)ImVR(!;~q) (1{42) Theconstraintonenergytransfers(0
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Asafunctionofq,Fhasatleasttwocharacteristicscales.Oneisprovidedbytheinternalstructureoftheinteraction(screeningwavevectorfortheCoulombpotential)orbykFwhicheverissmaller.Thisscale,Q,doesnotdependon!andprovidestheultra-violetcutointhetheory.Inadditionthereisasecondscalej!j=vF,and,since!isboundedfromaboveby"andforlowenergies("!0),onecanassumeQj!j=vF.Thusinadimensionlessform ImVR(!;q)=! QDUq Q;j!j TheangularintegrationoverImGRyieldsonthemassshell vFq); wherethesubscriptDstandsforthedimensionality,andA3(x)=(1jxj);A2(x)=(1jxj) 1{45 andEq. 1{44 intoEq. 1{42 ,oneobtains ImR(")Z"0d!!ZQqj!j=vFdqqD2Uq Q;j!j NowifthemomentumintegralisdominatedbylargemomentaoftheorderofQ,thenthefunctionUtoleadingordercanbeconsideredtobeindependentoffrequency(sinceQj!j=vF),andonecanset!=0inU,andalsoreplacethelowerlimitoftheqintegralbyzero.Themomentumandfrequencyintegralsthendecouple,(themomentumintegralgivesapre-factorandthefrequencyintegralgives"2),andoneobtainsananalytic"2dependenceforIm.Thenthelinearin"terminRecanbeobtainedbyusingtheKramers-Kronigrelation.Thus

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weseethatlargemomentum(andenergyindependent)transfersanddecouplingofthemomentumandfrequencyintegralareessentialtoobtainaFLbehavior.The"2resultseemstobequitegeneralundertheassumptionsmade.Whenandwhyaretheseassumptionsviolated?Long-rangeinteraction,associatedwith Figure1{6. Kinematicsofscattering.(a)\Any-angle"scatteringleadingtoregularFLtermsinself-energy;(b)Dynamicalforwardscattering;(c)Dynamicalbackscattering.Processes(b)and(c)areresponsiblefornonanalytictermsintheself-energy small-anglescattering,isknowntodestroytheFL.Forexample,transverselongrange(current-current[ 51 ]orgauge[ 52 ])interactionswhich,unliketheCoulombinteractionarenotscreened,leadtothebreakdownoftheFermi-liquid.Buttheseinteractionsoccurunderspecialcircumstances(e.g.,nearhalf-llingforgaugeinteractions).Foramoregenericcase,itturnsoutthatevenifthebareinteractionisofthemostbenignform,e.g.,adelta-functioninrealspace,therearedeviationsfromaFLbehavior.Thesedeviationsgetampliedasthedimensionalityisreduced,and,eventually,leadtoacompletebreakdownoftheFLin1D.Alreadyforthesimplestcaseofapoint-likeinteraction,thesecondorderself-energyshows

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Figure1{7. Nontrivialsecondorderdiagramsfortheself-energy anontrivialfrequencydependence.Foracontactinteractionthetwoself-energydiagramsofFig.canbelumpedtogether(theseconddiagramis1=2therstone).Twogivenfermionsinteractviapolarizingthemediumconsistingofotherfermions.Hencetheeectiveinteractionatthesecondorderisproportionaltothepolarizationbubble,whichjustshowshowpolarizablethemediumis, ImVR(!;q)=U2ImR(!;q): Forsmallanglescatteringq2kF;!EF,theparticle-holepolarizationbubblehasthesamescalingforminallthreedimensions[ 53 ], vFqBD! vFq; whereD=aDmkFD2isthedensityofstatesinDdimensions(a3=2;a2=1;a1=2=)andBDisadimensionlessfunctionwhosemainroleistoimposeaconstraint!vFqin2Dand3D,and!=vFqin1D.TheaboveformofthepolarizationoperatorindicatesLandaudamping:Collectiveexcitations(spinandchargedensitywaves)decayintoparticle-holepairs,thisdecayoccursonlywithintheparticle-holecontinuumwhoseboundaryforD>1isat!=vFqforsmall!;q,therefore,decayoccursfor!
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getsin3D, ImR(")U2Z"0d!ZQkFj!j=vFdqq21 vFqU2Z"0d!![kF|{z}FL! vF|{z}beyondFL];a"2bj"j3; wherethersttermoriginatesfromthelargemomentumtransferregimeandistheFermi-liquidresultwhereasthesub-leadingsecondtermoriginatesfromthesmall-momentum-transferregimeandisnonanalytic.Thefractionofphasespaceforsmallanglescatteringissmall:mostoftheself-energycomesfromlarge-anglescatteringevents(qQ),butwealreadystarttoseetheimportanceforsmallangleprocesses.ApplyingKramers-Kronigtransformationtothenon-analyticpart(j"j3)inImR,wegetacorrespondingnon-analyticcontributiontotherealpartas(ReR)non-an/"3lnj"jand,nally,usingthespecicheatformula(seeEq. 3{14 inchapter3)wegetanonanalyticT3lnTcontributionwhichhasbeenobservedexperimentallybothinmetals[ 54 ](mostlyheavyfermionmaterials)andHe3[ 55 ].Similarlyin2D ImR(")U2Z"0d!ZkFj!j=vFdqq1 vFqU2"2lnEF andReR(")/"j"jandthisresultsintheT2non-analyticityforthespecicheatwhichhasbeenobservedinrecentexperimentsonmonolayersofHe3adsorbedonsolidsubstrate[ 56 ]. In1D,asweshowinchapter3,thesituationisslightlydierent.EventhoughthesamepowercountingargumentsleadtoImR/j"jandReR/"lnj"jforthesecondorderself-energy,C(T)islinear(analytic)inTatsecondorderandthenonanalyticTlnTshowsuponlyatthirdorderininteractionandonlyforfermionswithspin.Thisdierenceisduetothefactthatin1D,smallmomentumtransfers(hereparticle-holecontinuumshrinkstoasingleline!=vFq,sodecayof

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collectiveexcitationsispossibleonlyonthisline)donotleadtothespecicheatnonanalyticitywhichoccurssolelyfromthenonanalyticityofthebackscattering(atq2kF)particle-holebubbleortheKohnanomaly.Thus,wehavethesamesingularbehaviorofthebubble(responsefunctions)andtheresultsfortheself-energydiersbecausethephasevolumeqDismoreeectiveinsuppressingthesingularityinhigherdimensionsthaninlowerones. Inadditiontotheforwardscatteringnonanalyticity,thereisalsoanonanalyticcontributiontotheself-energyandthermodynamicsarisingfromq2kF,partoftheresponsefunction,i.e.,theKohnanomaly.Usually,theKohnanomalyisassociatedwiththe2kFnonanalyticityofthestaticparticle-holebubbleanditsmostfamiliarmanifestationistheFriedeloscillationinelectrondensityproducedbyastaticimpurity(seesection1.1.2,ofthisthesis).HerethestaticKohnanomalyisofnointerestaswearedealingwithdynamicalprocesses.However,thedynamicalbubbleisalsosingularnear2kF,e.g.,in2D ImR(q2kF;!)/! Duetotheone-sidedsingularityinImRasafunctionofq,the2kFeectiveinteractionoscillatesandfallsoasapowerlawinrealspace.Bypowercounting,sincethestaticFriedeloscillationfallsoassin(2kFr) ~U/!sin(2kFr) DynamicalKohnanomalyresultsinthesamekindofnon-analyticityintheself-energy(andthermodynamics)astheforwardscattering.Thesingularitynowcomesfromjq2kFj!=vF,i.e.,dynamicbackscattering.Thereforethenonanalytictermintheself-energyissensitiveonlytostrictlyforwardor

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backscatteringevents,whereasprocesseswithintermediatemomentumtransferscontributetotheanalyticpartoftheself-energy. Figure1{8. Scatteringprocessesresponsiblefordivergentand/ornonanalyticcorrectionstotheself-energyin2D.(a)\Forwardscattering"-ananalogoftheg4processin1D(b)\Forwardscattering"withanti-parallelmomenta-ananalogoftheg2processin1D(c)\backscattering"withantiparallelmomenta-ananalogoftheg1processin1D Wewillnowperformakinematicanalysisandshowthatthenonanalytictermsintheself-energyandspecicheatin2Dcomesfromonly1Dscatteringprocesses.Considertheself-energydiagramofFig.1-7.(a).Thenonanalytic"2ln"termintheself-energycamefromtwoq1singularities:onefromtheangularaverageofImGRandtheotheronefromthedynamic,!=vFqpartoftheparticle-holebubble.Thisformofthebubblearisesonlyinthelimit!vFq, ImR(!;q)ImZZdD~pd"G("!;~p~q)G(";~p)! vFqZd(cos! vFq);! vFqq vFq(for!vFq): Fromthedeltafunction,cos=!=vFq1,whichmeansthattheanglebetween~pand~qis=2or~pand~qareperpendiculartoeachother.SimilarlytheangularaveragingofImGR(~k~q;";!)alsopinstheanglebetween~kandqto90degrees.hImGR(~k~q;";!)i1Zd1("!qvFcos1)=)cos1"! vFq! vFq1=)1=2 Thus~pand~k(thetwoincomingmomentaofthefermions)arealmostperpendiculartothesamevector~q.In2D,thismeansthattheyareeitheralmostparalleltoeachotheroranti-paralleltoeachother,andsincethemomentumtransferiseither

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small,~q0ornear2kF,i.e.,jq2kFj0,weessentiallyhavethree1Dscatteringprocesses(seeFig. 1{8 )whichareresponsibleforthenonanalyticcorrectionstotheself-energy.Thesethreeprocessesare(a)twofermionswithalmostparallelmomenta(~k1~k2)collideandtransferasmallmomentum(~q0)andleavewithoutgoingmomentumwhicharealmostparalleltoeachother(~k10~k20)andparalleltotheirincomingmomenta(~k10~k1;~k20~k2):analogoustothe\g4"scatteringmechanismin1D(seeFig. 1{8 (a)andchapter3)(b)twofermionswithalmostanti-parallelmomentacollide(~k1~k2)andtransferasmallmomentum(~q0)andleavewithoutgoingmomentumwhicharealmostanti-paralleltoeachother(~k10~k20)butparalleltotheirincomingmomenta(~k10~k1;~k20~k2):analogoustothe\g2"scatteringmechanismin1D(seeFig. 1{8 (b)andchapter3),(c)twofermionswithanti-almostparallelmomentacollide(~k1~k2)andtransferalargemomentum~q2kFandleavewithoutgoingmomentumwhicharealmostanti-paralleltoeachother(~k10~k20)andalsoanti-paralleltotheirincomingmomenta(~k10~k1;~k20~k2):analogoustothe\g1"scatteringmechanismin1D(seeFig. 1{8 (c)andchapter3).Thereforethenonanalytic"2ln"termintheself-energyin2Dcomesfrom1Dscatteringevents,theonlydierenceisthat2Dtrajectoriesdohavesomeangularspread,whichisoftheorderofj!j=EF.Itturnsout(Ref.[ 44 ]),thatoutofthethree1Dprocesses,theg2processandg1processaredirectlyresponsiblefornonanalyticcorrections(NAC)toC(T)in2Dandonlytheg1processleadstoNACtoC(T)in1D.Theg4processalthoughleadstoamass-shellsingularityintheself-energyinboth2Dand1D,butdoesnotgiveanyNACtothermodynamics. In3Dthesituationisslightlydierent,~p?~qand~k?~qmeanthatboth~pand~klieinthesameplane.However,itisstillpossibletoshowthatforthethermodynamicpotential,~pand~kareeitherparalleloranti-paralleltoeachother.Hence,thenonanalyticterminC(T)alsocomesfromthe1Dprocesses.In

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addition,thedynamicforwardscatteringevents(markedwithastarinFig.1-9.)which,althoughnotbeing1Dinnature,doesleadtoanonanalyticityin3D.ThustheT3lnTanomalyinC(T)comesfromboth1Dandnon-1Dprocesses[ 47 ].Thedierenceisthattheformerstartalreadyatthesecondorderininteractionwhereasthelatteroccuronlyatthirdorder.In2D,theentireT2nonanalyticityinC(T)comesfrom1Dprocesses.Thenonanalyticcorrectiontothespinsusceptibilitywillbethesubjectofdiscussioninchapter4ofthisthesis,wherewewillshowthatthenonanalyticityins,bothin2Dand3Dcomesfromboth1Dandnon1Dscatteringprocesses. Typicaltrajectoriesoftwointeractingfermions Ourkinematicargumentscanbesummarizedinthefollowingpictorialway.Supposewefollowthetrajectoriesoftwofermions,asshowninFig. 1{9 .There

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areseveraltypesofscatteringprocesses.First,thereisa\any-angle"scatteringwhich,inourparticularexample,occursatathirdfermionwhosetrajectoryisnotshown.Thisscatteringcontributesaanalytic,FLtermsbothtotheself-energyandthermodynamics.Second,therearedynamicforwardscatteringevents,whenqj!j=vF.Thesearenon-1Dprocesses,asthefermionsentertheinteractionregionatanarbitraryangletoeachother.In3D,athirdorderinsuchaprocessleadstoaT3lnTterminC(T).In2Ddynamicforwardscatteringdoesnotleadtoanonanalyticity.Finallythereare1DscatteringprocessesmarkedwithaSiriusstarwherefermionsconspiretoaligntheirmomentaeitherparalleloranti-paralleltoeachother.TheseprocessesdeterminethenonanalyticpartofandC(T)in2Dand1D. Thereforethenonanalytictermsinthetwo-dimensionalself-energyandthermodynamicsarecompletelydeterminedby1Dprocesses,2Dscatteringdoesnotplayanyroleinthenonanalyticterms.Asaresult,ifthebareinteractionhassomeqdependence,onlytwoFouriercomponentsmatter:U(0)andU(2kF)e.g.,in2D ImR(")/[U2(0)+U2(2kF)U(0)U(2kF)]"2lnj"j; ReR(")/[U2(0)+U2(2kF)U(0)U(2kF)]"j"j; whereaandbaresomecoecients.TheseperturbativeresultscanbegeneralizedfortheFermi-liquidcase,wheninteractionisnotweak.ThentheverticesU(0)andU(2kF),occurringintheperturbativeexpressionsarereplacedbyscatteringamplitude()atangle=, ^(~p;~p0)=c()^I+s()~:~0;

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wherecandsrefertothechargeandspinsectorsrespectively.Thusin2D[ 45 ], Theadditional(lnT)2factorinthedenominatorcomesfromtheCooperchannelrenormalizationofthebackscatteringamplitude[ 47 48 ].In3D,theT3lnTnonanalyticityinthespecicheatarisesfromboth1D(excitationofasingleparticle-holepair)andnon-1D(excitationofthreeparticle-holepairs)scatteringprocesses[ 47 ]. {z }1D,onep-hpair+a;12a;0+3a;1+:::| {z }non1D,threep-hpairs wheresubscripta=c;sand0;1;2:::indicatetheharmonicsoftheexpansion.Again,theadditional(1+glnT)2factorinthedenominatorcomesfromtheCooperchannelrenormalizationofthebackscatteringamplitude[ 47 48 ]. WesawthatthenonanalyticcorrectionstothespecicheatinD=2;3,arisefromonedimensionalscatteringprocesses,(andtheyshowupatsecondorderinperturbationtheory),andthedegreeofnonanalyticityincreaseswithdecreaseindimensionality.Thispredictsthatthestrongestnonanalyticityinthespecicheatshouldoccurin1D.However,itwasshowninRef.[ 57 ],thatthespecicheatin1DislinearinT,atleastinsecondorderinperturbationtheory.Inaddition,thebosonizationsolutionofaone-dimensionalinteractingsystem,predictsthattheC(T)islinearinT.Weresolvethisparadoxbyshowing(inchapter3)thatthegeneralargumentfornonanalyticityinD>1atthesecondorderininteractionbreaksdownin1D,duetoasubtlecancelationandthenonanalyticTlnTterminthespecicheatoccursatthirdorderandonlyforelectronswithspin.ThisisveriedbyconsideringtheRGowofthemarginallyirrelevantoperatorintheSine-Gordontheory(whichappearsinthebosonizationschemeforfermionswithspin).Forspinlesselectronsweshowthatthenonanalyticitiesinparticle-particle

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andparticle-holechannelscompletelycanceloutandtheresultingspecicheatislinearinT(thebosonizedtheoryisgaussian).Thesingularityintheparticle-holechannelresultsinanonanalyticbehaviorforthespin-susceptibilitys/lnmax[jQj;jHj;T],presentatthesecondorder. Thespinsusceptibilitybothin2Dand3Dgetsnonanalyticcontributionsfromboth1Dandnon-1Dprocesses.ThesecorrectionswillbedescribedindetailinChapter4ofthisthesiswherewealsostudythenonanalyticcorrectionsnearaferromagneticquantumcriticalpoint. 58 ]derivedaLandau-Ginzburg-Wilson(LGW)functionalforthistransitionbyconsideringasimplemodelofitinerantelectronsthatinteractonlyviatheexchangeinteraction.HertzanalyzedthisLGWfunctionalbymeansoftherenormalizationgroup(RG)methodsthatgeneralizetheWilson'streatmentofclassicalphasetransitions.Heconcludedthattheferromagneticorderinanitinerantsystemsetsinviaacontinuous(or2ndorder)quantumphasetransitionandtheresultingstateisspatiallyuniform.Furthermore,heshowedthatthecriticalbehaviorinthephysicaldimensionsd=3andd=2ismean-eld-like,

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sincethedynamicalcriticalexponentz=3,(whicharisesduetothecouplingbetweenstaticsanddynamicsinaquantumproblem),decreasestheuppercriticaldimensionfromd+c=4fortheclassicalcasetod+c=1inthequantumcase.Hertz'stheorywhichwaslaterextendedbyMillis[ 59 ]andMoriya[ 60 ],(itiscommonlyreferredastheHertzMillisMoriya(HMM)theory),isbelievedtoexplainthequantumcriticalbehaviorinanumberofmaterials[ 61 ];however,thereareothersystemswhichdonotagreewiththeHMMpredictionsandshowarstordertransition,(e.g.,UGe2),totheorderedstate.Thiscontradictionmotivatedthetheoriststore-examinetheassumptionsmadeintheHMMtheory. TheHMMscenarioofaferromagneticquantumphasetransitionisbasedontheassumptionthatfermionscanbeintegratedoutsothattheeectiveactioninvolvesonlyuctuationsoftheorderparameter.Thisassumptionhasrecentlybeenquestioned,asmicroscopiccalculationsrevealnon-analyticdependencesofthespinsusceptibilityonthemomentum(q),magneticeld(H),andforD6=3,temperature(T)[ 42 44 ]bothawayandnearthequantumcriticalpoint(seethediscussioninsection1.2).Forexample,in2D s(H;Q;T)=const.+max(jHj;jQj;T); andin3D s(H;Q)=const.+(q2;H2)ln[max(jHj;jQj)]; whereH,qandTaremeasuredinappropriateunits.ThedependenceonTisnonanalyticinthesensethattheSommerfeldexpansionfortheFermigascanonlygenerateevenpowersofT.Ofparticularimportanceisthesignofthenonanalyticdependence:s(H;Q)increasesbothasafunctionofHandq(at2ndorderinperturbationtheory)forsmallH,q.Ass(H;Q)mustdenitelydecreaseforHandqexceedingtheatomicscale,thenaturalconclusionisthenithasamaximum

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atniteHandq.Thismeansthatthesystemshowsatendencyeithertoarstordertransitiontoauniformferromagneticstate(themetamagnetictransitionasafunctionoftheeld),ororderingatniteq,(toaspiralstate).Thechoiceoftheparticularscenarioisdeterminedbyaninterplayofthemicroscopicparameters.InChapter4ofthisthesis,wewillobtainthenonanalyticcorrectionstos(H)insecondandthirdorderinperturbationtheoryandshowthatthesecorrectionsoscillatebetweenpositiveat2ndorder,(whichpointstowardsametamagnetictransition),andnegativeat3rdorder(whichpointstowardsacontinuoussecondorderphasetransition)values.Thusitisimpossibletopredictthenatureofthephasetransitionbyinvestigatingthenonanalytictermsatthelowestorderinperturbationtheory.Furthermore,inrealsystemsinteractionsarenotweakandonecannotterminatetheperturbationtheorytoafewloworders.Tocircumventthisinherentproblemwithperturbativecalculationsandtomakepredictionsforrealisticsystems(e.g.,He3),weobtainthenonanalyticelddependenceforagenericFermiliquidbyexpressingourresultintermsofthelowestharmonicsoftheLandauinteractionparameters.Wealsodescribethenonanalyticelddependencenearthequantumcriticalpointusingtheself-consistentspin-fermionmodel,andshowthatthesignofthecorrectionsismetamagnetic.Here,intheintroduction,webrieyreviewHertz'stheoryofthesecondorderphasetransition. ThepartitionfunctionisobtainedbyperformingaHubbard-Stratonovichtransformationtodecouplethefour-fermioninteractioninthechargeandspinchannel.Thechargechannelisassumedtobenon-criticalandisthusdiscarded,

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whereasthepartitionfunctionforthespinchanneltakesthefollowingform; where Theeldlistheconjugateeldtothenl"nl#,whichcanbeconsideredasthemagneticeldactingonthefermions.Performingthefunctionalintegrationoverthefermionoperators(C;Cy)hearrivedatthepartitionfunctionZ=ZDeSeff(); TheMean-eld-theorywouldcorrespondtothesaddlepointapproximationtothefunctionalintegrationwithrespectto.Todeduceaneective(LGW)functional,oneexpandstheinteractionterm(Trlnterm)inl.Thematrix(M)intheTrlnterminEq. 1{66 intheFourierspacebecomes (M)(k;i!n;;k0;i!m;0)=0[(i!n+k)!n;!m~k;~k0+U

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ThersttermontherighthandsideoftheaboveequationistheinverseGreen'sfunctionforfreefermions(G10),thesecondtermisthe\interaction"(V).ThenTrln[M]=Trln[G10(1G0V)]=Trln[G10]+Trln[1G0V]=Trln[G10]1Xn=11 2X~q;i!lv2(~q;i!l)(~q;i!l)(~q;i!l)+1 4VX~qi;i!iv4(~qi;i!i)(~q1;i!1)(~q2;i!2)(~q3;i!3)(~q4;i!4)(4Xi=1~qi)(4Xi=1!i): ThecoecientsvminEq. 1{68 aretheirreduciblebarem-pointverticesinthediagrammaticperturbationtheorylanguage.Thequadraticcoecientisv2(~q;i!l)=1U0(~q;i!l),where0(~q;i!l)isthefreeelectronsusceptibilitygivenbytheLindhardfunction(Polarizationbubble),whichatsmallqandsmall!=qvFbehavesas 3q Hertzassumedtheallthehigherordercoecientsvmstartingwithv4canbeapproximatedasconstantsastheyvaryonthescaleofq2kFand!EF.InappropriateunitsHertz'sformoftheeectiveLGWfunctionalis 2X~q;i!m(r0+q2+j!mj (1{70)

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wherer0=1UF2(isthecorrelationlengthwhichdivergesatthephasetransition),isthedistancefromthecriticalpointandu0=U400F=12isaconstant.Thus,Hertz'seectiveactionisalmostofthesameformastheclassicalLGWfunctional(forthe4theory),exceptforthepresenceofthefrequencydependentterminv2whichcontainstheessentialinformationaboutthedynamics.Theactionthereforedescribesasetofinteracting,weaklyLandau-damped(duetothej!mj=qvFterm)excitations:paramagnons. HertzthenappliedWilson'smomentumshellrenormalizationgrouptransformationtotheabovequantumfunctional.Here,qand!havetobere-scaleddierently.Thisisduetothefactthatintheparamagnonpropagator(v21),qandj!mjappearinanon-symmetricway.Therefore,thesystemisanisotropicinthetimeandspacedirections.Asaresultitbecomesnecessarytointroduceanewparameter,thedynamicalcriticalexponentzforscaling Forthequantumferromagnetictransitionwhichwestudyhere,z=3.IntheRGprocedureconsistsofthefollowingsteps(a)highenergystates(withqand!)inthe"outershell"(>q>=b;>!>=b;b>1;isacut-o)areintegratedout;(b)variablesqand!,arere-scaledasq0=qeland!0=!ezl,withlbeinginnitesimal.(c)eldsarealsore-scaledsothatintermsoftheneweldsandre-scaledqand!,theq2andj!j=qtermsinthequadraticpartoftheactionlookslikethoseintheoriginalfunctional.Performingallthesesteps,HertzobtainedthefollowingRGequations dl=2r+12uf2; dl=u18u2f4;

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where=4(d+z)andtheexpressionsforf2andf4canbefoundinRef.[ 58 ].ThesecondRGequationshowsthattheGaussianxedpoint,withu=0,isstableifisnegative,thatis,ifd>4z.Forz=3,weshouldthereforeexpectastableGaussianxedpointandLandauexponentsind=2;3. ThetwomainassumptionsthatHertzmadeinarrivingathisLGWfunctional(Eq. 1{68 and 1{70 )were:(1)thecoecientsvm;m4arenonsingularandcanbeapproximatedbyconstantsand(2)thestaticspinsusceptibilityhasregularq2momentumdependence.Forthe2Dferromagnetictransition,nonanalytictermsinvmwerefoundbyChubukovetal.,[ 62 ],however,theauthorsclaimedthatthesenonanalyticitiesdonotgiverisetoananomalousexponentinthespinsusceptibilityandthereforewerenotdangerous.Inchapter4ofthisdissertationweexaminethesecondassumption(2)morecarefully.ThereasoningbehindHertz'ssecondassumptionwasthebeliefthatinitinerantferromagnetstheqdependenceofthe2termcomessolelyfromfermionswithhighenergies,oftheorderofEForbandwidth,inwhichcasetheexpansioninpowersof(q=pF)2shouldgenerallyholdforqpF.ThisreasoningwasdisputedinRefs.[ 42 44 ].Theseauthorsconsideredastaticspinsusceptibilitys(q)inaweaklyinteractingFermiliquid,i.e.,farawayfromaquantumferromagnetictransition,andarguedthatforD3andarbitrarysmallinteraction,thesmallqexpansionofs(q)beginswithanonanalyticjqjd1term,withanextralogarithminD=3.Thisnonanalyticityoriginatesfromthe2pFsingularityintheparticle-holepolarizationbubble[ 42 { 44 ]andcomesfromlowenergyfermions(inthevicinityoftheFermisurface),withenergiesoftheorderofvFqEF.ThesenonanalytictermsarisewhenoneconsidersthereferenceactionS0astheonewhichcontainstheparticle-holespinsingletchannelinteraction(chargechannel)andtheCooperchannelinteraction,whichwereneglectedintheHertzmodel(Hertz'sreferenceactionwasjustthenoninteractingone).Furthermore,thepre-factorofthis

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termturnsouttobenegative,whichindicatesthebreakdownofthecontinuoustransitiontoferromagnetism.ThusaccordingtoRef.[ 42 63 ]themodiedeectiveactionnearthecriticalpointshouldbe 2X~q;i!m(r0jqjD1+q2+j!mj withanextralogarithminD=3.TheweakpointofthisargumentisthatwithintheRPA,oneassumesthatfermionicexcitationsremaincoherentatthequantum-criticalpoint(QCP).Meanwhile,itisknown[ 64 ]thatuponapproachingtheQCP,thefermioniceectivemassm?divergesaslninD=3and3Dinsmallerdimensions.Itcanbeshownthatm=m?appearsasaprefactorofthejqjD1term;whichwouldmeanthatthenonanalytictermvanishesattheQCP.ThisstilldoesnotimplythatEq. 1{70 isvalidatthetransitionbecause,asweshowinchapter4,thedivergenceinm?doesnotcompletelyeliminatethenonanalyticterm,itjustmakesitweakerthanawayfromtheQCP. Ourapproachwillbetousethelow-energyeectivespin-Fermionhamiltonian,whichisobtainedbyintegratingthefermionswithenergiesbetweenthefermionicbandwidthWandalowercut-o(withW),outofthepartitionfunction[ 64 65 ]: HereSqdescribethecollectivebosonicdegreesoffreedominthespinchannel,andgisresidualspin-fermioncoupling.InHertz'sapproach,allfermionswereintegratedout,whereasintheSpin-Fermionmodelonlythehigh-energyfermionsareintegratedoutwhilekeepingthelow-energyones.Thiswillturnouttobeimportantbecausethespinuctuationpropagatorisrenormalizedbythefermions,andthefermionselfenergyisrenormalizedbyinteractionwithbosons.Thismodelhastobesolvedself-consistentlyasittakesintoaccountthelow-energy(mass)

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renormalizationofthespinuctuationpropagator.Inchapter4ofthisdissertationweusethismodeltoobtainthemagneticelddependenceofthespinsusceptibilitynearthequantumcriticalpoint,andanalyzethestabilityofthesecondorderquantumphasetransition.

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One-dimensionalsystemsexhibituniquephysicalpropertieswhichreecttheinuenceofstrongcorrelations.Theeectivedimensionalityofchargecarriersinabulkmetalmaybereducedfrom3Dto1Dbyapplyingastrongmagneticeld.Ithasrecentlybeenshownthatthisreductionleadstoformationofastronglycorrelatedstate,whichbelongstotheuniversalityclassofaLuttingerliquid[ 5 ].Thetunnelingdensityofstatesexhibitsacharacteristicscalingbehaviorforthecaseoflong-rangerepulsiveinteraction[ 5 6 ].Thiseectismostpronouncedintheultra-quantumlimit(UQL),whenonlythelowestLandaulevelremainsoccupied.Here,inthischapterweinvestigatetheeectofdiluteimpuritiesonthetransportpropertiesofthesystem.Forgoodmetals,thequantizingeldistoohigh(oftheorderof104Tesla),butsemi-metalsanddopedsemiconductorshavealowcarrierdensityandquantizingeldsoftheorderof110Teslaandallowforaexperimentaltestofthetheoreticalpredictionsmadehere. Insection2.1wediscusslocalizationeectsfornon-interactingelectronsintheUQL.Wendthatthelocalizationbehaviorisintermediatebetween1D(D:stronglocalization)and3D(D:weaklocalization).Weshowthattheparticle-particlecorrelator(Cooperon)ismassiveinthestrongmagneticeldlimit.It's\mass"(inunitsofthescatteringrate)isoftheorderoftheimpurityscatteringrate.Therefore,localizationinthestrong-eldlimitproceedsasifastrongphase-breakingprocessisoperatingasfrequentlyasimpurityscattering.EvenatT=0,thisphase-breakingexistsasitisprovidedbythemagneticeldandasaresultcompletelocalizationneveroccursin3DUQL.Ontheother 43

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hand,theparticle-holecorrelator(thediuson)remainsmassless,whichmeansthatnormalquasi-classicaldiusiontakesplace.Ourndingsareinagreementwithpreviousworkonthissubject[ 66 67 ],wherethelocalizationproblemwasanalyzedforthecaseoflongrangeddisorder,whereasinourstudywehaveanalyzedthecaseofshortrangeddisorder.OurresultforconductivityintheUQLiscoop+Drude=Drude=2. Insection2.2wecalculatethecorrectionstotheconductivityduetoelectron-electroninteractionsusingnite-temperaturediagrammatictechniquewheredisorderistreatedintheballisticlimit.Duetothisreducedeectivedimensionality,torstorderininteraction,theleadingcorrectionsarelogarithmicintemperature.Anotherwayofobtainingtheconductivityistocalculatetheinteractioncorrectiontothescatteringcross-sectionthroughanimpurity(inaHartree-Fockapproximation)anduseaDruderelationbetweenthecross-sectionandtheconductivity.Weshowinsection2.3that,torstorderintheinteraction,thetwoapproachesareequivalent.Thisisimportantsince,whileahigherordercalculationusingthediagrammatictechniquewouldbeextremelylengthy,theinteractioncorrectiontothecross-sectionisobtainedtoallordersviaanexactmappingontoa1Dproblemoftunnelingconductanceofinteractingelectronsthroughabarrier[ 21 ].WendthattheDrudeconductivitiesparallel(=+1)andperpendicular(=1)tothemagneticeldexhibitthescalinglaws/T2,whereisafunctionofthemagneticeld.Thephysicalreasonforsuchabehavioroftheconductivityisanearly1DformoftheFriedeloscillationaroundanimpurityinthestrongmagneticeld. ThegroundstateofrepulsivelyinteractingelectronsintheUQLisknowntobeunstabletotheformationofacharge-densitywave(CDW)[ 1 { 3 ].Thishasbeenconrmed,forexample,byexperimentsongraphiteinhighmagneticelds[ 4 ].BoththeHartree-Fockandthediagrammaticcalculationspresentedherearedone

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withouttakingintoaccountrenormalizationcorrectionsfortheinteractionverticesthemselves.ThisisjustiedatenergiesmuchlargerthantheCDWgapbutbreaksdownatlowenoughenergies.Inorderforourresultstohold,thereshouldexistanintermediateenergyintervalinwhichtherenormalizationoftheinteractionverticesduetoCDWuctuationsisnotyetimportantbutthepower-lawrenormalizationofthescatteringcross-sectionisalreadysignicant.Thatsuchanintervalexistsforthecaseoflong-rangeelectron-electroninteractionwasshownbysolvingthefullRGequationsfortheverticesandforthecross-section.Wehavenotincludedthisdiscussionhereforbrevity.Wediscusspossibleexperimentalvericationofourresultsinsection2.4andconcludeinSection2.5. 66 68 ],atleastforshort-rangeimpurities.Inthiscase,whilescatteringatanimpurity,anelectronmovestransversetotheeldbyadistanceoftheorderofthemagneticlengthlH=1=p

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followingsubsectionweillustratethedierentscenariowhicharisesfor3DelectronsintheUQL. 22 ]pz;m(;;z)=eipzzeim where(r?=;).ThereasonforthisseparabilityisthedegeneracyoftheLandaulevel;theenergydoesnotdependonthetransversequantumnumbers.The1Dpart,G1D,isinthemomentumspaceandthetransversepart,G?,hasbeenkeptinrealspace.ThedisorderaveragedGreen'sfunctionisobtainedbydoing

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perturbationtheoryintheimpuritypotentialU(forweakdisorderkF`1,sothatthesmallparameteris1=kF`)andemployingstandardcrosstechnique[ 18 ]fordisorderaveraging.Theperturbative(inU)solutionoftheSchrodingerequationfortheGreen'sfunctionisG(r;r0;i")=G0(r;r0;i")+Zd3r1G0(r;r1;i")U(r1)G0(r1;r0;i")+ZZd3r1d3r2G0(r;r1;i")U(r1)G0(r1;r2;i")U(r2)G0(r2;r0;i")+::: Theleadingcontributiontotheself-energycomesfromthesecondorderdiagram.TherstandthirdordercorrectionsarezeroashU(r)i=0.WeworkintheBornlimit,neglectingprocesseswhereanelectronscattersfrommorethantwo(same)impurities.ThesecondordercorrectionishG2(r;r0;i")i=ZZd3r1d3r2G0(r;r1;i")G0(r1;r2;i")G0(r2;r0;i")hU(r1)U(r2)i; (i"p)2#"1Xm=0eim(0) (i"p)3#"1Xm=0eim(0)

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wherethetransversepartineachoftheseexpressionsissimplyG?(r?;r0?).Atfourthorder,therearethreediagrams,therainbowdiagram(Fig.2-1,(b))andtheintersectingdiagram(Fig.2-1,(c))aresmallbyafactor1=kF`comparedtotheleadingone(hG4i)(Fig.2-1,(a))forshort-rangeweakdisorder.IntheBorn Figure2{1. Diagram(a)istheleadingcontributiontotheselfenergyatfourthorder approximation,thescatteringrateinamagneticeld,is1==2niu02H,whereH=1=(22vFlH2)isthe3Ddensityofstatesinthepresenceofamagneticeld,andtheself-energyis=isgn(")=2.ThefullDyson'sseries(Fig. 2{2 )canbesummedtogive: 2!G?(r?;r0?) (2{3) Therefore,theeectofimpurityscatteringentersonlyinthe1DpartoftheGreen'sfunction.UsingtheaboveformoftheGreen'sfunctionandtheKuboformulawenowevaluatetheDrudeconductivity.TheKuboformulaforthelongitudinald.c.conductivity(EkHkz)inthekineticequationapproximationiszz=lim!!01 2{3 ).Duetothe

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Figure2{2. Dyson'sseries Figure2{3. Drudeconductivity factorizabilityoftheGreen'sfunction,theconductivityisalsoseparableintoa1Dandatransversepart:zz=1d?.The1DpartisinthestandardformandgivesthefamousEinstein'srelationforthed.cconductivity1d=e21D=e2`=,whereD=v2Fisthediusioncoecientand1=1=(vF)isthedensityofstatesin1D.Usingtheorthonormality:Rd[Rm()]2=1andcompleteness:Pn[Rn()]2=1=(lH2)propertiesofthewavefunction,thetransversepartcanbeshowntobeequalto1withthedegeneracyfactorofthelowestlandaulevel.?=1Xm;n=0Zd00Zd0eim(0) 2l2H1:

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magneticeldandevaluatetheCooperoncorrectiontotheconductivity.TheDiusonremainsmasslesswhichmeansnormalquasiclassicaldiusionoccursintheparticle-holechannel.Implicationsoftheseresultsonelectronlocalizationwillbediscussed. Figure2{4. Thirdandsecondorderfandiagram. WecalculatethesecoecientsforthelowestorderCooperondiagrams(2ndand3rdorderfandiagramshowninFig. 2{4 )explicitlyandthenstatethegeneralargumentbywhichthesenumberscanbeobtainedforallhigherorderdiagrams.

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Forthesecondorderfandiagram, Theone-dimensionalpartofEq.( 2{4 )isgivenby whereR(q)istheonedimensionalrungintheparticle-particlechannelandX(q;!)isthepartcontainingthevertices: Weusethelinearspectrumapproximation,Rdp1 Forthevertex,linearizingandusingqpF,(wecannotsetq=0inthevertexaprioribecauseourcooperonwillacquireamass)followedbythepoleintegrationin,andthe"-integration,weobtainfor!>0,

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andnallyforthe1Dpartoftheconductivity Notethat1dD,soperturbationtheorybreaksdownintheUQL.Thetransversepartoftheconductivityis, wherer?=(;)andG?isdenedinEq.( 2{2 ).Afterperformingtheazimuthalintegrations,weobtain (2)3Zd00R2l(0)Zd11Rm(1)Rn(1)Rl(1)Rn0(1)21Xl;m=0l+mXn=0R2m() (2)3[Almnn0]2: wheren0=l+mn.Noticethatthesecondorderfandiagramhastworadialintegrations,likewisethirdorderfandiagramswillhavethreeradialintegrations,andsoon.UsingtheintegralrepresentationoftheGammafunction,Almnn0is 2l2H1 2m+l1 andthetransversepartoftheconductivitybecomes: 42lH231Xl;m=0l+mXn=0 lH2me2 2lH2

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Thesumovernisdoneusingthebinomialpropertyandthesumoverlisatabulatedsum[ 69 ], 42lH231Xm=02 2lH2 =1 22lH23: Thecoecientofthetransversepartofthesecondorderfandiagramfortheconductivityisc2=1=2.CombiningEq.( 2{10 )and( 2{16 ),theQCCfromthefandiagramatsecondorderiszz=e2DH=8. Similarly,thehigherorderdiagramscanbeevaluated.ThethirdorderfandiagramhasthesamevertexasthesecondorderonebuthasoneextrafactoroftherungR(q).Thisgivesfortheonedimensionalpart:1D=(3e2`=16)(2l2H)3.Thetransversepartnowhasthreeradialintegrations(3factorsofA's)isgivenas: (2)4[Amnqs0][Anss0n0][Asqmn0]: wheres0=m+qnands=m+qn0.TheradialintegrationscanbeperformedasbeforetoobtaintheA's.Performingthesums,wend (2)42lH231Xm;q=0R2m()(m+q)! 42lH24: Thusforthethirdorderfandiagram,c3=1=4,andtheQCCiszz=3e2DH=64.Thenth-orderfandiagramhasnradialintegrations,eachofwhichgivesafactorof1=2sothatonehasacoecient(1=2)n.Thesummationoverangularmomentumindicesgivesafactor2regardlessofthediagram'sorder.So,theoverallcoecientinthenth-orderfandiagramduetothetransverseintegrationiscn=1=2n1.

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Figure2{5. Cooperonsequencefor3DelectronsintheUQL.Unlikein1D,eachtermintheseriescomeswithadierentcoecientcn. WeconstructtheCooperonsequenceinUQLasshowninFig.( 2{5 ),withtheprefactorsindicatingcn,thenumbersobtainedaftertransverseintegrationateachorder.ThesenumbersareresponsibleforthemassoftheCooperon.TheDOSfactoratnthorderis1=2lH2n+1.Thedashedlineintheguredenotesg(g1),thecorrelatorin3D(1D),whereg=1=2H=niu02=2lH2=(21)=g12lH2.Ristheonedimensionalrungintheparticle-particlechannel(smalltotalmomentum)evaluatedinthediusivelimit. For3DelectronsintheUQL,thecooperonsequencegives: andusingEq.( 2{19 ),thisbecomes, Inthelimitq;!!0,Cbecomesaconstant.Therearenoinfrareddivergence,becausewehaveamassiveCooperon.Themassinunitsofthescatteringrateisapurenumber(1/2).Itcanbeinterpretedas=H,sothatHisoftheorderoftheimpurityscatteringtime.Thisindicatesthatlocalizationinastrongeldproceedsasifastrongphase-breakingprocessisoperatingsimultaneouslywithimpurityscattering.ThisisdephasingbytheeldanditpersistsevenatT!0.

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WenowcontrastthissituationwhicharisesintheUQLwiththatofanyotherdimensions(1D,2D,3D)withoutthemagneticeld.Intheabsenceoftheeldthecn'sareallone(foralldimensions)andthecooperonsequenceissingular(thereisadiusionlikepoleforrealfrequencies,forq;!0). Dq2+j!j: Thisgivestheweak-localizationcorrectiontotheconductivity(WLQCCdiscussedinchapter1)in2Dand3D[ 12 ].In1Dalthoughthecooperondiagramhasapole,allnon-cooperondiagramsarealsoofthesameorder,andoneneedstosumoverallthediagramstogetstronglocalization[ 15 ]. Intheultraquantumlimitthetransversenumbersfortheparticle-holediusionpropagator(thediuson)areallequaltounity(cn=1).Thereforediusonremainsmasslessinastrongeldandnormalquasi-classicaldiusionoccursintheparticle-holechannel.Wewillnowevaluatethetransversepart Figure2{6. Firstandsecondorderdiuson oftherstandsecondorderdiusoncorrectionofFig.( 2{6 ),assumingalongrangedimpuritypotentialsuchthat1d6=0.Wedonotattempttocalculatethelongitudinalpartoftheconductivity,(1d)asthiswillbemorecomplicatedduetothelongrangedisorderpotential.Wejustassumethatthelongitudinalpartisnite.Intheshortrangedimpuritycasethediusoncorrectiontoconductivityiszero(becausethe1d=0).ThetransversepartfortherstorderDiuson

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correctionis, whereG?isdenedinEq.( 2{2 ).Performingtheazimuthalintegrations, (2)21Xm;n=0[Rm()]2Zd11[Rm(1)]2[Rn(1)]2Zd00[Rn(0)]2: Usingorthonormalityandcompleteness,weget?=1=2lH221,sothatc1=1.Forthesecondorderdiusonweperformtheazimuthalintegrationsandobtain, (2)31Xl;k;n=0[Rl()]2Zd11[Rk(1)]2[Rl(1)]2Zd22[Rk(2)]2[Rn(2)]2Zd00[Rn(0)]2; andusingorthonormalityandcompleteness,weobtain?=1=2lH231,andc2=1.Anynth-orderdiagramcanbecalculatedinthesameway,givingcn=1.Thereforethelongitudinaldiusionisfreeandthediusonremainsmassless. Figure2{7. Interferencecorrectiontoconductivity Nextwecalculatethequantuminterferencecorrectiontotheconductivityintheultraquantumlimit(seeFig.( 2{7 )):

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wherethetransverseintegrationshavebeenperformed.ThevertexpartofthisdiagramhasalreadybeenevaluatedinEq.( 2{9 ).UsingEq.( 2{21 ),Zdq 2lH2=HD TheaboveresultindicatesthatperturbationtheoryfailsintheUQL(coop=HD=1=2)inthesamemannerasitdoesinaonedimensionalsystem.However,contrarytowhathappensin1D,thereisnostronglocalizationintheUQL.Thecrosseddiusondiagram(nextorderin1=kF`,seeFig. 2{8 ),isalsononsingularandmassive.Thetransversecoecientforthelowestordercrosseddiusondiagramisc=1=3p 2<1:

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Thelowerboundforisbasedonthefactthatcoop+HD=1 2HD.Thenon-cooperontypediagrams,atleastinthelowestorder,havetheoppositesignasthatofthecooperon.Theyarealsoofthesameorderasthecooperon,soitisnotclearwhattheywilladdupto.Itmayhappenthatallthenon-Cooperondiagramsmodifyourpredictionforandmaymakeanywherefrom0!1.ToobtainabetterestimateforoneneedstogeneralizeBerezinskii's[ 15 ]diagramtechnique(developedforthe1Dlocalizationproblem)to3DUQL.OurresultsalsoagreewiththoseobtainedbytheauthorsofRef.[ 66 67 ].Theseauthorsconsideredlongrangedisorder,`?lH,andobtainedk(`? Inthenextsectionweusethenitetemperaturediagrammatictechniquetocalculatethecorrectionstotheconductivityduetoelectron-electroninteractions(interactionQCC).Wewillshowthatthesecorrectionsarelogarithmicintemperatureandthustheyconrmthatthesystembehaviorisone-dimensional. Figure2{8. Crosseddiusondiagrams.Left,adouble-diusondiagram,whichalsoacquiresamass.Right,athird-ordernon-cooperondiagramwhich,uptoanumber,givesthesamecontributionasthethirdorderfandiagram. (2{29)

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forthesingle-electronwavefunction,where (2nn!p HerelH=1=p (2{31) wherethesumisoverallLandaulevels,n(pz)=(p2zp2F)=2m+n!c,and!c=eH=mistheelectroncyclotronfrequency.WewillneedonlyexcitationsneartheFermilevelforourcalculation,sointheUQL(EF0termsinthesuminEq.( 2{31 )arenegligibleduetothelargemassterm(oforder!c)inthedenominator.Neglectingtheseterms,thetotalGreen'sfunctioniswrittenastheproductofalongitudinalandaperpendicularpart withG?(px;yy0)=0(y+pxl2H)0(y0+pxl2H).Asshownintheprevioussectionthedisorder-averagedlongitudinalGreen'sfunctioncorrespondstoG1D(";pz)=1=(i"pz+isgn(")=2)where1==2Im.CalculatingtheconductivityusingthisGreen'sfunctiongivestheDrudeformula,withdensityofstatesH=1D=2l2H. The(dynamically)screenedCoulombpotentialintheultra-quantumlimitisgivenby[ 70 ] wherethescreeningwavevectorisrelatedtothedensityofstatesviatheusualrelation2=4e2HandR(!;qz)isthepolarizationbubbleof1Delectrons.Inwhatfollows,wewillneedonlysomelimitingformsofthepotential.For

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1=!EFand1=`qkF; independentofthetemperature(upto(T=EF)2-terms).Inthestaticlimit,whenthetransversemomentaaresmall(q2?l2H1),thepotentialreducestoanisotropicform whichdiersfromacorrespondingquantityinthezeromagneticeldonlyinthatscaleswithHasp 5 6 ])comeswithprocesseswithq?l1H:Therefore,theGaussianfactorinthedenominatorofEq.( 2{33 )canbereplacedbyunityforallcasesofinterest. Thepolarizationbubbleexhibitsa1DKohnanomalyforqznear2kF.SuchlargemomentumtransfersareimportantonlyinHartreediagrams,wherethepotentialistobetakenat!=0intheballisticlimit.NeartheKohnanomaly,thestaticpolarizationbubblecanbewrittenas 2kF(0;qz)=1 21DlnEF tologarithmicaccuracy. Finally,thepoleofthepotentialinEq.( 2{33 )correspondstoacollectivemode{magnetoplasmon.For!;qvFEFandq?lH1;thedispersionrelationofthemagnetoplasmonmodeisgivenby where!p0=p

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transverseones,jqzjq?;sothatonecanwrite Figure2{9. Firstorderinteractioncorrectionstotheconductivitywhereeectsofimpuritiesappearonlyinthedisorder-averagedGreen'sfunctions. Wenowproceedtocomputetherstorderinteractioncorrectiontotheconductivityintheballisticlimit(T1).ThisincludescontributionsfromdiagramsshowninFig. 2{9 ,whereeectsofimpuritiesappearonlyinthedisorder-averagedGreen'sfunctions.Italsoincludesdiagramswithoneinteractionlineandoneextraimpurityline.Thesecanbeseparatedfurtherintoexchange(Fig. 2{10 )andHartree(Fig. 2{11 )diagrams.Inthissectionweshowthatdiagrams 2{10 (a), 2{10 (b), 2{11 (a)and 2{11 (b)givealeading{jln(T=EF)j{correctiontotheconductivity,whereasallotherdiagramsgivesub-leadingcontributions. 2{9 (a), 2{10 (a), 2{10 (c),and 2{11 (a)involvecorrectionstotheself-energyduetoelectron-electroninteraction.Diagram 2{9 (a)describesinelasticscatteringofanelectrononacollectivemode(plasmon),whichwouldhaveexistedevenforasystemwithoutdisorder.Astheelectron-electroninteractioncannotleadtoaniteconductivityinthetranslationallyinvariantcase,thisdiagramiscanceledbythecounter-correctionofthevertextype[Fig. 2{9 (b)].DiagramsFig. 2{10 (a), 2{10 (c),and 2{11 (a)describecorrectiontotheself-energydueto

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Figure2{10. Exchangediagramsthatarerstorderintheinteractionandwithasingleextraimpurityline.TheGreen'sfunctionsaredisorder-averaged.Diagrams(a)and(b)givelnTcorrectiontotheconductivityandexchangediagrams(c),(d)and(e)givesub-leadingcorrectionstotheconductivity. interferencebetweenelectron-electronandelectron-impurityscattering.Ageneralformofthecorrectiontotheconductivityforalldiagramsoftheself-energytypecanbewrittenas m"Zdpz where1D("n;pz)isthecorrectiontothe(Matsubara)self-energyoftheeective1Dproblem,towhichtheoriginalproblemisreduceduponintegratingouttransversecoordinates.ThisispossibleduetothefactthattheGreen'sfunctionsarefactorizedintoa1Dandatransversepart,asshowninEq.( 2{32 ),andtheintegrationsovertransversevariablescanbecarriedoutandsimplygivethe

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Figure2{11. Hartreediagramsthatarerstorderintheinteractionandwithasingleextraimpurityline.TheGreen'sfunctionsaredisorder-averaged.BothdiagramsgivelnTcorrectiontotheconductivity. degeneracyfactor1=2l2H.Inthiseective1Dproblem,electronsinteractviaaneectivepotential whereaseachimpuritylinecarriesafactorniu20=2l2H=vF=2,whereniistheconcentrationofimpuritiesandu0istheimpuritypotential.Theoverallfactorof2inEq.( 2{39 )isthecombinatorialcoecientassociatedwitheachdiagramoftheself-energytype. Substituting( 2{33 )into( 2{40 )andusingtheconditionlH1;weobtain PerformingtheanalyticcontinuationinEq.( 2{39 ),weobtain whereGR1D=1=("p+i=2)andR1Distheinteractioncorrectiontotheretardedself-energyofthenon-interactingelectronswhichis0=i=2:(Forbrevity,wesuppressedtheargumentsofGR1DandR1D;whichare";p):

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2{10 (a) 2{10 (a)whichcorrespondstoacorrectionintheself-energyasshowninFigure 2{12 .As!andqzareexpectedtobelargecomparedto1=and1=`,respectively,itsucestoreplacetheGreen'sfunctionsintheself-energybythoseintheabsenceofdisorder.Intherestofthediagramfortheconductivity,theGreen'sfunctionsaretakeninthepresenceofdisorder.In1D,itisconvenienttoseparatetheelectronsintoleft-andright-moversdescribedbytheGreen'sfunctionsG("n;p)=1=(i"nvFp+isgn"n=2),wherep=pzpF:Accordingly,therearealsotwoself-energies;forleft-andright-movingelectrons.Thecontributionfor+isshowninFig. 2{12 .TheGreen'sfunctionsofright/leftelectronsarelabeledbyinthediagram.Processesinwhichanelectronisforward-scatteredtwiceatthesameimpuritydonotcontributetotheconductivityandarethereforenotconsideredinthiscalculation.ThediagramwithbackscatteringbothatanimpurityandotherelectronsinvolvesstatesfarawayfromtheFermisurfaceandisthusneglected.TheonlyimportantdiagramistheoneshowninFig. 2{12 wheretheelectronisbackscatteredbyanimpurityandforwardscatteredbyotherelectrons. Figure2{12. Theself-energycorrectioncontainedindiagram 2{10 (a),denotedinthetextas( 212 )+. Atrst,weneglectthefrequencydependenceofthepotential.Themomentumcarriedbytheinteractionlineissmall,qz'"=vF'T=vF,andatlowtemperatures,suchthatT=vF,onecanneglectqzcomparedtoinV1DandreplaceV1Dbya

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constant,V1D!2g0vF,where isadimensionlesscouplingconstant.Theperturbationtheoryisvalidforg01: 212 )+("n;p)=2ig0vF [i!+vFqz][i("n!m)vF(pqz)]; ( 212 )+("n;p)=2g0 Nowweseethattologarithmicaccuracyitissafetocutthesumat!MvFqmaxEFandomitthefactorinthecurlybracketsinEq.( 2{44 ): ( 212 )+("n;p)=2g0 =g0 2+"nivFp ReR( 212 )+(";p)=g0 (2{46) ImR( 212 )+(";p)=g0 2i"+vFp

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ToobtaintherealpartinaformgiveninEq.( 2{46 )weusedanidentity1 2ix2==coshx;whereasthelastlineinEq.( 2{47 )isvalidtologarithmicaccuracy.Theself-energyofleft-movingelectronsisobtainedfromEqs.( 2{46 ),( 2{47 )bymakingareplacement"+vFp!"vFp: 2{46 )and( 2{47 ).Eq.( 2{46 )showsthatthecorrectiontotheeectivemassisT-dependent:forj"+vFpjT;m/T1:Inprinciple,suchacorrectionmightresultinanadditionalT-dependenceoftheconductivity.However,thisT-dependenceoccursonlyinthenext-to-leadingorderintheparameter(T)11oftheballisticapproximation.Theleadingcorrectiontotheconductivitycomesfromtheimaginarypartoftheself-energy,Eq.( 2{47 ).Thiscorrectionexhibitsacharacteristic1Dlogarithmicsingularity,whichsignalsthebreakdownoftheFermiliquid(tothelowestorderintheinteraction). Themaincontributiontotheconductivitycomesfromthecorrectiontotheimaginarypartoftheself-energy[Eq.( 2{47 )].SubstitutingEq.( 2{47 )intoEq.( 2{42 )andaddingasimilarcontributionfromtheleft-movingelectrons,weobtain 210 Wenotethattheaboveresultwasobtainedusingthestaticformoftheinteractionpotential.WenowreturntothefulldynamicpotentialandshowthatthefrequencydependenceofthepotentialdoesnotchangetheresultsgivenbyEqs.( 2{46 )and( 2{47 ),tologarithmicaccuracy.Foradynamicpotentialitismoreconvenienttoperformtheintegrationoverq?attheveryendsothatthepotentialenteringthecalculationisofthe3Dform

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whereweusedthatqzq?andintroduced2(q?)=q2?=(q2?+2):Theintegraloverp0givesthesameresultasforthestaticpotential.Integratingoverqz;weobtainfortheeective1Dself-energyinsteadof( 2{45 ),~( 212 )+("n;p)=e2 2+"nivFp 212 2{46 )and( 2{47 ).ComingbacktoEqs.( 2{49 )and( 2{50 ),wecaninterpretthisresultinthefollowingway.Thedierencebetweenthedynamicpotentialandthestaticoneisinthepresenceofthedynamicpolarizationbubblemultiplying2inthedenominatorofEq.( 2{49 ).Ifthepotentialistakeninthestaticform,typicalfrequenciesarerelatedtotypicalmomentaas!vFqz;whichmeansthatthisfactorisoforderofunityandmustbereplacedby:Butbecausethenalresultfordependsononlyviaa(large)logarithmicterm,log(jln`Bj);sucharenormalizationofisbeyondthelogarithmicaccuracyofthecalculation. 2{11 (a). Theself-energycorrectioncontainedindiagram 2{11 (a),denotedinthetextas( 213 )+

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DiagramFig. 2{11 (a)isaHartreecounter-partoftheexchangediagramofFig. 2{10 (a).Separatingthecontributionsofleft-andrightmovers,thediagramcorrespondingtobackscatteringatthestaticimpuritypotentialisshowninFig. 2{13 .Again,diagramscorrespondingtoforwardscatteringattheimpuritypotentialdonotcontributetotheconductivityanddonotneedtobeconsideredhere.ThediagramofFig. 2{13 alsoincludesbackscatteringataFriedeloscillation.Althoughthisdiagramcontainsaparticle-holebubble,itismoreconvenienttolabelthemomentaasshowninFig. 2{13 ,integrateoverp0rst,andthenoverk.Forbackscattering,the1DpotentialofEq.( 2{41 )becomes wherethelastlineisvalidtologarithmicaccuracy.Asarstapproximation,weneglecttheq-dependenceoftheinteractionpotential,replacingV2kF1DinEq.( 2{51 )byaconstantV2kF1D!2g2kFvF.Thisresultsin R( 213 )+("n;p)=2g2kF =g2kF 2+"nivFp which,uptoasignandoverallfactorofthecouplingconstant,isthesameastheexchangecontributionR( 213 )+inEq.( 2{45 ). WhenthedependenceofV2kF1Donq0isrestored,theresultinEq.( 2{53 )changesonlyinthatthecouplingconstantacquiresaweakTdependence

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CalculatingthecontributionofEq.( 2{53 )withEq.( 2{54 )totheconductivity,wendthecorrectiontotheconductivityfromdiagramFig. 2{11 (a)tobe: 211 2lnEF NoticethatinthelimitofverylowTand/orverystrongelds,thescreeningwavevectordropsoutoftheresultandthenetT-dependenceoftheconductivitybecomeslnxln(lnx);wherexEF=T: 2{10 (b)andFig. 2{11 (b).Thesearethevertexcorrectionscounterpartsoftheself-energydiagramsinFig. 2{10 (a)andFig. 2{11 (b),correspondingly. 2{10 (b) 2{10 (b)canbeshowntogivethesamecontributionas 2{10 (a).InthisSectionweshowthisbyreducingdiagram 2{10 (b)to 2{10 (a)withoutdoingexplicitintegrationsoverqzandMatsubarasummations. Decomposingdiagram 2{10 (b)intocontributionsfromleftandrightfermions,weobtain 210 2l2Hlim!0e2v2F whereMarethevectorverticesM=Zdp (2{57)

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wehave (2{59) respectively.Forallothercases,theresultscanbeshowneithertovanishbecauseofthelocationsofthepolesortocanceleachother.Intheballisticlimit,theproductM+Mcanbesimpliedinbothcasesto (m+1=)2[!2l+(vFqz)2]:(2{61) Thesubsequentintegrationofthisexpressiongivesaj!lj1-singularityanditisthissingularitywhichgivesthelnT-dependenceofthecorrectiontotheconductivity. Nowwegobacktodiagram 2{10 (a).InSec. 2.2.1.1 ,wefoundthecontributionofthisdiagrambyevaluatingtheself-energyrstandthensubstitutingtheresultintotheKuboformula.Toprovetheequivalenceofdiagrams 2{10 (a)and(b)itisconvenienttoconsiderthefulldiagram 2{10 (a)withoutsinglingouttheself-energypart.Summingupthecontributionofleftandrightfermions,weobtain 210 2l2Hlim!0e2v2F whereP=Zdp 2{62 )isobtainedonlyforthecasegiveninEq.( 2{57 ),when

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withQ=i v2F1 (i!lvFqz+i=)2(i(!lm)vFqz)(i!+vFqz+i=)(qz!qz): 2{63 ),weseethatitcoincideswithEq.( 2{61 ).TheMatsubarasummationgoesoveratwicesmallerintervaloffrequenciescomparedtothatinEq.( 2{56 ).WeseethatEqs.( 2{62 )and( 2{56 )givethesameresultandthus 210 210 2{11 (b) Diagram 2{10 (b)vsdiagram 2{11 (b). ThediagraminFig. 2{11 (b)isavertexcorrectioncounterpartoftheHartreeself-energydiagramFig. 2{11 (a),anditgivesthesamecontributionasFig. 2{11 (a).Toseethis,wecomparethediagramsinFigs. 2{10 (b)and 2{11 (b)

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labelingthemasshowninFig. 2{14 .Foraq-and!-independentinteraction,diagramFig. 2{14 (b)isofthesamemagnitudebutoppositesignasdiagramFig. 2{14 (a).Foraq-dependentinteraction,theT-dependentpartsofthesediagramsdieralsointheoverallfactorofthecouplingconstant:diagram 2{14 (a)containsg0whereasdiagram 2{14 (b)containsg2kF:Electron-electronbackscatteringindiagram(a)andelectron-electronforwardscatteringindiagram(b)giveeithersub-leadingorT-independentcontributions.Thus 211 210 210 211 2{10 (c)givesaself-energytypecontributiontotheconductivitysoweuseEq.( 2{42 ).Iftheinteractionpotentialistakentobestatic,thecontributionfromthisdiagramiszero.Usingthedynamicalpotential,theleadingcontributionfromthisdiagramisalnT-correctiontotheconductivity 210 2e2 ThiscontributionissmallerthanthatfromdiagramsFig. 2{10 (a)[Eq.( 2{48 )]andFig. 2{11 (a)[Eq.( 2{55 )](anddiagramsFig. 2{10 (b)and 2{11 (b))byaT-independentlog-factor. Diagrams 2{10 (d)and 2{10 (e)givemutuallycancelingcontributionsoftheform: 210 (2{67) 210 Eachofthesecontributionsissmallsinceweareintheballisticlimit(T1).

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Allthecalculationsshownherearedoneconsideringthedynamicinteractionpotentialatsmallfrequencies.Athighfrequencies,i.e.,atfrequenciesclosetothemagnetoplasmonfrequency,thecontributionsfromalltheconductivitydiagramscancelout.ThatthishastobethecasewaspointedoutrecentlyinRef.[ 19 ].Thisisaveryusefulresultbecauseeachindividualdiagram,takenseparately,maygivesingularcorrections.Inourcasewehavealsoexplicitlycheckedthatthiscancellationindeedoccurs.Contributionsfromdiagrams( 2{9 a),( 2{10 a)and( 2{10 c)canceleachother.Contributionfrom( 2{9 b)cancelsthatof( 2{10 b),andnally( 2{10 d)and( 2{10 e)canceleachother. 2{48 ),( 2{55 ),( 2{64 ),and( 2{65 ),wendtheleadingcorrectiontotheconductivity =( 210 211 210 211 2lnEF Eq.( 269 )isthemainresultofthisSection. 2{10 (a,b)andFigs. 2{11 (a,b),determinetheleadingcorrectiontotheconductivitysuggeststhattheremustbesomesimplereasonforthesediagramstobethedominantones.Indeed,onlythesediagramsariseifoneconsidersscatteringofelectronsby\eective"impuritiesthatconsistofacombinationofbareimpuritiesandtheCoulombeldsofelectronssurroundingthebareimpurities.Forweakdelta-functionbareimpurities,theeectiveimpuritypotentialcorrespondsto\dressing"theimpuritywiththemeaneldofHartreeandexchangeinteractions(seeFig. 3{1 ).~V0(";p;p0)=V0+VH(pp0)+Vx(";p;p0):

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Therstterminthisequationisthestrengthofabareimpurity,thesecondoneistheCoulombpotentialofelectronswhosedensityismodulatedduetothepresenceofthebareimpurity,andthethirdoneisanexchangepotentialforelectronsinteractingandscatteringthroughaweakimpurity. Figure2{15. Eectiveimpuritypotential Duetotheexchangecontribution,theeectiveimpuritypotentialisnon-local,anditmaydependontheenergy,iftheinteractionisdynamical.Performingtheimpurityaveraging,weobtainthecorrelationfunctionoftheeectiveimpuritypotential whereg=e2=vFistheinteractionstrength.Diagrammatically,Ccorrespondstoadashedlineofthecross-technique[ 18 ].Therstterm(bareimpurities)istakenintoaccountintheleadingorderin1=EF1bysumminginniteseriesforthesingle-particleGreen'sfunctionandthenusingtheKuboformulafortheconductivity.Becausethebareimpuritiesareshort-range,thereisonlyonediagramfortheconductivity{theusual\handle"diagram;thevertexcorrectiontothisdiagramvanishes.CorrectionstotheconductivityresultfromtheHartreeandexchangetermsinEq.( 2{70 ).Torstordertherearetwodiagrams,showninFig. 2{16 .Althoughthebareimpurityispoint-like,theHartreeandexchangepotentialsitgenerateshaveslowlydecayingtailsandalsooscillateinspace.Thusthevertexcorrection,Fig. 2{16 (b),isnotzero.Theself-energydiagram,Fig.

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2{16 (a),correspondstotwodiagrams:Fig. 2{10 (a)andFig. 2{11 (a).DiagramFig. 2{16 (b)correspondstothediagramsinFig. 2{10 (b)andFig. 2{11 (b).Foranarbitraryimpuritypotential,itcanbeshownthatcontributionsof 2{16 (a)and 2{16 (b)comingfromforwardscatteringcancelseachother.Forbackscattering,thecontributionfrom 2{16 (a)and 2{16 (b)arethesame. Figure2{16. Thehandlediagramcorrespondstodiagrams 2{10 (a)and 2{11 (a)andthecrossingdiagramcorrespondsto 2{10 (b)and 2{11 (b). 2.2.5 wedemonstratedthat,torstorderintheinteraction,theonlydiagramswhichareimportantforcorrespondtoscatteringataneectiveimpuritypotential.Thissuggeststhattheresultforcanbeobtainedbycalculatingtheinteractioncorrectiontotheimpurityscatteringcross-sectionandthensubstitutingthecorrectedcross-sectionintotheDrudeformula.InthissectionweshowthattorstorderthisproceduregivesaresultidenticaltothatofthediagrammaticapproachofSec. 2.2 .Unlikethediagrammaticseriesintheinteractionfortheconductivity,theperturbationtheoryforthescatteringcross-sectioncanbesummeduptoallordersviaarenormalizationgroupprocedure.ThiswillleadtoaLuttinger-liquid-likepower-lawscalingoftheconductivity,discussedattheendofthissection.

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withmz=0;1;2:::. ElectronsarerestrictedtothelowestLandaubandandthereforethereareonlytwotypesofscatteringevents:forwardandbackward.Onlybackscatteringeventscontributetothescatteringcross-section,whichcanbewrittenasA_N J,where_NisthenumberofelectronsbackscatteredperunittimeandJisthetotaluxofincomingelectrons.UsingaLandauer-typescheme,thescatteringcross-sectionineachchannelofconservedmzcanberelatedtoareectioncoecientinthischannelviaAmz=2l2Hjrmzj2:Thetotalcross-sectionisobtainedfromthesumofthecross-sectionsineachchannel[ 71 ]:

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Thecoecientsrmzarethereectionamplitudesof1Dscatteringproblems,givenbyasetof1DSchrodingerequations1 2m@2 witheective1DimpuritypotentialsVmz(z)=hmzjVimp(r)jmziobtainedbyprojectingtheimpuritypotentialontheangularmomentumchannelmz.Thekineticenergyoftheelectronisdenotedby"=p2z=2m.Thecross-sectionAisrelatedtothebackscatteringtimeviatheusualrelation,1=H=nivFA,whereniisthedensityofimpurityscatteringcenters.WhentheelectriceldisalongthemagneticeldandforT=0,thecorrespondingcomponentoftheconductivityisrelatedtoHvia AnimpurityofradiusalHcanbemodeledbyadelta-function:Vimp(r)=V0(r).Foradelta-functionpotential,onlythemz=0componentofVmz(z)isnon-zero,Vmz(z)=(V0=2l2H)mz;0(z).Inthiscase,thescatteringcross-sectionfornon-interactingelectronsissimply wherevz=pz=m:Consequently,atT=0theconductivityisgivenby nie2 2l2H"1+2l2HvF IntheBornlimit(whenV02l2HvF)werecovertheresultfortheconductivityasfoundbyusingtheKuboformulaforweak,delta-correlateddisorder[Eq.( 2{73 )].Intheopposite(unitary)limitA=2l2Hand

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wherefarawayfromtheimpuritysite,theasymptoticformofthez-componentoftheunperturbedwave-functionis: (2{77) Bycalculatingtheelectron-electroninteractioncorrectiontothewavefunction,oneobtainsthecorrectiontoamplitudest0andr0;andthereforetothescatteringcross-sectionviaEq.( 2{72 ).Sincenowtheproblemhasbeenmappedontoa1Dscatteringproblem[ 21 24 ],onecananticipatethatthisinteractioncorrectionhasaninfraredlogarithmicsingularity,asitdoesinthepure1Dcase. The1DnatureofthesystemintheUQLisalsoclearlymanifestedbythebehavioroftheFriedeloscillationsaroundtheimpurity.Theproleoftheelectrondensityaroundtheimpuritysiteisgivenbyn(r)=Rdr0(r;r0)Vimp(r0),where(r;r0)isthepolarizationoperator.Foraweakdeltaimpuritypotential,weobtain 2l2Hsin(2pFz) whichshowsonlyaslow,1=zdecay(seeFig. 2{17 ),characteristicofone-dimensionalsystems(incontrasttothe1=r3decayin3Dsystems).Correspondingly,theHartreeVH(r)andexchangeVex(r;r0)potentials,thatanincomingelectronsfeels

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whenbeingscatteredfromanimpurity,alsoexhibit2pF-oscillationsanddecayas1=zawayfromtheimpurityandalongthemagneticelddirection.Inthetransverseplane,thedensity,andthusthepotentials,haveGaussianenvelopeswhichfalloonthescaleofthemagneticlength(seeFig. 2{17 ). Figure2{17. ProleoftheFriedeloscillationsaroundapointimpurityina3DmetalintheUQL.Theoscillationsdecayas1=zalongthemagneticelddirectionandhaveaGaussianenvelopeinthetransversedirection. TheinteractioncorrectiontothewavefunctionduetotheHartreeandexchangepotentialsis Asdiscussedintheprevioussection,fortheUQLtheGreen'sfunctionistheproductofalongitudinal(1D)andatransversepart,G(r;r0;E)=G1D(z;z0;pz)G?(r?;r0?),wheretheasymptoticformofthelongitudinalpartasz!1is ipz8><>:t0eipz(zz0);z0<0eipz(zz0)+r0eipz(z+z0);z0>0(2{80)

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and,inthesymmetricgauge,thetransversepartis 2l2Hexp(jj2+j0j220) 4: Forz>0,Eq.( 2{79 )directlygivesthecorrectionforthetransmissionamplitudet.Werstconsidertheexchangepotential, whichcanbefactoredas (2{83) where Fromtheformoff(z;z0)onecanseethattheexchangepotentialalsohastermswith1=(zz0)and1=(z+z0)decay.Forexample,forz;z0>0, 2i(z+z0)r0(eipF(zz0)1) 2i(z+z0) (2{85) The1=(z+z0)decayleadstoalog-divergentcorrectiontojtj.DecomposingthescreenedCoulombpotentialV(rr0)intoFouriercomponents,allthedependenceofEq.( 2{79 )onthetransversecoordinatesr?canbecollectedintothefactor PerformingtheintegralswhichappearinEq.( 2{86 )fortheexchangecontribution,wendthatthepartcontainingperpendicularcoordinatessimplyenterstheinteractioncorrectionasaformfactor:

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Therefore,thetransversepartofthefreewavefunctionmz=0(r?)simplyremainsunchangedintherhsofEq.( 2{79 ).Theremainingexponentialtermappearsinthedenitionoftheeective1Dpotential,asinEq.( 2{40 ) ThesameresultisobtainedforthetransversepartoftheHartreecontributioninEq.( 2{79 ).Oncethetransversepartissolvedandtheeective1Dpotentialisdened,therestofthecalculationisexactlyequivalenttothecalculationofYueetal.[ 21 ]fortunnelingofweakly-interacting1Delectronsthroughasinglebarrier.Theinteractioncorrectiontothetransmissionamplitudetisdirectlyobtainedfromthecorrectiontothewavefunction,Eq.( 2{79 ).Justasin1D,alogarithmicallydivergentcorrectionfortisobtainedfromthelongitudinalpartofthisequation,afterintegratingoverzandz0. Itisstraightforwardtoseewhythereisalog-divergentterm.TheHartreetermofEq.( 2{79 ),afterintegrationofthetransversecoordinates,is (2{89) where (2{90) Let'sconsiderforsimplicityjr0j1andjt0j1,inwhichcaseEq.( 2{80 )givesG1D(z;z0)=(2m=pz)exp(ipzz)sinpzz0.TheHartreepotentialbehavesasVH(z)'V1D(2pF)sin(2pFz)=zsothatEq.( 2{89 )gives The1=ztermgivesalogarithmicsingularityonlyinthelimitpz!pF,sothatt=t0/V1D(2pF)ln[1=(pzpF)].TheHartreecontributioncorresponds

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toenhancementoft0.TheexchangecontributionhasoppositesignandisproportionaltoV1D(0).Thegeneralanswer,canbewrittenas[ 72 ] t0=jr0j2ln1 where=(g0g2kF)=2,andg0andg2kFaredenedinEqs.( 2{43 )and( 2{54 ),respectively. Figure2{18. Renormalizedconductivitiesparallel(zz)andperpendicular(xx)tothedirectionoftheappliedmagneticeld.Power-lawbehaviorisexpectedinthetemperatureregion1=TW. Thesecond-ordercontributiontothetransmissionamplitudewascalculatedexplicitlyinRef.[ 21 ].Thehigher-ordercontributionscanbesummedupbyusingarenormalizationgroup(RG)procedure.WithoutrepeatingallthestepsofRef.[ 21 ],wesimplystateherethatinourcasethetransmissionamplitudesatisesthesameRGequation,asinthepurely1Dcase.i.e., d=t(1jtj2);(2{93)

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whereln(1=jppFjlH)andt(=0)=t0:ThesolutionofEq.( 2{93 )ist0=t(0)0j(pzpF)lHj 2{72 ),butnowwrittenintermsoftherenormalizedreectioncoecientjrj2=1jtj2.ThenalresultfortheconductivitycanbecastinaconvenientformbyexpressingthebarereectionandtransmissioncoecientsviabareconductivitiesintheBornandunitarylimits,0zzand0zz;U,givenbyEqs.( 2{75 )and( 2{76 ),respectively: W2;(2{94) whereWisanultravioletcut-ooftheproblemand=(g0g2kF)=2,andg0andg2kFaredenedinEqs.( 2{43 )and( 2{54 ),respectivelywhichshowsthatscaleswiththemagneticeldasHlnH.Weareinterestedintemperaturedependenceoftheconductivitiesduetoelectron-electroninteractioncorrectionsandweassumeherethatthebareconductivities0zzand0zz;UhaveonlyweakT-dependencewhichcanbeneglected. Eq.( 2{94 )isthemainresultofthisSectionandisshowninFig. 2{18 .Ithasasimplephysicalmeaning:AtT=W;theconductivityisequaltoitsvaluefornon-interactingelectrons.AttemperaturesTW;theconductivityapproachesitsunitary-valuelimit,whichmeansanyweakimpurityiseventuallyrenormalizedbytheinteractiontothestrong-couplingregime.However,iftheimpurityisalreadyattheunitarylimitatT=W;itisnotrenormalizedfurtherbytheinteractions.WeemphasizethatEq.( 2{94 )isapplicableonlyforhigh-enoughtemperatures,i.e.,Tmax[1=;]:Therstconditionsisnecessarytoremainintheballistic(single-impurity)regime,thesecondoneallowsonetoconsideronlytherenormalizationoftheimpurity'sscatteringcross-sectionsbytheinteractionwithoutrenormalizingtheinteractionvertex.Thelatterprocessleads

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eventuallyforacharge-density-waveinstabilityatatemperatureT,whereisthecharge-density-wavegap[ 1 { 3 ].Forthepower-lawbehavior[Eq.( 2{94 )]tohavearegionofvalidity,thereshouldbeanintervalofintermediateenergiesinwhichtherenormalizationoftheinteractioncouplingconstantsduetoCDWuctuationsisnotyetimportantbutthecorrectionstothecross-sectionleadingtotheformationofpower-lawisalreadysignicant.Suchanintervalexistsforalong-rangeCoulombinteraction(jlHj1)bothfortheconductivityandtherenormalizationofthetunnelingdensityofstates[ 6 ]. Thedissipativeconductivityinageometrywhenthecurrentisparalleltotheelectriceldbutbothareperpendiculartothemagneticeld,xx;occursviajumpsbetweenadjacentcyclotrontrajectories.Intheabsenceofimpurities,electronsarelocalizedbythemagneticeldandxx=0:Inthepresenceofimpurities,xxisdirectly,ratherthaninversely,proportionaltothescatteringrate.Inparticular,forshort-rangeimpurities,xx/1=/1zzandthetemperaturedependenceofxxisoppositetothatofzz.Inthescalingregime,zz/T2andxx/T2:ThissituationisillustratedinFig. 2{18 ,where=(g0g2kF)=2,andg0andg2kFaredenedinEqs.( 2{43 )and( 2{54 ),respectivelywhichshowsthat(H)HlnH.Inthenextsectionwediscusspossibleexperimentalstudiesforobservingthelocalizationandcorrelationeectsmentionedintherstthreesections.

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likebehavioristhatthesystemsberelativelyclean,sothatthereisasizablerangeoftemperaturesinwhichthesystemisintheballisticregime(1=TEF).Thisrulesoutdopedsemiconductors[ 73 { 75 ]sincethechargecarrierscomefromdopantswhichactasimpuritycentersinthesystem.Anadditionalconditionforoccurrenceofthepower-lawscalingbehaviorandformationofcharge-density-waveorWignercrystal,isthattheelectron-electroninteractionisstrongenough.Bismuthcrystalscanbemadeextremelypure;however,thechargecarriersinbismuthareextremelyweaklyinteractingduetoalargedielectricconstant(100)oftheionicbackground.Therefore,thelog-correctionscalculatedherecanbeestimatedtobeverysmallandwouldbediculttobeobservedexperimentally.Charge-densitywaveinstabilityhavebeenobservedingraphite[ 4 ]suggestingthatinteractionofchargecarriersinthissystemisimportantinstrongmagneticeldsandatverylowtemperatures.Thusgraphitewouldbeanidealmaterialtoobservethecorrelationandlocalizationeectsmentionedhere.Belowwepresentsomerecentexperimentalresultsoftransportmeasurementsingraphiterstinweakmagneticelds[ 76 ]andtheninultraquantumregimeandtrytointerprettheminviewofourndings. Graphitehasalowcarrierdensity,highpurity,relativelylowFermi-energy(220K),smalleectivemass(alongthec-axis)andanequalnumberofelectronsandholes(compensatedsemi-metal).ThemetallicTdependenceofthein-planeresistivityinzeroeldturnsintoaninsulatinglikeonewhenamagneticeldoftheorderof10mTisappliednormaltothebasal(ab)plane.UsingmagnetotransportandHallmeasurements,thedetailsofthisunusualbehaviorwereshown[ 76 ],tobecapturedwithinaconventionalmultibandmodel.Theunusualtemperaturedependencedisplayedin(Fig. 2{19 )canbeunderstoodforasimpletwo-bandcase

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Figure2{19. Temperaturedependenceoftheab-planeresistivityxxforagraphitecrystalatthec-axismagneticeldsindicatedinthelegend wherexxisgivenby[ 77 ], withi,andRi=1=qini(i=1;2)beingtheresistivityandHallcoecientofthetwomajorityelectronandholebands,respectively.Atnottoolowtemperatures(wherethemeasurementswereperformed)electron-phononscatteringisthemainmechanismfortheresisitivityintheband.Assumingthat1;2/Tawitha>0,wendthatforperfectcompensation,(R1=R2=jRj),Eq. 2{95 canbedecomposedintotwocontributions:aeld-independentterm/Taandaeld-dependentterm/R2(T)H2=Ta.AthighT,thersttermdominatesandmetallicbehaviorensues.AtlowT,R(T)/1=n(T)saturatesandthesecondtermdominates,givinginsulatingbehavior/Ta.Althoughthisinterpretationexplainsthequalitativefeaturesoftheeldinducedmetal-insulatorbehaviorshowninFig. 2{19 ,theactualsituationissomewhatmorecomplicatedduetothepresence

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ofathird(minority)band,Tdependenceofthecarrierconcentrationandimperfectcompensationbetweenthemajoritybands.FormoredetailsseeRef.[ 76 ]. LetusnowdirectourattentionontransportmeasurementsintheultraquantumregimeinwhichweexpecttoseethepowerlawconductivitybehaviorsimilartowhatisshowninFig. 2{18 .Belowwepresentsomerecentdataonthesamegraphitesamplesonwhichtheweak-eldmeasurementswereperformed. Figure2{20. Temperaturedependenceofthec-axisconductivityzzforagraphitecrystalinamagneticeldparalleltothecaxis.Themagneticeldvaluesareindicatedontheplot,withtheeldincreasingdownwards,thelowestplotcorrespondstothehighesteld Withintheexperimentallystudiedtemperaturerange(5K
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Figure2{21. Temperaturedependence(log-logscale)oftheab-planeresistivityscaledwiththeeldxx=B2foragraphitecrystalatthec-axismagneticeldsindicatedinthelegend eld-inducesLuttingerliquidmodeltheexponentofthepowerlawshoulddependontheeld(seeEq.2.94,HlnH).Also,theexponentsoftheTscalinginzz(whichis1),andxx(whichis1=3)aredierent(asseeninexperiments)whereastheywerepredictedtobethesameintheLuttingerliquidmodel. Wearegoingtoarguethattheunusualtemperaturebehaviorofzzandxx,canbeunderstoodwithinamodelwhichincludesphonon-induceddephasingofone-dimensionalelectrons(intheUQL)andthecorrelatedmotioninthetransversedirectionduetothememoryeectofscatteringatlongrangeddisorder.Beforewegetintothedetailsofthemodel,letuskeepinmindafewnumbersforthesystemweareabouttodescribe.ForourgraphitesamplestheFermienergyisEF=220K,theBloch-Gruniesentemperature(whichseparatestheregionofTandT5contributiontotheresistivity)is!0=2kFs10KandtheDingletemperature(whichgivestheimpurityscatteringrate)is3K.Alsothetransportrelaxationtimeismuchlonger(byafactorof30),thanthetotalscatteringtime(orlifetime)tr,indicatingthelongrangenatureoftheimpurities.

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Figure2{22. Temperaturedependence(onalog-logscale)oftheab-planeresistivityxx=B2atthehighestattainedc-axismagneticeldof17:5Tforthesamegraphitecrystal WerstoutlineanargumentbyMurzin[ 73 ]whichshowsthatthetransversemotionoftheelectroniscorrelatedduetodriftmotioninacrossedmagneticandelectriceld.Thedisordermodelisassumedtobeionizedimpuritytypeandisthereforelongranged.Thetransversedisplacement(afterasinglescatteringact)isassumedtosatisfyr?lHrD(rDbeingthescreeningradius).Electronsareassumedtodiuseinthezdirection.Anelectronre-enterstheregionwheretheimpurity'selectriceldisstrong,(r?
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intheeldofagivenimpurityisWDrD HWD=cerD H2NDzz1=2t3=2: H4=3N2=3rD2=3Dzz1=3;xxce3F

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resistivityxx1 wherec=4=3 Nowwewillshowthatthec-axistransportbehavior(onlythetemperaturedependence)canbeexplainedwithinthecontextofapower-lawhoppingmechanism,inwhichphononscauselocalizedelectronstohopoverdistancesoftheorderofthelocalizationlengthwithafrequencyof1 2{96 ,oneobtainstheT1=3powerlawtemperaturedependenceforxx.Inastronglylocalizedsystem(happensonlyin1D)oneshouldexpecttheabsenceofstaticconductivity.Wheninelasticscatteringisallowedthesituationchangesconsiderablyasjumpsbetweenindividuallocalizedstatesbecomepossibleandareaccompaniedbyphonon(orelectron-holepair)emissionorabsorption.ViolationoflocalizationleadstoaniteconductivityofadiusiontypewhichisthecalledthesuperdiusionregimeorPowerlawhoppingregime[ 78 ].Inthisregimeelectron'smovebydiusionbutaretrappedinsidealocalizationlength`.Inelasticscatteringallowstheelectrontojump/hopoverdistancesoftheorderofthelocalizationlengthwithafrequencyof((T))1,thephasebreakingrate.ThediusionconstantisthenD=`2=((T))andtheconductivity(T)=e2D.Thisoccursfori(T),whereiistheelasticscatteringtime.AsthetemperatureisfurtherloweredsuperdiusiongiveswaytoMott'svariablerangehoppingtransportregimewhere(T)/Deq

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Figure2{23. PhasebreakingratevsTduetoelectron-phononscattering Thebehaviorofthephasebreakingtimeduetoelectron-phononscatteringisillustratedschematicallyinFig. 2{23 .InthelowtemperatureregimeT2kFs10K,electron-phononscatteringisinelasticandthephasebreakingtimeisoftheorderoftheelectron-phononscatteringtime1=1=/T3.ButtheexperimentalplotswerenotinthisTrange.Inthehightemperatureregime,T>2kFs,theelectron-phononscatteringtimeis1 76 ],duetoitslowdensityandsmalleectivemass.ThehighTregimecanbefurthersubdividedintotheballisticregime(epht1 2kFs=)2kFsT2kFs A)andthediusiveregime(epht1 2kFs=)2kFs ATEF),wheret,isthedurationofasinglecollisionact,whereasistherelaxationtime.Noticethattheballisticregimeexistsonlyfor(semi-metals)materialswithA1.FormetalsA1andtheballisticregimedoesnotexist.Inthehightemperaturediusivelimitthephase

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breakingrateisgivenby[ 26 ] 1 whichcorrespondstotheT1=3temperaturedependenceinFig. 2{23 andoccursatveryhightemperatures(T>200K)insemi-metalsduetothesmallvalueofA.Intheballisticregimeepht(!D)1,where!DistheDebyefrequency.Thismeansthationsperformmanyoscillationsduringasingleactofelectron-phononscattering.Thusscatteringatmovingionsworksasdephasingduetodynamicpotential[ 79 ]Thephase-breakingtimeisthenoftheorderoftherelaxationtime,i.e.,eph.Therefore,thegraphitesamplesintheballisticregime(5K
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thebareDrudeconductvityitself.Thereforeperturbationtheorybreaksdownjustasitdoesin1D.However,unlikeinthe1Dcase,theconductivityremainsniteatzerotemperature.Therefore,wecallthisregimeintermediatelocaliza-tion.Thesecondimportantmanifestationofelectroniccorrelationsandlowerdimensionalityisthattherstorderinteractioncorrectiontotheconductivityislogarithmicallydivergentintemperature,justasfor1Dsystems.Wethencalculatetheinteractioncorrectiontothecross-sectionofelectronsscatteringoasingleimpurity.UsingaDruderelationtoobtaintheconductivityfromthecross-section,wendthatthisresultisequaltotheresultobtainedfromthefulldiagrammaticcalculation.Thissuggeststhatthedominantdiagramsfortheconductivitycanbedescribedintermsofscatteringoaneective\dressed"impuritypotential.Arenormalization-groupcalculationforthecross-sectionallowsforthesummationofaseriesofmostdivergentlog-correctionsatallordersintheinteraction.Justasin1Dthissummationinourcaseleadstopower-lawscaling.Howeverinthesystemof3DelectronintheUQLthisisahigh-energybehaviorwhichexistsprovidedthattheelectron-electroninteractionissucientlylong-ranged.Somerecenttransportmeasurementsingraphitewerecomparedwiththeabovetheoreticalndingsandshowntodisagree.Toresolvethedisagreement,weinvokedamodelwithlongranged-disorderandphononinduceddephasingtoexplaintheexperimentalobservations.

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TherehasbeensubstantialrecentinterestinthethermodynamicsofaFermiliquid[ 41 { 47 80 ].Therevivalofinterestintheproblemistwofold.Ontheexperimentalside,technicaladvancesnowallowonetomeasurethetemperaturedependenceofthethermodynamicparameterssuchasspecicheatandspinsusceptibilityofatwo-dimensional(2D)Fermiliquidwithshortrangeinteraction,suchasmonolayersofHe3[ 56 ],aswellas,two-dimensionalsemiconductorstructureswithlongrangeinteractionandrelativelylowFermitemperatures(1K).Onthetheoryside,itturnsoutthattheleadinginteractioncorrectionsarenonanalyticfunctionsoftemperatureormagneticeld,makingthesubjectparticularlyinteresting.Thefateofthesenonanalyticcorrectionsinthespinsusceptibility,nearaquantumcriticalpointisimportantforourunderstandingofthenatureofparamagnetictoferromagneticquantumphasetransitionandwediscussthisindetailinchapter4. Asithasbeenmentionedintheintroduction,naivepowercountingargumentssuggestthatthetemperaturedependenceofanythermodynamicquantity,includingthespinsusceptibilityandthespecicheatcoecientC(T)=T=,shouldstartwithtermsquadraticintemperature.ThisconjectureisbasedontheobservationthatathermodynamicquantityatnitetemperaturecanbewrittenasRa()nF()d,wherenF()istheFermidistributionfunctionanda()issomefunction.Ifthelatterissmooth,thetemperaturedependencestartswithatermoforderT2[ 28 ].Suchacorrectioniscalled"analytic".Thisisalsoconsistent 95

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withtheintuitiveexpectationoftheone-to-onecorrespondencebetweenthenoninteractingFermigasandtheinteractingFermiliquidsinceintheFermigas,theSommerfeldexpansionleadstoasimplequadratictemperaturecorrections. However,wealsosawthattheassumptionabouttheanalyticityofthefunctionsinvolvedinthecalculationofvariousthermodynamicpropertiesoftheFermiliquidisquitegenerallynotjustied,becauseinanyFermiliquid,thedynamicinteractionbetweenparticlesgivesrisetoanonanalyticenergydependenceofa().ThisleadstotemperaturecorrectionsthatdonotscaleasT2andarethereforecalled"nonanalytic".Collectingthesenonanalyticcorrectionsisasubtletheoreticalproblemandinthischapterwewillevaluatethesecorrectionsforaone-dimensionalinteractingsystem. InagenericFermiliquid,thefermionicself-energybehavesasReR(";k)=A"+Bk+:::andImR(";k)=C("2+2T2)+:::.Thisformoftheself-energyimpliesthatquasiparticlesarewelldened,andthedominanteectoftheinteractionsatlowenergiesistherenormalizationofthequasi-particlemassandtheresidueofthequasiparticleGreen'sfunction.Thisbehaviorhasaprofoundeectonobservablequantitiessuchasthespecicheatandstaticspinsusceptibility,whichhavethesamefunctionaldependenceasforfreefermions,e.g.,thespecicheatC(T)islinearinT,whilethesusceptibilitys(Q;T)approachaconstantvalueatQ=0andT=0. Ithasbeenknownforsometimethatthesub-leadingtermsinthe"andTexpansionsofthefermionself-energydonotformregular,analyticseriesin"2orT2[ 81 ].Inparticular,powercounting(dimensionalanalysis)showsthattherstsubleadingtermintheretardedon-shell("=k)self-energyatT=0,isR(")/"3ln(i")in3DandR(")/"2ln(i")in2D.Thesesingularitiesintheself-energygivecorrectionstotheFermi-liquidformsofthespecicheatandstaticspinsusceptibility(CFL/T;FLs=const.),whicharenonanalyticinD3

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andscaleasC(T)/TDands(Q)/QD1,withextralogarithmsinD=3and1.Itwasshownrecently[ 44 ]thatthenon-analyticcorrectionstothespecicheatandspinsusceptibilityin2Doriginatefromessentiallyone-dimensionalscatteringprocesses,embeddedinahigherdimensionalphasespace.Inparticular,these1Dscatteringevents(strictlyforwardorbackscattering)givenon-analyticsub-leadingtermsintheelectronself-energy,withthedegreeofnon-analyticityincreasingasthedimensionalityislowered,simplybecausethephasevolume(qD)ismoreeectiveinsuppressingthesingularityinhigherdimensionsthaninlowerones.Thusnon-analyticitiesinhigherdimensionscanbeviewedasprecusorsof1DphysicsforD>1andthestrongestnonanalyticityshouldoccurin1D. Thepurposeofthisworkistoobtainthenonanalytic,TlnT,correctiontothespecicheatin1D,andtoelucidatethesimilaritiesanddierencesbetweenhigherdimensionalandonedimensionalnon-analyticities.Thenonanalyticterminthespecicheatin1D,wasmissedinanearlierwork[ 57 ],astheauthorsanalyzedtheself-energyuptosecondorderinperturbationtheory.Weshowthatthespecicheatandspinsusceptibilityin1Dacquirenonanalyticcorrectionsfromthesingularitiesintheonedimensionalbosonicresponsefunctions,justastheydidforhigherdimensions.Themajordierencebetweenthenon-analyticitiesinC(T)inD>1andD=1isthattheformeroccursatthesecondorderininteraction,whereasthelatterstartsonlyatthirdorder(contrarytotheexpectationthatthedegreeofnonanalyticityincreaseswithreductionofdimensionality),andthenonanalyticityin1Doccursonlyforfermionswithspin.Inhigherdimensionsthespecicheatisnonanalyticevenforspin-less(i.e.,fullyspinpolarized)fermions.Naivepowercountingbreaksdownin1DbecausethecoecientinfrontoftheTlnTterminC(T)vanishesinsecondorder,andonehastogotothirdorder.AlthoughbosonizationpredictsthatC(T)/Tin1D,thisistrueonlyforspin-lessfermions,inwhichcasethethirdorderdiagramsforthenon-analytic

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temperaturedependenceexactlycancelout.Forfermionswithspin,thebosonizedtheoryisofthesine-Gordontypewiththenon-gaussian(cos)termcomingfrombackscatteringoffermionsofoppositespins.Eventhoughthistermismarginallyirrelevantandowsdowntozeroatthelowestenergies,atintermediateenergiesitleadstoamultiplicativelnTfactorinC(T)andalnmax[Q;T;H]correctiontos.Thespinsusceptibilityisnonanalyticatsecondorderininteractionasinhigherdimensions.Theadvantageofusingthefermionicdiagrammaticapproachin1D,isthatinadditiontocorrectlypredictingthenon-analyticitiesitalsoelucidatestheunderlyingphysics:thenon-analyticitiesarisefromuniversalsingularitiesinthebosonicresponsefunctions,thusestablishingtheconnectionwithhigherdimensions. Thischapterisorganizedasfollows.Insection 3.1 ,wediscussthemainphysicsofaonedimensionalinteractingsystemandstatethe1DmodelweusedinourcalculationswhichistheTomonaga-Luttingermodelwithbackscattering.Insection 3.2 ,weevaluatethespecicheatforaone-dimensionalinteractingsystem.Insubsection 3.2.1 ,weexplicitlyobtainthenonanalyticformsofthe2ndorderself-energyin1D,butweshowthatthesenon-analyticitiesofthefermionself-energydoesnotleadtoanonanalyticcorrection(henceforthreferredtoasNAC)tothespecicheatatsecondorderininteraction.ThisisanimportantdierencebetweenhigherdimensionalandonedimensionalNACtothespecicheat,andweshowthisintwodierentways,rstinsubsection 3.2.1 ,wherewecalculatethespecicheatfromthefermionself-energyandthenagainfromthethermodynamicpotentialinsubsection3.2.2,tormlyestablishthisdierence.Insubsection3.2.3,weobtainthenonanalyticTlnTterminthespecicheatusingthethirdorder(nonanalytic)self-energyandshowthatthistermispresentonlyforfermionswithspin.TheabsenceofthenonanalyticTlnTtermat2ndorderanditspresenceatthe3rdorder(forspinfulfermions)isconsistentwiththe

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renormalizationgrouptreatmentoftheSine-Gordonmodel,andthisisshowninsubsection3.2.4.Insection3.3,weshowthatthenonanalyticlnmax[Q;H;T],terminthestaticspinsusceptibilityispresentatsecondorderinperturbationtheorysimilartohigherdimensions.Wediscusspossibleexperimentalobservationsofthequantitiesstudiedinthischapterinsection3.4,andgiveourconclusionsinsection3.5. 28 ].Theyareinone-to-onecorrespondencewiththebareparticlesand,specically,carrythesamequantumnumbersandobeyFermi-Diracstatistics.ThefreeFermigas,thusisthesolvablemodelonwhichFermiliquidtheoryisbuilt.Theelectron-electroninteractionhasthreemainaects:(1)itrenormalizesthekinematicparametersofthequasi-particlessuchastheeectivemass,andthethermodynamicproperties(specicheat,spinsusceptibility),describedbytheLandauparametersFa;sn;(2)itgivesquasiparticlesanitelifetimewhichdiverges,however,as(EEF)2astheFermisurfaceisapproached,sothatthequasi-particlesarerobustagainstsmalldisplacementsawayfromEF;(3)itintroducesnewcollectivemodes.Theexistenceofquasi-particlesformallyshowsupthroughanitejumpzKF,ofthemomentumdistributionfunctionn(k)attheFermisurface,correspondingtoaniteresidueofthequasi-particlepoleintheelectron'sGreensfunction. Incontrast,thepropertiesoftheone-dimensionalinteractingsystem,theLuttingerliquid,arefundamentallydierentfromtwoorthree-dimensionalFermiliquids.Inparticular,theelementaryexcitationsarenotquasi-particlesbutratherbosoniccollectivechargeandspindensityuctuationsdispersingatdierentvelocities.Anincomingelectrondecaysintochargeandspinexcitationswhich

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thenspatiallyseparatewithtime(spin-chargeseparation)[ 82 ].Thecorrelationsbetweentheseexcitationsareanomalousandshowupasinteraction-dependentnon-universalpowerlawsinmanyphysicalquantitieswhereas,thoseofordinarymetals(FermiLiquids)arecharacterizedbyuniversal(interactionindependent)powers. Tobemorespecic,alistofsalientpropertiesofsuch1Dinteractingsystemsinclude:(1)acontinuousmomentumdistributionfunctionn(k)varyingasjkkFjwithaninteraction-dependentexponent,andapseudogapinthesingle-particledensityofstates/j!j,bothofwhicharetheconsequencesoftheoftheabsenceoffermionicquasi-particles;(2)similarpowerlawbehaviorinallcorrelationfunctions,specicallyinthoseforsuperconductingandspinorchargedensitywaveuctuations;(3)nitespinandchargeresponseatsmallwavevectors,andaniteDrudeweightintheconductivity;(4)spin-chargeseparation.Allthesepropertiescanbedescribedintermsofonlytwoeectiveparameters(K;uinEq. 3{2 )perdegreeoffreedom,(spinandcharge)whichplaytheroleofLandauparametersin1D. Thereasonforthesepeculiarproperties,istheveryspecialFermisurfacetopologyof1Dfermions,producingbothsingularparticle-holeresponsefunctionsandsevereconservationlaws.Ina1Dsystem,therearetwoFermi"points"kF,andonehasperfectnesting,namelyonecompleteFermipointcanbetranslatedintotheotherbyasinglewavevector2kF.Thisproducesasingularparticle-holeresponseat2kF.ThistypeofresponseisassumedniteinFermiliquidtheorybut,in1D,isdivergentforrepulsiveforwardscattering,leadingtoabreakdownoftheFermiliquiddescription.Inadditionwehave,asin3D,theBardeen-Cooper-Schrieer(BCS)singularityforattractiveinteractions.Infact,noneoftheseinstabilitiesoccur,thecompetitionbetweentheparticle-particle(BCS)andparticle-hole(at2kF)channelleadstoacriticallike(powerlaw)

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behaviorofthe1Dcorrelationfunctionsatzerotemperatures.Theone-dimensionalelectrongasisthusalwaysatthevergeofaninstabilitywithoutbeingabletoorder.Onetheotherhand,thedisjointFermisurfacegivesawelldeneddispersion,i.e.,particle-likecharactertothelowenergyparticle-holeexcitationsinafreesystem.Theseparticle-holeexcitationsarewelldened,meaningtheyhavewelldenedmomentaandenergy.Theynowcanbetakenasthebuildingblocksuponwhichonecanconstructadescriptionofthe1Dlow-energyphysics.Thedensityoperatorwhichisasuperpositionoftheparticle-holeexcitations((q)=Pkcyk+qckbq),isusedasabosonicbasisinwhichtheoriginalfourfermioninteractinghamiltonian(Eq. 3{1 )becomesquadraticandthereforeexactlysolvable.ThisistheessenceofthebosonizationtheorywhichwasusedbyMattisandLiebtosolvethe1DTomonaga-Luttingermodel[ 83 ].Thenotionofa\Luttingerliquid"wascoinedbyHaldanetodescribetheseuniversallowenergypropertiesofgapless1Dquantumsystemsandtoemphasizethatanasymptotic(!!0;q!0)descriptioncanbebasedontheLuttingermodelinmuchthesamewayastheFermiliquidtheoryin3DisbasedonthefreeFermigas. Figure3{1. Interactionvertices OurmodelhamiltonianforcalculatingthenonanalyticcorrectionstothespecicheatandspinsusceptibilitywillbethestandardTomonaga-Luttingermodel,extendedtoincludebackscatteringvertices[ 82 ], ^H=Xk;r=+;vF(rkkF)cyr;kcr;k+1 2Xr;k;k0;qV(q)cyr;k+qcyr;k0qcr;k0cr;k

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wherecyk(ck)istheelectroncreation(destruction)operatorandV(q)istheinteractionpotential.Thelinearizationofthespectrum(whichisessentialformakingtheparticle-holeexcitationswelldened)forcesonetointroducetwospeciesoffermions:rightmovers(r=+1)andleftmovers(r=1).OnehastokeepinmindthatthemostecientprocessesintheinteractionaretheoneswhichactsclosetotheFermisurface.Thefactthatinone-dimensiontheFermisurfaceisreducedtotwopoints(+pFandpF)allowsonetodecomposetheimportantlowenergyprocessesoftheinteractionintothreedierentsectors.ThesethreeinteractionprocessesareshowninFig. 3{1 ,wheresolidlinesrepresentrightmoversanddashedlinesdenoteleftmovers.Therstprocessg4couplesfermionsonthesamesideoftheFermisurface.Thesecondoneg2couplesfermionsfromdierentbranches.However,eachspeciesstaysonthesamesideoftheFermisurfaceaftertheinteraction(bothforwardscattering).Finally,thelastprocessg1correspondsto2kFscattering(backscattering)wherethefermionsexchangesides.Weassumethattheinteractionpotentialisniteranged(g16=g2),andforgeneralityweallowfordierentinteractionsbetweenleftandrightmovingfermions(g26=g4)butweneglectthemomentumdependenceoftheinteractioncoecients,treatingthemasconstants.TheinteractionpartoftheHamiltonianiswrittenintermsofoperatorsc+;k(c;k)denotingright(left)movingfermionsas[ 84 ],^Hint=1 2LXk1;k2;pX;g4k+g4?;(cy+;k1cy+;k2c+;k2+pc+;k1p++cy;k1cy;k2c;k2+pc;k1p):

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Forspinlessfermionswithonlyforwardscattering(g2andg4vertices,g1=0),theHamiltoniancanbebosonizedandtransformedtoaquadraticform[ 82 ]: 2ZdxhuK(r(x))2+u K(r(x))2i; whereandarethebosoniceldswhichsatisfythecanonicalcommutationrelations [(x1);r(x2)]=i(x2x1); anduandKaretheparametersrenormalizedbytheinteraction, 1+y4=2+y2=21=2; withy=g=(vF)beingadimensionlesscouplingconstant.Thusthephysicsoftheone-dimensionalinteractingspin-lessfermionicsystemiscompletelydescribedbyfreebosons.Theenergyspectrumischangedfrom(p)=vFjpj(forafreefermionicsystem)to(p)=ujpjfortheinteractingsystem.Thespecicheatis dT=d dTXp(p)fB((p))=u2 u(L=3): Thespecicheatislinearintemperatureevenforaninteractingsystem(forfreefermionsC(T)=T(L=3vF)).Forfermionswithoutspin,includingbackscatteringamountstoreplacingg2withg2g1.Allthepreviousresultsstillholdaftermakingthechangeg2!g2g1inuandK.Thespecicheatwillremainlinearintemperaturewithanewcoecient.Forfermionswithspin,includingbackscatteringwillleadtoasine-Gordonterminadditiontothequadraticterminthespinpartofthehamiltonian.Thechargepartretainsitsquadraticformbutwithnewcoecientsuc;Kc.Thespecicheatforthismodelisanalyzedindetailinsection3.2.4.

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Therearethreedistinctnon-analyticitiesinthebososnicresponsefunctions,inonedimensionsatT=0(asinanyotherdimensions[ 53 ]).Thesearethesingularitiesin(1)theparticle-holepolarizationbubbleforsmallmomentumandfrequencytransfers,in(2)theparticle-holepolarizationbubbleformomentumtransfernear2kFandin(3)theparticle-particlepolarizationoperatorforsmalltotalmomentum.ThefreeGreen'sfunctionforleft()andright(+)moversare Herek=ppFisthemomentumcountedfromthecorrespondingFermipoint.Theparticle-holepolarizationbubbleforleftmoversandsmallq;!is (q;i!)=Zdk andsimilarlyforrightmovers ++(q;i!)=Zdk wherevF=1.Boththepolarizationoperatorshavethesamesingularityinq:(q;!)! q,forqkFandq!.TheaboveformofthepolarizationoperatorindicatesLandaudamping:Collectiveexcitations(spinandchargedensitywaves)decayintoparticle-holepairs,thisdecayoccursonlywithintheparticle-holecontinuum,whichin1D,shrinkstoasingleline!=vFq.Asitwasshownintheintroduction,thissingularityintheparticle-holepolarizationbubbleresultedinnon-analytic,sub-leadingtermsintheselfenergyandthermodynamics.Wewillshowbelow(insubsection3.2.1),thattheforwardscatteringresponsefunctionsin1Dgivesnonanalytictermsintheself-energy,butthesedonotleadtoNACtoC(T)ors. ThedynamicalKohnanomaly,whichisthesingularityinthe(2)particle-holeresponsefunction(forq2kF),alsogivesnon-analyticsub-leadingtermsinthe

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self-energyandthermodynamics[ 53 ].In1D, 2kF+(q;i!)=Zdk 4lnj(q+i!)(qi!) (2)2j; whereistheultravioletcut-o.Finally,the(3)particle-particleorCooperbubble(forsmalltotalmomentum)hasthesamenon-analyticitybutwithanoppositesignasthe2kFparticle-holechannelin1D pp(q;i!)=Zdk 4lnj(q+i!)(qi!) (2)2j: Wewillshowthattheabovesingularitiesgiverisetononanalyticsub-leadingtermsintheone-dimensionalself-energy,fore.g.Im+R(";k=0)/j"jatsecondorderandIm+R(";k=0)/j"jlnj"j Fromthisfunction,onecandeterminetheentropybyusingthethermodynamicrelation@N @T=(@S @)T.FollowingthestepsofRef.[ 18 ]weisolatethe

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temperaturedependenceofthenumberdensityandobtainfortheentropy,S V=2@ @TTX"Zd~p V=2Zdp 2iTZ1d""@nF wherenFistheFermidistributionfunction.GR(GA)istheretarded(advanced)Green'sfunctionatzerotemperature.UsingDyson'sequation(G=1 @TS V=CFG+C(T)theinteractioncorrectiontothespecicheat(tolowestorderin)is[ 18 ]: @ @T1 whereR(A)istheretarded(advanced)selfenergyevaluatedatzerotemperature,andG0R(G0A)isthefreeretarded(advanced)Green'sfunction.Strictlyspeakingtheaboveformulaforthespecicheatisvalidonlyfortheleadingtemperaturedependence(seeadiscussionaboutthisinRef.[ 45 ]).However,wearejustiedinusingthezerotemperatureformalismin1D,sincethenon-analyticTlnTterminthespecicheatgrowsfasterthantheanalyticTtermforlowenoughtemperatures.Insub-section 3.2.1 ,weevaluatethespecicheatfromthesecondorderself-energy(atT=0)andndonlyaregular,linearinTcontribution.Insub-section3.2.2,weagainevaluatethespecicheat,onlythistimeusingthethermodynamicpotential(forT6=0)alsoatsecondorder,toverifytheabsenceofthenonanalytictemperaturedependenceatsecondorderinperturbationtheory.Thisisonemajordierencefromthehigherordernonanalyticites.Insubsection3.2.3,weshowthatthenon-analyticTlnTtermarisesonlyatthirdorderininteraction,andonlyforfermionswithspin.Inthissectionthenonanalyticcorrectiontothespecicheatisobtainedfromthethirdorderselfenergy

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evaluatedatzerotemperature.Weveriedour3rdorderresultbyperformingarenormalizationgroupanalysisofthesine-Gordontheoryinsubsection3.2.4. 3{2 .Thedashed(solid)linesrepresenttheGreen'sfunction,G(G+)forleft(right)movingfermionsandthewigglylinesdenotetheinteractionvertices.Therestofthesecondorderandrstorderselfenergydiagrams[ 18 ]areconstantandleadtoatrivialshiftofthechemicalpotentialandthusresultinalinearTdependenceforC(T). Figure3{2. Non-trivialsecondorderselfenergydiagramsforrightmovingfermions Thesingularitiesinthe2kFparticle-holepolarizationbubbleandtheparticle-particlechanneldonotaectthe2ndorderself-energy(andthisisthereasonwhywegetananalyticcontributionforthespecicheatin1Dat2ndorder),whichcanbesolelywrittenintermsoftheforwardscatteringpolarizationbubblesand++.Thusthereareonlytwodistinctcontributionsfromalloftheabovediagrams,theoneinFig. 3{2 (a)and 3{2 (c).ThediagramsinFig. 3{2 (b),(d)and(e)whichclearlyhaveabackscatteringparticle-holebubblecanbeshownequaluptoaconstantpre-factortothediagraminFig. 3{2 (a).ThediagraminFig. 3{2 (f)

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issameasthatofFig. 3{2 (c).Fortheself-energydiagraminFig. 3{2 (a)wehave,+ 32 3{14 oneneedstheimaginaryandrealpartoftheretardedselfenergy.ApplyingthespectralrepresentationfortheGreen'sfunctionandthepolarizationoperator[ 85 ],followedbyasimplepoleintegration(forT=0)andanalyticcontinuationtorealfrequencies,onegets ImR+ 32 Thisisthezerotemperatureform.Atnitetemperatures,onecansumoverthebosonicMatsubarafrequenciestoget ImR+ 32 However,itwillbeshownthatthenitetemperatureformdoesnotchangetheresultforthespecicheat.Now,substitutingImR(!;q)=q(!+q)=2fromEq. 3{8 ,andImG+R("!;kq)=("!k+q)toEq. 3{15 ,andperformingtheintegrations,weget ImR+ 32 Therealpartoftheself-energyisthenobtainedfromtheKramers-Kronigrelation ReR+ 32 whereisacut-o.Noticethattheselfenergy(realpart)iszeroonthemass-shell("=k)contrarytohigherdimensions.In1D,theentireself-energycomesfromtheprocesses!=q(becausetheparticle-holecontinuumhasshrunktoasingle

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line,!=q).Thisistheultimatecaseofforwardscattering,whoseprecursorsinhigherdimensionsleadtonon-analyticitiesinthespecicheat[ 44 ].Howeverin1D,thesenonnalyticitiesinthesecondorderselfenergywillleadtoalineartemperaturedependenceforthespecicheat.AlsonotethatImR(";k=0)/j"j,whichisindicativeofthepoorlydenedquasi-particlesin1D(ImRscaleswith"inthesamewayastheenergyofafreeexcitationabovetheFermilevel).InaconventionalFermiliquid,theconditionforwelldenedquasi-particleexcitationsis,ImR(")"2ReR".In1D,ReR(";k=0)"lnj"j,whichmeansthateectivemassdivergesas:m?lnj"jand,tothisorderthebehaviorisreminiscentofamarginalFermiliquid[ 86 ]. Theself-energydiagramofFig. 3{2 (c)is, ImR+ 32 SubstituteImR++(!;q)=1 2q(!q),fromEq. 3{9 andtheGreen'sfunctionandperformingtheintegrationsweobtain ImR+ 32 Weseethattheself-energydivergesonthemass-shell.Thisistheinfra-redcatastrophe[ 87 ]in1D.Theonedimensionalelectronscanemitinnitenumberofsoftbosons:quantaofdensityexcitations.Therealpartoftheself-energyisfoundagainfromtheKramers-Kronigrelation ReR+ 32 where(A=g42

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Whenthisformoftheself-energyissubstitutedintheGreen'sfunction,therearetwopoles,whichcorrespondstodispersionofthethespinandchargemode.This,essentiallynon-perturbativeand1Deectisthespin-chargeseparation,whichisconrmedbyanexactsolution(Dzyaloshinskii-Larkin[ 88 ]solutionofTomonaga-Luttingermodel).Therestoftheself-energydiagramsinFig. 3{2 canbereducedtoeitherofthetwoselfenergiesevaluatedabove(uptoapre-factor)byrelabelingthedummyvariables,e.g.,theselfenergyinFig. 3{2 (b)is (b)=g12ZqG(kq)2kF(q)=g12ZqG(kq)Zk0G+(k0)G(k0q)=g12Zk0G+(k0)ZqG(kq)G(k0q)=g12Zk0G+(k0)(kk0)=g12Zq0G+(kq0)(q0)=g12 (a) (f)=1 2 (c),thenegativesignisduetooneextraclosedloopandthefactoroftwoisduetospinsuminclosedloopofFig. 3{2 (c).Also (d);(e)=g1 (a).Sothenetself-energyatsecondorderininteractionis:+R(";k)g22g12+g1g21+g422;Im1=sgn(")("k)(j"jjkj);Re1=("k)lnjk2"2 3{14 .AlthoughthelogsingularityinRe1,suggestsanonanalyticterminC(T);however,acarefulanalysisshowsthatthisisnotthecase.NotethatinEq. 3{14 ,ReismultipliedbyImGR+(";k)whichis("k),sodoingthekintegration

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projectstheRe1(";k)onthemassshellwhere,itiszero(Re1(";k=")=0).ConsidernowthecontributionfromtoIm1.Themomentumintegrationgives,PZdk1 @T1 @"'Tg22g12+g1g2: ThelineartemperaturedependenceinEq. 3{23 ,canbeseenbyre-scaling"byTandbringingtheintegralinadimensionlessformwhichgivesanumberoforderone.Boththerealandimaginarypartof2willcontributeequallytothespecicheat.Considerthemomentumintegralfor2'scontributiontoC(T),ZdkImGRReR2+ReGRImR2=Zdk("k)Ak2 44 ]).Thelineartemperaturebehaviordoesnotchangeevenifweusethenitetemperatureformulafortheself-energy.TheresultforthediagraminFig. 3{2 (a)atnitetemperaturefromEq. 3{16 is, ImR (a)/("k)coth"k (3{24) ThetemperaturedependencecomessolelyfromtheMatsubarasum,thepolarizationoperatorisstillevaluatedatzerotemperature.ThemomentumintegrationinthespecicheatformulaEq. 3{14 givesalinear"dependence(thesameresultbothfor

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zeroandnitetemperature).PZdk"k "kcoth"k 3{2 (c)is ImR (c)(";k)/"2+2T2("k)=2(";k;T): UsingtheaboveformofthenitetemperatureselfenergythemomentumintegrationinEq. 3{14 givesatermproprotionalto",whichagainleadstoC(T)/TThus,1Disdierentfromhigherdimensionsbecausethespecicheatisanalyticatsecondorderininteraction.Beforeconsideringthethirdorderself-energycontributiontothespecicheat,wewillcalculatethespecicheatfromthethermodynamicpotentialatsecondorder(seeFig. 3{3 )andshowthatanapparentTlnTcontributiontothespecicheatgetscanceled,whenweconsiderthetemperaturedependenceofthepolarizationbubble. Secondorderdiagramsforthethermodynamicpotentialwithmaximumnumberofexplicitparticle-holebubbles

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AnotherwaytoobtainC(T)beyondtheleadingterminTistondthethermodynamicpotential(T)withintheLuttinger-Wardapproach[ 89 ],andthenusethethermodynamicrelationC(T)=T@2 85 ](itisanapproximatemethod,sinceonlyacertainsubsetofthediagramsaresummed).Hereinthissectionwewillevaluatethethermodynamicpotentialdiagramsdirectlytosecondorderinperturbationtheory,andverifythelinearspecicheatobtainedviatheselfenergycalculationoftheprevioussub-section.Fig. 3{3 showsthesecondorderdiagrams,whichhavemaximumnumberofexplicitparticle-holebubblesforthethermodynamicpotential.Onceagaindashed(solid)linesdenoteleft(right)movers.Wewillshowthattheforwardscatteringdiagrams[(Fig. 3{3 (a)and(c))]givealinearspecicheat.Usingthezerotemperatureformforthe2kFparticle-holepolarizationbubbleinthediagraminFig. 3{3 (b),onewouldgetaTlnTterminthespecicheat,however,suchatermgoesawaywhenweusethefullnitetemperatureresultforthebubble.Therstorderandtheothersecondorderdiagrams(nonRPAtype)forthethermodynamicpotentialgivealineartemperaturebehaviorforthespecicheat.ForthediagraminFig. 3{3 (c),=Lg22 WeomittheconstantfactorofL=2andsumoverthebosonicMatsubarafrequenciesusingacontourintegration[ 85 ]andget (c)=g22Zdq whereIm++RRshouldnowbeevaluatedatanitetemperature.Forforwardscatteringprocesses,boththenitetemperatureandzerotemperatureformsofthepolarizationoperator(++;)arethesame.Atnitetemperature

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fromEq. 3{8 ,(q;i!)=1 3{8 andEq. 3{9 toobtain,ImR++R=q (c)=g22 (c)=+g222T (c)(T)/T: NowforthediagraminFig. 3{3 (a), (a)=g42Zdq UsingEq. 3{9 ,onegets 2(!q); evenforniteT.Thusthethermodynamicpotentialbecomes, (a)=+4g42

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andonceagaintheentropyandthespecicheatarelinearintemperature. (a)=+g42T (a)(T)/T: ThebackscatteringdiagramofFig. 3{3 (b)is, (b)=g12Zdq Wenowshowthatusingthezerotemperatureformofthe2kFbubblefromEq. 3{10 ,wegetaTlnTterminthespecicheat;howeverthisnonanalytictermdropsoutwhenweusethenitetemperatureformofthebubble.AtT=0, (b)=g12 j! (b)=Tg12 j2 (b)(T)/TlnT Thisnon-analyticityisarticialandisremoved(exactlycanceled)whenwesubstitutethenitetemperatureIm2kF2inthethermodynamicpotential.Thatsuchacancelationmustoccurcanbeseeneasily:thediagraminFig. 3{3 (b)canbeshownequivalenttothediagraminFig. 3{3 (c),(uptoanoverallmultiplicativeconstant)bypairingdierentGreen'sfunctiontoformthebubble,interchangingtheorderofintegrationofthedummyvariablesandrelabelling.SincethediagramofFig. 3{3 (c)givesalinearspecicheat(seeEq. 3{27 ),thedoublebackscatteringdiagramofFig. 3{3 (b)mustgivealinear(regular)temperaturecorrectionforthespecicheataswell.Toresolvetheapparentcontradiction,wecalculateexplicitly

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thenite-Tformofthebackscatteringbubble Im22kF(q;!)R=+1 4n!q Thethermodynamicpotentialnowbecomes =APZd!coth! (2)2j22T2 (!q)2+1 (!+q)2; whereAisanumericalconstant.Evaluatingthespecicheat,wendthatthenonanalyticTlnTtermdropsoutandthespecicheatremainslinearintemperature.Thiscalculationistedious,sowedonotpresentitinthethesis.Thissectionveriestheresultobtainedinprevioussection,thatthespecicheatin1Disanalyticat2ndorderinperturbationtheoryunlikeinhigherdimensions(D=2;3)[ 44 47 ].Inthenextsectionweevaluatethespecicheatfromthethirdorderself-energyandobtainagenuinenonanalyticTlnTcontributionforspin-fullfermionsin1D. 3{4 .Thesediagramsexplicitlycontaintwoparticle-holepolarizationbubblesinthem.Therearefourdistinctpossibilities

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whichcanoccurindiagramswithtwopolarizationbubbles:(1)bothpolarizationbubblescanbebackscatteringones(2kF2)asinFig. 3{4 (a);(2)bothpolarizationbubblesareforwardscattering,withoneofthembeing++andtheotherasinFig. 3{4 (b);(3)bothpolarizationbubblesareforwardscatteringandasinFig. 3{4 (c);and,nally,(4)bothforwardscatteringpolarizationbubblesare++,asinFig. 3{4 (d).Allthirdorderself-energydiagrams,whichhavetwoparticle-holepolarizationbubblescanbeclassiedintotheabovefourcategories.Wewillexplicitlyevaluateallfourofthesediagramsandshowthatonlywhenboththeparticle-holepolarizationbubblesareofthebackscatteringtype(2kF2),onegetsanonanalyticTlnTdependenceforthespecicheat.Theparticle-particlechannelhasthesamenonanalyticmomentaandfrequencydependenceastheparticle-holebackscatteringbubble,soweexpectanonanalyticTlnTterminthespecicheatfromdiagramswhichhavetwoCooperbubblesaswell.Allthe3rdorderdiagramswhichgiveanon-analyticspecicheatareshowninFig. 3{9 .Inthissectionwewillbeusingthezerotemperatureformsofthebosonicresponsefunctionsinevaluatingtheselfenergyandthespecicheat.Weremindourselvesthattheg4couplesfermionsonthesamesideoftheFermisurfacewherasg2couplesfermionsfromdierentbranches.However,eachspeciesstaysonthesamesideoftheFermisurfaceaftertheinteraction(bothforwardscattering).Finally,theg1processcorrespondsto2kFscattering(backscattering)wherethefermionsexchangesides.Onceagain,solidlinesrepresentrightmoversanddashedlinesdenoteleftmovers. Figure3{4. Thedierentchoicesforthe3rdorderdiagram.

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(1)Twobackscatteringbubbles: + 34 (a)(k;i")=g13Zdq UsingthespectralrepresentationfortheGreen'sfunctionandthepolarizationoperator, ImR (a)=2g13 FromEq. 3{10 ,wegetIm2kF2(q;!)=1 8lnj!2q2 ImR (a)(k;")=g13 Therealpartoftheself-energyisobtainedfromKramers-Kronigrelationandonthemassshellitis ReR (a)=PZd!(!") (3{38) becausetheintegralisanoddfunctionof!.Thustherealpartoftheself-energydoesnotcontributetothespecicheat.ThecontributiontothespecicheatfromImRisnon-analytic,C(T) (a)=b2T @ @T1 2T @ @T1 j;

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one. (a)/g13Tln(T=): (2)Twoforwardscatteringbubbles,withonebubblebeingandtheother++; ImR (b)(k;")=2g22g4 Usingthezerotemperatureformsfortheaboveresponsefunctionsoneobtains, ImR (b)(k;")=+g22g4 Theaboveformfortheself-energyconsistsoftwopartseachofwhichwereearlierobtainedforthe2ndorderselfenergydiagramsinEq. 3{17 andEq. 3{20 .Thisisnotunexpectedbecausethis3rdorderdiagramismadeupofsecondorderpieces(seeFig. 3{4 (b)).Thenfromthesecondorderspecicheatanalysiswecansurelysaythattheaboveformoftheself-energygivesalinearspecicheat. (3)Twopolarizationbubbles,bothbeing, ImR (c)=2g22g4 Onceagaintheselfenergyisthesameasthatforthesecondorderdiagram,(seeEq. 3{17 ), ImR (c)(k;")=g22g4 Fromthe2ndorderanalysisweknowthatthisformoftheself-energydoesnotgiveanonanalyticcontributiontothespecicheat.

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(4)Twopolarizationbubbles,bothare++; ImR (d)=2g43 Theaboveformoftheself-energyisthesameasthatobtainedforthesecondorderdiagramwithamassshellsingularity(seeEq. 3{20 ).ThisalsoresultsinalinearinTcontributiontothespecicheat.ThereforewehaveshownthatthethreeforwardscatteringdiagramsofFig. 3{4 (b),(c)and(d)allgiveonlyalinearinTcontributiontospecicheat.TheonlydiagramwhichgivesanonanalyticTlnTcorrectiontothespecicheatistheonewithtwobackscatteringbubbles,Fig. 3{4 (a).Fromhereonwewillfocusonlyonthosediagramswhichgiveanonanalyticcontributiontothespecicheat. TheselfenergydiagramsinFig. 3{4 containstwoexplicitparticle-holebubbles.Thereareseveralother(seven)selfenergydiagrams(seeFig. 3{5 )whichdonotcontainexplicitparticle-holebubbles,buttheycanbeshownequivalenttotheonesshowninFig. 3{4 ,bytriviallyre-labelingthedummyvariables.Thesearetheselfenergydiagramswhichimplicitlycontaintwoparticle-holebubblesinthem,Fig. 3{5 (b)-(h).Thusallthirdorderselfenergydiagramswhichhavetwoparticleholebubbles(explicitorimplicit)fallintothefourcategoriesstudiedabove.Weshowthisnext,onlyforthecaseoftwobackscatteringbubblesbecausethenonanalyticTlnTtermarisesonlyfromtwo2kF,bubbles.ConsiderthediagraminFig. 3{5 (b),whichwewillshowisequal(uptoanumericalpre-factor)tothediagram

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inFig. 3{4 (a) (b)(k;i")=+g12g2ZdqZd!Zdk1Zd"1Zdq1Zd!1G(kq;i("!))G(k1;i"1)G+(k1+q;i("1+!))G+(k1+q+q1;i("1+!+!1))G(k1+q1;i("1+!1))=+g12g2ZdqZd!G(kq;i("!))Zdk1Zd"1G(k1;i"1)G+(k1+q;i("1+!))Zdq1Zd!1G+(k1+q+q1;i("1+!+!1))G(k1+q1;i("1+!1))=+g12g2ZdqZd!G(kq;i("!))[2kF(q;i!)]2: (b)=g2 (a),andthespecicheatisC(T) (b)/g12g2TlnT.Similarlyitcanbeshownthatthealltheself-energydiagramofFig. 3{5 withtwo2kFbubblesgiveaTlnTcontributiontothespecicheat.ThediagramsofFig. 3{5 (b)-(h),canalsobedrawnwithforwardscatteringbubbles(analogoustothediagramsinFig. 3{4 (b)-(d)).However,thesediagramsgivealinearTcorrectiontothespecicheatandwehaveomittedtheminthischapterforlackofspace.Thenonanalyticcontributionsfromallthoseselfenergydiagramswithtwoparticle-hole(backscattering)bubblesare: (a)!C(T)/g13TlnT (b)+ 35 (c)+ 35 (d)!C(T)/3g12g2TlnT (e)!C(T)/g23TlnT (f)+ 35 (g)+ 35 (h)!C(T)/+3g22g1TlnTThenetnonanalyticcontributionisC(T)/(g1g2)3TlnT.Thenon-analyticities

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Figure3{5. All3rdordersediagramsforrightmoverswhichhavetwo2kF. seemtoarisefromg1(exactbackscattering)andg2(exactforwardscattering)interactionvertices,similarto2D,asshowninRef.[ 44 ].Thisresultsuggeststhatevenifg1=0(longrangeinteractionpotential),thespecicheatremainsnonanalyticin1D.ThiscontradictsthebosonizationresultwhichstatesthatforaGaussiantheory(withg4andg2interaction,seeEq. 3{2 )thespecicheatremainslinearintemperature.Thereforeourresultcannotbecorrectandwemusthaveoverlookedsomediagramswhichmustcancel(atleast)theg2dependenceoftheNACtoC(T).Thesearetheparticle-particleorCooperdiagrams,whichhavethesamenonanalyticbehaviorasthe2kFparticle-holebubble(compareEq. 3{11 andEq. 3{10 ).ThereforeallthethirdorderselfenergydiagramswithtwoCooperbubblesinthem(explicitorimplicit),alsogiveaTlnTterminthespecicheatandmaycancel(someorall)thenonanalyticcontributionarisingfromthebackscatteringparticle-holebubbles.ThesediagramsareshowninFig. 3{6 .OnecanreducetheseCooperselfenergydiagramstoa2kFself-energydiagramtoobtainaTlnTcontributionforthespecicheat. Theself-energydiagraminFig. 3{6 (e)hastwoparticle-particlebubblesanditwillbeshowntobeequaltotheself-energydiagramwithtwobackscattering

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Figure3{6. AllthirdorderselfenergydiagramscontainingtwoCooperbubbles bubblesandthuswillgiverisetoaTlnTcontributiontothespecicheat. (e)(k;i")=g23ZdqZd!G(qk;i(!"))Zdk1Zd"1G+(k1+q;i("1+!))G(k1;i"1)Zdk2Zd"2G+(k2+q;i("2+!))G(k2;i"2);=g23ZdqZd!G(qk;i(!"))pp(q;i!)2;=g23ZdqZd!G(kq;i("!))[2kF(q;i!)]2= (e)=)C(T)/g32TlnT: 3{10 andEq. 3{11 ).Sincetheselfenergiesareequalandopposite,theTlnTcorrespondingtermsinthespecicheatareexactlycanceled.ThisconrmsourpreviousexpectationthattheNACtoC(T)fromtheCoopertypeselfenergydiagramswillcancelsome(orall)ofthenonanalyticcontributionfromthebackscatteringparticle-holeselfenergydiagrams. Inordertosystematicallylistallthethirdorderselfenergydiagrams,onemuststartfromthesecondorderselfenergydiagramsandreplaceoneoftheinteractionlineswithavertex(seeFig. 3{7 ).Allsecondorderverticeswithg2

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Figure3{7. Eectivethirdorderself-energydiagrams(thedoublelineisavertex). andg1interactionlinesareshowninFig. 3{8 .Wehaveleftouttheg4verticesastheydonotresultinanon-analyticself-energy.Thisisbecauseag4interactionlinecanonlybepairedwithanotherg4linetoformag42vertex.Therecannotbeag4g2org4g1vertexeither.Therefore,althougheachoftheverticesshowninFig. 3{8 ,canbedrawnwithg4lines,theycanbeonlybecombinedwithanotherg4lineintheself-energymakingtheoverallcoecientinfrontofthediagramg43.Thesediagramscanonlyhavetwoforwardscatteringparticle-holebubblesinthemandthuscannotleadtoanonanalyticity.AlsonoticethatthevertexinFig. 3{8 (f)couldhavebeendrawnwithtwog2processes;however,suchavertex,whenincludedinaselfenergydiagramcomeswithtwoforwardscatteringbubblesonebeingandtheother++andwehaveseenthatthiscombinationgivesalineartemperaturedependenceofthespecicheat.InFig. 3{9 weshowallthethirdorderself-energieswhicheitherhavetwop-h(2kF)bubblesortwop-pchannels,andallofthemgiveTlnTnonanalyticitytothespecicheat.However,forfermionswithoutspinthereisanexactcancelationamongthediagrams,makingthespecicheatlinearintemperature.ThenonanalyticTlnTtermsurvivesonlyforfermionswithspin. ThersteightdiagramsofFig. 3{9 ((a)..(h))arisewhenwereplacethevertex(doubleline)inFig. 3{7 (c)byeachoftheeightverticeslistedinFig. 3{8 .ThenexteightdiagramsofFig. 3{9 ((i)..(p))arisewhenwereplacethe(double)interactionlineinthesecondorderselfenergydiagramofFig. 3{7 (a)byeachofoftheeightverticesinFig. 3{8 .Nowletuswritethetotalnonanalyticspecicheat

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Figure3{8. Allg2andg1verticesat2ndorder. contributionfromallthediagramsinFig. 3{9 (a)+(d)+(e)!C(T)/3g22g1TlnT; (k)+(l)+(m)!C(T)/+3g12g2TlnT; (c)!C(T)/g13TlnT; (i)!C(T)/+g23TlnT; (b)+(g)+(h)!C(T)/+3g22g1TlnT; (f)+(p)+(o)!C(T)/3g12g2TlnT; (j)!C(T)/g23TlnT; (n)!C(T)/+g13TlnT: 3{9 (a)+(d)+(e)andthoseinFig. 3{9 (k)+(l)+(m)withtwoCooperbubblescancelwiththediagramsinFig. 3{9 (b)+(g)+(h)andFig. 3{9 (f)+(p)+(o);correspondingly.SimilarlythediagraminFig. 3{9 (i)cancelswiththeoneinFig. 3{9 (j)andthediagraminFig. 3{9 (c)cancelswiththeoneinFig. 3{9 (n).Thusthespecicheatisperfectly

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Figure3{9. Allthirdorderself-energydiagramswithtwoCooperbubblesortwo2kFbubbles. regularwithalineartemperaturedependenceevenatthe3rdorderininteraction.Thisisconsistentwiththebosonizationtreatmentsinceforfermionswithoutspin(evenwithbackscatteringvertexg1)onehasaquadratichamiltonian(seeEq. 3{2 )whereinK;uonehastoreplaceg2!g2g1. ForfermionswithspinthecancelationbetweenFig. 3{9 (c)andFig. 3{9 (n)isincompletebecauseofextraspinsuminthepolarizationbubble.Theparticle-particlediagramofFig. 3{9 (c)canonlyhaveg1kinteractionvertexsoC(T) (c)/g1k3TlnT,butthedoublebackscatteringparticle-holediagramofFig. 3{9 (n)canhavetwochoices.Itcan(1)haveallthreeg1kinteractionlines,forwhichC(T) (n)/g1k3TlnTwhichwillcancelthepreviouscontribution,butitcanalsohave(2)twog1?linesandoneg1kline,andsoC(T) (n)/g1?2g1kTlnT,anonanalyticcontributionwhichsurvives.Furthermore,ifweassume,g2k6=g2?,thenthethreediagramsinFig. 3{9 (f),(o),(p)andthecorrespondingonesinFig. 3{9 (k),(l),(m)donotcancel.IntherstsetFig. 3{9 (f),(o),(p),thecoecient

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infrontoftheTlnTtermcanbeeither(g1k)2g2kor(g1?)2g2kwhereasforthesecondsetFig. 3{9 (k),(l),(m),thecoecientscanbe(g1k)2g2k(whichcancelout)or(g1?)2g2?,whichdonotcancel.Alltherestofthediagramscanceloutcompletelyevenwithspin.Thusthemainresultofthissectionisthatthespecicheatatthirdorderininteractionis: (3{45) (3{46) Inthenextsectionweshowthatthisresultisconsistentwitharenormalizationgroupanalysisofthesine-Gordonmodelwhichariseswhenoneusesthebosonrepresentationtotreatfermionswithspinandwithbackscatteringinteractionvertexg1.Inthebosonizationdescription,theg1?term(backscatteringwithantiparallelspins)inthehamiltoniangivesrisetoacosterminspinpartoftheaction.Wewillshowthatthissine-Gordontermleadstoanonanalytic(TlnT),temperaturedependenceinthespecicheat,withthesamecoecient(g1?2g1k)wepredictedusingthediagrammaticanalysis. 82 ], where 2Zdxu 2Zdxu

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where=("+#)=p 2KZdxZ0d(@)2+(@x)2; Wehavesetu=1,becauseitis2ndorderinthecouplingconstants,whereasK,whichislinearinthecouplingconstantsiskeptnite.Below,wedropthesuxfortheelds,aswewillbesolelyconsideringthespinelds.Treatingthesine-Gordontermperturbatively(g1?1),onecanevaluatecorrectionstothespecicheat.ConsidertheFree-energy,F=TlnZ=TlnZDexp([S0+S1])=TlnZDexp(S0)Tln1+RDexp(S0)S12 2RDexp(S0)S21

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where,A(jxj;jj)=2g1? 82 ], 2h(Pj(Aj(rj)+Bj(rj)))2i: Wethenget, (1=T)2sinh2xT+sin2(T)!2K; byusingthenitetemperatureformofthecorrelationfunction[ 82 ], 4ln1 ToevaluatethetemperaturedependenceofthecorrectiontotheFreeenergywewillgototherelativeandcenterofmasscoordinatesystemforbothandx,andscaleoutthetemperaturedependencebybringingtheintegraltoadimensionlessform, (=)2sinh2(x=)+sin2(=)!2K; wherewehavesetK1=g1kg2k+g2?,forg1.NowC(T)=T@2F @T2.ThereforethereisanonanalyticTlnTterminthespecicheat,withthesame

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coecient(g1?2(g1kg2k+g2?))asobtainedusingthediagrammaticapproach. Weseethatthespecicheatin1Dacquiresnonanalytictemperaturedependencestartingatthirdorderininteraction(unliked=2;3wheretheyoccurevenat2ndorder).Thisnonanalyticpieceexistsonlyforfermionswithspin,becausethenthe1Dhamiltonianwithspinandbackscatteringinteractionhasasine-Gordonterminadditiontothegaussianterm.ThisresultwasearlierobtainedbyJaparidzeandNersesyan[ 49 ]byanexactsolutionoftheSU(2)Thirringmodel.Inarecentworkonthissubject,AleinerandEfetov[ 48 ]usedasupersymmetricapproachandobtainedthenonanalyticcorrectionstothespecicheatinarepulsiveFermigasinalldimensions.Howevertheirworkseemtosuggestthattheone-dimensionalNACtoC(T),startatfourthorderinperturbationtheorywhereaswejustshowed(usingtwodierentmethods:diagrammaticallyaswellasusingbosonization)thattheTlnTnonanalyticityin1Dshouldoccuratthirdorderininteraction. Next,weshowthatthesingularityinthebackscatteringparticle-holechannelcausesanon-analyticityinthespinsusceptibilityalreadypresentatthesecondorderininteractionjustasinhigherdimensions. ^H=hZdx1 2["(x)#(x)]=h p whereh=gBH,withHthemagneticeld,BtheBohrmagnetonandgistheLandefactor.Ifweassumethatg1?=0,thenthespinHamiltonianisquadratic(seeEq. 3{49 withg1?=0)andtheaboveelddependenttermcanbeabsorbedintothequadraticpartbyshiftingtheeldby,~=+hKx=p

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thespinsusceptibilityis@h("#)i=@his Aconstantspinsusceptibilitywithrenormalized(byinteractions)coecientsis"Fermiliquid"like.Aswesawinthecaseofthespecicheat,makingg1?nite,ledtoanonanalyticcorrectiontotheFermiliquidform(CFL(T)T)forthespecicheat,inthesamewayweexpectthespinsusceptibilitytoacquirenonanalyticcorrectionswhichareproportionaltog1?.Indeed,wendthatthesingularityinthebackscatteringparticle-holechannel(thedynamical-Kohnanomaly)isresponsibleforanonanalyticityinthespinsusceptibility,/lnmax(jQj;jHj;jTj)atsecondorderininteraction,andverifyanearlierresultofDzyaloshinskiiandLarkin[ 90 ].Thechargesectorisstillgaussian,hencechargesusceptibilityremainsanalytic. Wewillevaluatethethermodynamicpotentialinanitemagneticeld,(atsecondorderperturbationtheory)andthenobtainthespinsusceptibilityusingthethermodynamicrelation,(s(H)=@2=@H2).ThefreeGreen'sfunctionin1D(nowinthepresenceofthemagneticeld)is ThediagramsforthesecondorderthermodynamicpotentialwereconsideredbeforeinFig. 3{3 .HerefermionshavespinthereforeeachofthediagramsofFig. 3{3 ,willcomeinthreevarietiesdependingonwhetherthebubbleshaveparallelspins(eitherbothbubbleshaveGreen'sfunctionwithupspins,orbothbubbleshaveGreen'sfunctionwithdownspins)orifthebubbleshaveantiparallelspins(onebubblehasbothspinsupinitsGreen'sfunctionandtheotherbubblehasbothspinsdowninitsGreen'sfunction).AllthesediagramsareshowninFig. 3{10 .Wewillshowthattheforwardscatteringparticle-holepolarizationoperatorsdonot

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haveanexplicitelddependence(theycanonlyhaveananalyticmagneticelddependencethroughthedensityofstates).Therefore,allthosethermodynamicpotentialdiagramswhichhaveforwardscatteringpolarizationbubblesinthemfore.g.,diagramsinFig. 3{10 (a),(b),(c)andFig. 3{10 (g),(h),(i)cannotgiveanon-analyticcontributiontothespinsusceptibility. Figure3{10. Secondorderdiagramsforthethermodynamicpotential.

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++""(q;i!)=Zdk Thereforethereisnononanalyticcontributiontothespinsusceptibilityfromdiagramsin 3{10 (a),(b),(c),(g),(h),(i).Thebackscatteringpolarizationoperatorhasanexplicitnonanalyticelddependence. 2kF##(q;i!)=Zdk 4lnj(q+hi!)(q+h+i!) (2)2j: Henceweexpectallthreebackscatteringthermodynamicpotentialdiagrams(Fig. 3{10 (d),(e),(f))togiveanonanalyticcontributiontothespinsusceptibility.However,itturnsout,thatonlythethermodynamicpotentialdiagramwithantiparallelspins(Fig. 3{10 (f))inthetwobackscatteringparticle-holebubblewillgiveanonanalyticelddependenceandtheparallelspindiagrams(Fig. 3{10 (d),(e))canagaingiveananalyticcontributiontothespinsusceptibility.

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Thethermodynamicpotentialwithparallelspinsinthetwobackscatteringbubblesis, (e)=L=g1k2Zdq Noticethatinthemomentumintegrationonecanrelabelthedummyvariable,q+h=q0,andthentheelddependencedropsoutandthereforethisdiagramcanatthemost,makeananalyticcontribution.Thesameargumentappliestothediagramwherebothbackscatteringbubbleshavespin-up,whichalsocannotgiveanonanalyticterminthespinsusceptibility.Thereforethenon-analyticityinthespinsusceptibility,canonlyarisesfromasinglediagramatsecondorder;theonewhichhasantiparallelspinsinthetwobackscatteringparticle-holebubbles,Fig. 3{10 (f).Thisargumentalsoexplainswhytheremainingsecondorderdiagramsforthethermodynamicpotential(singleloopwithtwointeractionlines),andtherstorderdiagram(singleloopwithoneinteractionline),donotgiveanonanalyticcorrectiontothespinsusceptibility;theycannothaveantiparallelspinsastheinteractionlinecannotipthespinintheloop.Theonlynonanalyticcontributiontothespinsusceptibilityat2ndorderisfrom, (f)=L=Zdq

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Hererelabelingthedummymomentumdoesnotgetridoftheelddependence,sotherewillbeanitenonanalyticcontributionto.Performingthemomentumintegrationweget, (f)=4cZ10d!coth(!=2T)4!+2!lnj!2h2 !hj; wherec=(g1?)2 (f)(0)=4cZ10d!coth(!=2T)2!lnj!2h2 !hj: Noticethattheintegraldivergeslogarithmicallyattheupperlimit.Wecancutitat!=EFanddotherestofthecalculationtologaccuracy: Analyzingtheaboveintegralinthetwolimitsa)hTandb)hT,wendtheleadingcontributiontothethermodynamicpotentialtobe (3{67) and ThiscontributionarisessolelyfromasinglediagramtheoneinFig. 3{10 (f).TheissueofapreciseformofthefunctioninterpolatinginbetweenhandTunderthelogisoutsidethelogaccuracy.Thusweseethatthespinsusceptibilityisanon-analyticfunctionofhandTalreadyatsecondorderininteraction.The

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nonanalyticmomentumdependenceisnotmanifestintheabovemethod,(asoneintegratesoverthemomentumtogetthethermodynamicpotential).Onecanshowthatthespinsusceptibilityisanonanalyticfunctionofthebososnicmomentum,magneticeldandtemperaturein1D[ 44 90 ].s/g1?2lnEF 54 ]aswellasbulkHe3[ 55 ].AnonanalyticT2,terminthespecicheatin2DhasbeenobservedrecentlyonmonolayersofHe3adsorbedonsolidsubstrate[ 56 ].However,tothebestofmyknowledge,theTlnTnonanalyticityintheone-dimensionalspecicheathasnotbeenobservedinexperiments.Themainreasonforthiscouldbetheinherentdicultyassociatedwithmakingspecicheatmeasurementsonreal1Dsystemsfore.g.,quantumwiresandcarbonnanotubeswhichareextremelysmall(mesoscopic)structures.Onewaytoavoidthisdicultymightbetomeasurethethermalexpansioncoecientofacarbonnanotube.TheGruneisenlawstatesthattheratioofthethermalexpansioncoecienttothespecicheatstaysconstantinthelimitT!0.Atlowtemperatures,thespecicheatisdetermined,mostly,byelectrons,therefore=Cel=const..Measuringthethermalexpansioncoecientofacarbonnanotube,onecantrytodetecttheTlnTbehavior.AdenitivetestofourtheorywouldbetoseethepolarizationdependenceoftheTlnTterm(whichshouldvanishforcompletespinpolarization).Graphiteintheultra-quantumlimit(whichclearly

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showssignaturesof1Dlocalization,chapter2)mightbeapossiblecandidateforobservingtheTlnTterminC(T).

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TheLandauFermiliquid(FL)theorystatesthatthelow-energypropertiesofaninteractingfermionicsystemaredeterminedbystatesinthevicinityoftheFermisurface,andaresimilartothatofanidealFermigas.Atthelowesttemperatures,whenthedecayofquasiparticlescanbeneglected,thespecicheatC(T),scaleslinearlywithTandspinsusceptibilitys(T),approachesaconstantvalue,astheydoinaFermigas,theonlydierencebeingtherenormalizationsoftheeectivemassandgfactor[ 18 ].However,thislowtemperaturelimitoftheFLtheory,consideredbyLandau,cannottellwhetherthesub-leadingtermsinTareanalyticornot,andwhethertheycomeonlyfromlow-energystates(andarethereforedescribedbytheFLtheory)orfromthestatesfarawayfromtheFermisurface. Fornoninteractingfermions,thesub-leadingtermsinC(T)=Tands(T)scaleasT2(comefromSommerfeldexpansion)andcomefromhigh-energystates.However,itwasfoundbackinthe1960sthatin3Dsystems,theleadingcorrectiontoC(T)=Tduetointeractionwitheitherphonons[ 37 ]orparamagnons[ 38 ]isnonanalyticinTandcomesfromthestatesinthevicinityoftheFermisurface.Thesameresultwaslatershowntoholdfortheelectron-electroninteractions[ 36 40 47 48 ].Morerecently,itwasshownbyvariousgroups[ 41 { 46 80 ]thatthetemperaturedependenceofC(T)=Tisalsononanalyticin2Dandstartswithalinear-in-Tterm.Furthermore,itwasshowninRef.[ 44 45 ]thatthenonanalytictermsinthespecicheatin2Doccursexclusivelyfromonedimensionalscatteringprocesses(wheretheincomingfermionmomentaareanti-parallel,andmomentum 138

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transfersareeither0or2kF)and,foragenericFermi-liquid,thecoecientinfrontofthenonanalytic(T)correction,isexpressedintermsofthespinandchargecomponentsofthescatteringamplitudeatthescatteringangle=(backscatteringamplitude).Howeverin3D,both1Dandnon-1DscatteringprocessescontributetotheT2lnT,correctiontoC(T)=TforagenericFL[ 47 ],andthecoecientinfrontcannotbeexpressedsolelyintermsofthebackscatteringamplitude.Therearecontributionsfromtheangularaverages(andnotjust=)ofthescatteringamplitude(orLandaufunction).Thenonanalyticcorrectionstothespecicheathavebeenobservedexperimentallybothin3D(heavyfermionmaterialslikeUPt3)aswellasin2D(monolayersofHe3adsorbedonasolidsubstrate). Inthischapterwewillbestudyingthenonanalyticcorrectionstothespinsusceptibilityinboth2Dand3D.Untilrecently,theprevailingopinionhadbeenthatthenonanalytic,T3lnTterminthespecicheatisnotparalleledbyasimilarnonanalyticityinsin3D.CrucialevidenceforthisviewwasprovidedbytheresultsofCarneiroandPethick[ 36 ]andBeal-Monodetal.,[ 91 ]whofoundthattheleadingterminthespinsusceptibilityscalesasT2in3D.However,inanimportantpaperBelitzetal.[ 42 ]demonstratedthattheapparentanalytictemperaturedependenceofsmaybemisleading.Theyperformedaperturbativecalculationofthemomentumdependentspinsusceptibilitys(Q;T=0)atsmallQandfoundanonanalyticQ2lnQbehavior.Later,itwasfound[ 92 ]thatthemagnetic-elddependenceofanonlinearspinsusceptibilityparallelstheQdependence,i.e.,s(Q=0;T=0;H)/H2lnjHjwhichnegatedanearlierresultofBeal-Monod[ 93 ],whofoundonlyananalyticmagneticelddependences/H2in3D. Nonanalyticityofthespinsusceptibilitywasalsofoundfor2DsystemsbyMillisandChitov[ 43 ]and,later,byChubukovandMaslov,[ 44 ],Galitski

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etal.[ 46 ]andBetourasetal.[ 92 ].Theseauthorsshowed,(usingsecondorderperturbationtheory),thats(T;Q;H)scaleslinearlywiththelargestoutofthethreeparameters(inproperunits).FurthermoreChubukovandMaslov[ 44 45 ]andGalitskietal.[ 46 ]haveshownthatin2D,thenonanalyticterminscanbesolelyexpressedintermsofthebackscatteringamplitude.Wearegoingshowthatthisresultisvalidonlyifthebackscatteringamplitudeissmall.Ingeneralthenonanalyticcorrectiontothespinsusceptibilityin2Dacquirecontributionsfromtheangularaveragesofthescatteringamplitude(andnotjust=,aswouldhavebeenthecaseforbackscatteringamplitude)(section4.3).Therefore,thereisanimportantdierencebetweenthenonanalyticcorrectionstothespecicheatandthespinsusceptibilityin2D.Theformersolelycomesfrom1Dscattering,expressedintermsofbackscatteringamplitudewhereasthelattergetscontributionsfromboth1Daswellasnon-1Dscattering. AsitwasmentionedintheIntroduction,thesignofthenonanalyticdependenceofs(H;Q)isimportantinunderstandingthenatureofthephasetransitiontotheferromagneticstate.Alltheknownresultsforthenonanalyticdependence(bothinD=2;3)givesanincreaseofs(Q;H)withQ;Hwhichpointstowardsametamagnetic(rstorder)transitionororderingatniteQ.Theseresultswereforthesecondorderininteraction.Weshow(insection4.1and4.2)thatthethirdorderininteractiongivesadecreaseofs(H)withH(oppositeto2ndorder)whichfavors,Hertz'ssecondorderphasetransitionpicture.Howeverthesesignsoscillateateveryorderinperturbationandingeneral,itisimpossibletodeterminetheorder(rstorsecond)ofthephasetransitionfromthesignofafewloworderinperturbationtheory.Toresolvethisissue,wecalculates(H)atthecriticalpointusingthespin-Fermionmodelandshowthatitisofthemetamagneticsign.Evenexperimentally,thesituationisnotclear,TheferromagneticmetallicalloyswithlowCurieTemperatures(\weakferromagnets")

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doseemtoshowavarietyofbehaviors:insomeofthem,theQCPisoftherstorder,e.g.,MnSi[ 94 ]andUGe2,whereasother,e.g.,NixPd1x[ 95 ]showasecondordertransitiontothelowesttemperaturemeasured. Inordertokeepthisdiscussionfocusedwewillonlytalkaboutthenonanalyticmagnetic-elddependences(H)inD=2;3.Thischapterisorganizedasfollows.Wewillobtainthenonanalyticcorrectionstos(H),in2Dinsection4.1.atbothsecondandthirdorderinperturbationtheory.Section4.2isdevotedtothenonanalyticcorrectionsins(H),in3D.Insection4.3,weobtains(H)foragenericFermiliquidin2D,andshowthatitcannotbeexpressedonlyintermsofthebackscatteringamplitude.Thesameargumentcanbeextendedtoshowtheimportanceofany-anglescatteringin3D(andnotjust=);however,duetolackofspacewedonotpresentthe3Dcalculationhere.Insection4.4,weanalyzethebehaviorofthenonanalytictermsinthespinsusceptibilitynearthequantumcriticalpointusingthelowenergyeectivespin-fermionmodel.Weconcludeinsection4.5. Wewillbeonlyinterestedinthespineectofthemagneticeld,butnottheorbitaleect.AmagneticeldsplitstheFermisurfacesforfermionswithspinsparallelandanti-paralleltotheeld.Wewillseethatthissplittingdoesnotaectthe! qnonanalyticity(seeIntroduction,section1.2)ofthepolarizationbubbleatsmallq,ifaparticleandholehavethesamespins(inthiscaseamagneticeldjust

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shiftsthechemicalpotentialwhichcanatmostgiveananalyticH2contribution).However,ithasanontrivialeectonabubblecomposedofaparticleandholeofoppositespins,"#(q;i!;H).Atsmallmomentumtransfer(q0),partoftheparticle-holepolarizationoperatorhasanexplicitmagneticelddependenceonlyifthespinsareantiparallel.Toseethis,rstconsiderthecaseofparallelspins""(q;i!)=Zd2k ""(q;i!)=FZdkZd Toarriveatthelastexpressionweperformastandardcontourintegration.Weseethattheelddependencedropsout.Similaranalysisshowsthat##(q;i!)isalsogivenbyEq. 4{2 .Thereforeallthosethermodynamicpotentialdiagramswhichcontainstheq0particle-holebubbleswithparallelspinsdonotgiveanynonanalyticelddependenceinthespinsusceptibility.Letusnowconsiderthesebubblesforanti-parallelspins."#(q;i!)=Zd2k

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Wenowgetanexplicitelddependence.FollowingthesamestepsusedtoobtainEq. 4{2 ,weget(atT=0) "#(q;i!)=FZ20d Wewillusethisresponsefunctiontoevaluatethethermodynamicpotentialatsecondandthirdorderininteraction. Figure4{1. Particle-holetypesecondorderdiagramforthethermodynamicpotential. ConsiderthesecondorderdiagramforthethermodynamicpotentialshowninFig. 4{1 .InthisdiagramonepairsaGreen'sfunctionfromthetopbubblewithaGreen'sfunctioninthelowerbubbletoform"#(q;i!).(H)=g2 (H)=g2F2

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wherewehaveintroducedultravioletcut-ostoregularizetheformallydivergentintegrals.Thenonanalytictermsarisefromthelowerlimitsoftheintegralsandarecut-oindependent. (H)=BjHj3; whereB=16(gF)2B3 Weseethatsincreasesasafunctionoftheeld.NowconsiderthethirdordercorrectionshowninFig. 4{2 Figure4{2. Particle-holetypethirdorderdiagramforthethermodynamicpotential. (H)=g3

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Keepingthenonanalyticpartoftheresponsefunction(Eq. 4{3 .)andrestrictingourcalculationstozerotemperature,weobtain (H)=(gF)3 (x2+(!ih)2)3=2;=(gF)3 where:::standforanalytictermsinh.Thenonanalyticcorrectiontothespinsusceptibility(at3rdorder)is, Herethespinsusceptibilitydecreases,astheeldisincreasedoppositetothebehavioratsecondorder.HoweverateveryhigherorderininteractiononegetsanindependentnonanalyticjHj3termsinthethermodynamicpotential.Thereforeonecannotpredictthenatureofthephasetransitionbylookingatrstfewordersinperturbationtheory.SincethenonanalyticcontributionoccursforqvF!,(seemomentumintegrationatbothsecondatthirdorder),theycomefromanyanglescatteringevents,(inordertohave1Dscatteringacts,thenecessaryconditionwasqvF!,i.e.,deepinsidetheparticle-holecontinuum). "#(q;i!)=Zd3k =FZdkZ11d(cos) 2nF(~k+h=2)nF(~k+~qh=2)

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Performingthesamemanipulationsasin2Dcase,wegetfortheparticle-holebubblein3DatT=0, "#(q;i!)=FZ11d(cos) 2(h+vFqcos) i!vFq+h): Usingthisformofthebubblewegetanonanalyticcontributionforthethermodynamicpotentialateveryorderininteraction,startingatsecondorder(seeFig. 4{1 ).(H)=g2 i!q+h)2; q=y;h q=~H)andwritetheintegrandinadimensionlessform(H)=(gF)2 (H)=AZEF0dqq3~H4Z1dy(6y5tan1(1=y)+3y46y3tan1(1=y)+4y2 Theyintegralconverges,sothatwecanextendtheintegrationlimitstoinnity,uponwhichitgivesanumber.Theremainingqintegralislog-divergent.Aswehadperformedanexpansionin~H=h=q1,itislegitimatetocutthedivergentintegralatq=h.Tologarithmicaccuracythearbitrarinessinchoosingthe

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numericalprefactorinthecut-odoesnotaecttheresult. (H)=2A whereA=(gF)2=(2vF)3.Thenon-analyticcorrectiontothespinsusceptibilityis, Followingtheaboveprocedureonecanevaluatethethirdorderthermodynamicpotential(seeFig. 4{2 ),(H)=+2(gF)3 i!q+h)3; Notethatthethirdordersignisoppositetothesecondordersign.However,ateveryhigherorder4th,5th,therewillbeanindependentnonanalyticcontributionwhosesignwilloscillateandthusonecannotdenitelypredictthesignofthenonanalytictermbylookingatfewlowordersinperturbationtheory.Furthermoreinrealsystemsinteractionsarenotweakandonecannotterminatetheperturbationseriesexpansiontothelowestorders.Tocircumventthisinherentproblemwithperturbativecalculationsandtomakepredictionsforrealisticsystems(e.g.,He3),weobtainthenonanalyticelddependenceofthespinsusceptibilityforagenericFermiliquidin2D. 89 ] =Tr(ln[G01]+G)+skel

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withbeingtheexactself-energy,andskelisthesetofskeletondiagramsforallinteractioncorrectionstothatarenotaccountedforbythersttwoterms.Thetraceistakenoverspace,time,andspinvariables.TheskeletondiagramforthethermodynamicpotentialisshowninFig. 4{3 ,wherethebareinteractionlinesU(q),arereplacedwithfullydressedvertices,(~p~k).Thevertex(~p~k),hasanexpansionintermsofparticle-holebubbles[ 18 ],whichtolowestordergivesadiagramwhichistheparamagnondiagram(at2ndorder,seeFig. 4{1 ),exceptwithmomentumdependentinteractionlines,theanalyticexpressionforwhichis (H)=TXi!Zd2~q (4{16) Figure4{3. Theskeletondiagramforthethermodynamicpotential. Anext-to-leadingorderexpansionofthevertex,givesadiagramwhichisthethirdorderparamagnondiagram,againwithmomentumdependentinteractionlines. (H)=TXi!Zd2~q (4{17)

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Belowweevaluatethespinsusceptibilityarisingfromtheselowestorderexpansionoftheskeletondiagrams.ForthesecondorderdiagramgivenbyEq. 4{16 ,weget(H)=(F)2Z1d! vFq vFq)Z20dk vFq vFq)2(pF(kp)) NowwewillexpandtheangledependentvertexinaFourierseries,(kp)=1Xn=nein(kp): (4{18) (4{19) Usingtherelation(cospib)1=sgn(b)iZ10deisgn(b)(cospib)

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UsingtheBesselfunctionproperty,Z20d Letusrstevaluate1(H).Itisconvenienttosplitthesumovertheintegersn;mintotwosums,onewithn+m>0andtheotherwithn+m<01(H)=(F 69 ], (4{20)

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Wepulloutthecommondenominatorfromboththesumswhichyields1(H)=(F vFq)2+1(!+ih vFq)2n+2m+1Xn;m=n+m<0(1)2n2ms vFq)2+1(!+ih vFq)2n2mnm(1)(n+m) 1(H)=(F EF)2BS wherethebackscatteringamplitude(scatteringamplitudeatthescatteringangle=)isdenedas,BS=Pn(1)nn.Similarlyonecanshowthat,2(H)=(F EF)2BS (H)=(F wherewehavemultipliedwithanoverallfactorof1=2,whichisthesecondordercombinatorialcoecient.Therefore,thesecondorderskeletondiagramforthespinsusceptibilitycanbeexpressedsolelyintermsofthebackscatteringamplitude.

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Thisresultissimilartothatforthespecicheat[ 45 47 ], wherecisapositivenumber.However,weshowthatthenext-to-lowestorderexpansionofthevertex,intheskeletondiagram,acquirescontributionswhichcannotbeexpressedintermsofthebackscatteringamplitude.Thissuggeststheimportanceofnon-one-dimensionalscatteringprocesses.Thethirdorderskeletondiagramreads(H)=TXi!Zd2~q Thenonanalyticpartatthirdordercomesfrom(H)=(F)3Z1d! vFq vFq)Z20dk vFq vFq)Z20dk1 vFq vFq)1Xn;m;l=nmlein(pk)+im(k1p)+il(kk1) (H)=1(H)+2(H); where, 1(H)=(F)3

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where,Anml=[(mn)(p 2(H)=(F)3 where,Bnml=[(mn)(1)nm(p 4{25 andEq. 4{26 ,hasapowerlawsingularity,soonecannotcutitatqvF>!+ihandsetqvF!+ihinAnmlandBnml,aswedidinsecondordercasewheretherewasalogsingularity.Thisistheessentialdierencebetweenthesecondorderandthirdordercalculationswhichmakesitextremelydiculttoarriveataclosedform(forarbitraryn;m;l)atthirdorder.WewillobtainthecoecientinfrontofthejHjterminthespinsusceptibility,forthefewlowestharmonics(0n;m;l1)andshowthatitisnotpossibletowritethethirdorderresultsolelyintermsofthebackscatteringamplitude.Severalcasesneedtobeconsideredseparately.Case(a):

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000(H)=(F0)3 wherethesuperscriptdenotestheharmonics. Case(b):m=1;n=0;l=0.ThenA100=(1)(p q2[q2+(!+ih)2]3=2(p 100(H)=3(F)3120 Case(e):m=1;n=1;l=0.ThenA110=(1)(p

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pre-factoris210.Theextrafactorof3arisesduetothecase(f):m=1;n=0;l=1andthecase(g):m=0;n=1;l=1whichgivesthesamecontribution. 110(H)=3(F)3021 Case(h):m=1;n=1;l=1,hereA111=B111=1.Theintegralsareidenticaltocase(a)and 111(H)=+(F1)3 Addingtheresultfortherstfewharmonicswegetforthespinsusceptibility, Notethatthenumericalfactorof2ln(2)(andalsothesigns)makesitimpossibletorepresentthe3rdorderresultintermsofbackscatteringamplitude.Ifthethirdorderresulthadanexpansionintermsof3BS,thenourlowestharmonicsresultswouldhavebeen(Pn(1)nn)3=303201+321031whichisdierentfromwhatweobtained. Parameters0,1,canbeestimatedusingtherelationbetweentheharmonicsoftheamplitudesandtheLandaufunctionsofa2DFermiliquid[ 47 ] n=Fn TheLandauparameterscanbemeasuredfromtherenormalizationsoftheleading(analytic)termsinthermodynamicquantities.InbulkHe3, (4{33) (4{34)

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inawideintervalofparameters[ 55 ].AssumingthattheFermi-liquidparametersarethesameinbulkHe3andina2Dlayer,wecanestimate0and1tobe 0=2:3 (4{35) 1=0:76 (4{36) (4{37) Inthisapproximation,thesquareofthebackscatteringamplitude 2BS=(Xn()nn)22:5 (4{38) whereasthecoecientofthethirdorderwhichwefoundis3=0:05.Therefore,inthisapproximationthebackscatteringamplitudestilldominatestheresult. 62 ], (H)=FG+1 2ZdD~q 2ln((s;0)1+g2); whereFGisthethermodynamicpotentialfortheFermigas,gisthespin-Fermioncouplingconstant,ands;0isthebarespinsusceptibility,whichisanalyticatqpFandisgivenbytheOrnstein-Zernickeform 2+q2: NoticethatA2istheq=0susceptibility,whichdivergesatthesecond-orderQCP.istheparticleholepolarizationoperator,inanitemagneticeldorinasystemwithnitemagnetization (q;i)=2ZdD~k

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where(G),arethedressedGreen'sfunctioncontainingthefermionself-energies.s;0isaphenomenologicalinputofthetheory.Itisassumedthatthehighenergystates(awayfromtheFermisurface)arealreadyintegratedout,andtheireectistodrivethesystemtothevicinityoftheQCP.TheSpin-Fermionmodelprovidesanaccuratedescriptionofthefeedbackfromthelow-energystatesonthepropertiesoftheQCP. Onecanperformthecalculationseitherinanitemagneticeld,inwhichcaseG!G"#,oronecanworkwithasystemwithnitepolarizationinazeromagneticeldassumingspinupandspin-downelectronshavedierentdensitiesnanddierentchemicalpotentials Thefermionself-energies(seeFig. 4{4 )arecalculatedwithintheEliashbergapproximation,whereoneassumesamomentumindependentself-energyandneglectsthevertexcorrections.ThedoublelineindicatesthedressedGreen'sfunctionandthedoublewavylineindicatesthespinuctuationpropagator,renormalizedbythebosonicself-energy.ItcanbeshownthattheEliashbergapproximationworksaslongas1[ 62 ],whereisthedenesasfollows,(!)=i~(!),with ~(!)=!; where(themassrenormalizationfactor,m?1)takesdierentfunctionalformsintheFermi-liquid(farfromtheQCP)andthenon-Fermiliquid(neartheQCP)regimes,andalsodependingonwhetheroneisin2Dor3D.In2D[ 62 ], (4{44) (4{45)

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whereg2A=EFisasmallparameter,isthecorrelationlengthwhichisassumedtobemuchlargerthantheinteratomicdistanceand!0=2EFisaconstant.In3D, wherecisasmallparameter,and0EFisagainaconstant. Figure4{4. Fermionself-energy(a)andBosonicself-energy(b). 4{41 in2D,wehave(q;i)=2F

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Inthenon-perturbativeregime(1)wehave~(!)!,sotheMatsubarafrequenciesi!andi!+iintheGreen'sfunctioncanbeneglectedcomparedto~(!)and~(!+).andPerformingasimplepoleintegrationink,wearriveat(for>0)(q;i)=2Fi (q;i)=2FZmax[0;]min[;0]d! Changingthesignof,isthesameasm!m,whichisequivalenttocomplexconjugation.Thenthethermodynamicpotential(Eq. 4{39 ,nonFermi-gaspart),becomes (m)=2BReZEF0dqqZEF0dlnq2 whereB=1=2(2vF)2isconstantpre-factorandwherewehavescaledoutthevFdependencebyrelabelingvFq!q.TherstterminsidethelogcomesfromtheOrnstein-Zernickeformofthebaresusceptibilityandwehaveset!1. First,weanalyzethenonanalyticpartofthethermodynamicpotentialintheFermi-liquidregime,where~(!)=!with=(pF).Then,=,andthe!integralistrivial.Keepingjustthedynamicpartofthebubbleweget FL(m)=2BReZEF0dqqZEF0dlnjg1

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whereg1=2g2F.Re-scaling,=1weget FL(m)=2B ReZEF0d1ZEF0dqqln1 ReZEF0d1(1im)2lnj1im wherewehavedroppedtheanalyticinmterms.Weseethatthenonanalyticterminthethermodynamicpotentialcomeswithanegativepre-factorwhichindicatesthepossibilityofarstordertransition.TheFreeenergyisgivenby, Wewillnowperformathermodynamicanalysisandndthedependenceonthemagneticeld.TheGibbsfreeenergyis and wherebandcareconstantsandwherewehavekeptthenextanalytic(m4)termsinF,whichstabilizesthephase.TheunitsarechoseninsuchawaythatFandmanddimensionless.TheminimumofG(m;H)withrespecttothemagnetizationgivestheequationofstate, @m=m 2cmjmj Solvingthisequation 5HjHjb8H3+:::

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Thereforethespinsusceptibilityisgivenby @H=2+c 5jHj3b8H2+::: whichdivergesattheQCPas!1. NextweanalyzethestabilityoftheQCP,wheretheelddependence(ofs),changestojHj3=2.ToanalyzethebehaviorneartheQCPwestartfromEq. 4{49 .Noticethatifweusethenon-FLformoftheself-energyin2D(~(!+)sgn(!+)j!+j2=3),in,wegetthefollowingscalingbehavior 2=3m; neartheQCP.Also,sinceinthequantumcriticalregime;()isafunctionof!,onecannotdothe!integrationtrivially.InEq. 4{49 ,thenonanalyticelddependencearisesfromthedynamicbubblesothatonecanscaleoutthestaticpart,(m)=2BReZEF0dqqZEF0dln1+2g2FA q3Z0d! (m)=2BReZEF0dqqZEF0dln[g q3Z0d!(11 2(im q)2)] (4{59) whereg=2g2FA, (m)=2BReZEF0dqqZEF0dln[g (4{60) Onceagainwescaleoutthersttermwhichdoesnotdependonthemagneticeld.Deninganaverageself-energysquare,((;m))2=1R0d!(

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(m)=2BReZEF0dZEFdqqln[1+()2 wherewewilltakethelimit!0intheend.Theaverageself-energysquareis, 2((;m))2=1 Z0d![(!0)1=3sgn(!+)j!+j2=3(!0)1=3sgn(!)j!j2=3im]2=4=3(!0)2=3Z10dx[(1x)2=3+x2=3ia]2 wherea=m=(2=3(!0)1=3).SubstitutingEq. 4{62 ,inEq. 4{61 ,weperformascalinganalysistoget (m)jmj7=2=)s(H)jHj3=2; WeseethattheFree-energy(thermodynamicpotential)becomenegative,whichsignalsaninstabilityofthesecondorderquantumcriticalpoint. 4{41 ),=F cos(m+i

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wherewehaveperformedthesamemanipulationsasin2D.Performingtheangularintegrationwhichgivesalog,andintegratingbypartsweobtainfortheleadingterminthepolarizationoperator,(>0), (q;i)=g2Fi (4{64) Usingtheaboveformofthepolarizationoperator(andalsoconsideringthecase<0),theformulaforthethermodynamicpotential,(nonFermigaspart),(Eq. 4{39 ). (m)=1 (2vF)32ReZEF0dqq2ZEF0dln[g2F(i wherewehavekeptjustthedynamicpartandhavedroppedthestaticpart(comingfromtheOrnstein-Zernickeformofthebaresusceptibility).IntheFLregime,~()=,where,(=cln(kF),isaconstant.Therefore,FL(m)=1 (2vF)32ReZEF0dqq2ZEF0dln[g2F(i Nowonecanre-scale=1toget, FL(m)=1 (4{66) Writingtheintegrandintermsofdimension-lessvariables,1 q=dandTaylorexpandingtheintegrandford1,andextractingthecoecientofthed4term(similartowhatwedidinperturbationtheory)weget, FL(m)=1 (4{67) q(42

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wherec1isapositiveconstant.Thereforeperformingthesamethermodynamicanalysisaswasdonefor2D,onegets wherewehavesubstituted,(ln).AsoneapproachestheQCP,becomesfrequencydependentandthatchangestheelddependenceatthecriticalpointto,sQCP(H)/H2ln(ln(jHj)).Toanalyzetheelddependenceinthenon-FLregime,westartfromEq. 4{65 anduse~()=cln(0 nFL(m)=2BReZEF0dqq2ZEF0dln[i (4{70) whereonceagainwehavere-scaledthedummyvariablesas,y==q;d=m=qanda=0=qandB1=(2vF)3isaconstantpre-factor.Toobtainascalingformforthethermodynamicpotentialweneglecttheiyln(y)termsinsidethelogarithm.nFL(m)2BReZEF0dqq3ZEF0dyln[iyln(1+d+iyln(a) whichisofthesameformasEq. 4{67 .OnceagainTaylorexpandford1andintegrate(overy1)thecoecientofthem4term. nFL(m)Bm4ZEFmdq qln(0=q)(42 (4{71) wherebisapositivenumber.ThesecondorderQCPisunstableevenin3D,withatendencytowardsrstordertransition.ThespinsusceptibilityattheQCPissnFL(H)/H2ln(ln(0=H)).

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Inthisworkwehaveinvestigatedtheroleofone-dimensionalelectroniccorrelationsinthetransportandthermodynamicpropertiesofhigherdimensional(D=2;3)systems. Intherstpartofthisworkwepresentedastudyoftransportpropertiesofathree-dimensionalmetalsubjectedtoastrongmagneticeldthatconnestheelectronstothelowestLandaulevel(UQL).Weshowedthatthenatureofelectrontransportisonedimensionalduetothereducedeectivedimensionalityinducedbytheeld.TherstsignofthisisthatthelocalizationcorrectionstotheconductivitywasoftheorderofthebareDrudeconductvityitself.Thereforeperturbationtheorybreaksdownjustasitdoesin1D.However,unlikeinthe1Dcase,weshowedthattheconductivityremainsniteatzerotemperature.Therefore,wecallthisregimeintermediatelocalization.Thesecondimportantmanifestationofelectroniccorrelationsandlowerdimensionalityisthattherstorderinteractioncorrectiontotheconductivityislog.divergentintemperature,justasfor1Dsystems.Thephysicalreasonforsuchabehavioroftheconductivityisanearly1DformoftheFriedeloscillationaroundanimpurityinthestrongmagneticeld.Arenormalizationgroupcalculationofthetransmissionamplitudethroughasinglebarrierallowedforasummationofaseriesofthemostdivergentlog-correctionsatallordersintheinteraction.Justasin1Dthissummationinourcaseledtopowerlaw(temperature)scalingbehaviorfortheconductivity.Somerecenttransportmeasurementsingraphitewerecomparedwiththeabovetheoreticalndingsandshowntodisagree.Toresolvethedisagreement,we 166

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invokedamodelwithlongranged-disorderandphononinduceddephasing(1Dphenomenon)toexplaintheexperimentalobservations. Previousworkonthethermodynamicpropertiesofhigherdimensionalsystemshadindicatedthespecialroleplayedbyone-dimensionalscatteringevents,inthenonanalyticcorrectionstothespecicheatandspinsusceptibility.Inthesecondpartofthiswork,wehaveshownthatthenext-to-leadingtermsinthespecicheatandspinsusceptibilityin1Darenonanalytic,inthesamewayastheyareinhigherdimensions(D=2;3).Thus,eventhoughthelowenergytheorywhichdescribesaone-dimensionalinteractingsystem(Luttingerliquidtheory)isdierentfromthehigherdimensionallowenergytheory(Fermiliquid),thesub-leadingtermsinthethermodynamicpropertiesgetnonanalyticcorrectionswhicharisesfromthesamesourcesinalldimensions.Theonlydierenceisthatthenon-analyticcorrectiontothespecicheatin1Dispresentonlyforfermionswithspin,anditoccursat3rdorderininteraction;C(T)/g1?2(g1kg2k+g2?)TlnT,whereasinhigherdimensionstheyoccurevenforspin-lessfermionsandstartat2ndorder.Thenon-analyticcorrectionstothespinsusceptibilityoccuratsecondorderininteractioninalldimensionsD=1;2;3.Thus,wehaveshownthat1Dsystemsaresimilartohigherdimensionalsystems,atleastinthecontextofnonanalyticcorrectionstothermodynamics. Inthethirdpartofthiswork,weperformedadetailedinvestigationofthenonanalyticmagneticelddependenceofthespinsusceptibility,inhigherdimensionalsystems,andshowedthattheyarisefromboth1D,aswellasnon-1Dprocesses.Weobtaineds(H)foragenericFermi-liquidin2D.Ourresultcanbecomparedwiththeexperimentalresultsonatwo-dimensionalHe3layer.Wealsostudiedthespinsusceptibilityinthevicinityoftheferromagneticquantumcriticalpointusingthespin-Fermionmodelandshowedthatthesecondordercriticalpointisunstablebothin2Dand3Dwithatendencytowardsrstordertransition.

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RonojoySahawasbornonJuly17,1978,inCalcutta,India.HespenttheearlyyearsofhischildhoodinCalcutta,andafewyearsinnorthBengal,beforemovingtoNewDelhiforhishighschool.Aftercompletinghighschoolin1995,hecamebacktoCalcuttaandjoinedthePhysicsDepartmentofPresidencyCollegeforhisundergraduatestudies.DuringhisundergraduatedaysinPresidencyCollege,hemethisfuturewifeSreya.HereceivedhisBachelorofSciencedegreein1998andmovedtoNewDelhitopursuehismaster'sinphysicsattheJawaharlalNehruUniversity(JNU),whichhecompletedinsummerof2000.Duringhismaster'satJNUhedevelopedakeeninterestintheoreticalphysics.InFall2000,hejoinedthegraduateprograminphysicsattheUniversityofFlorida.SinceFall2001,hehasworkedwithProfessorDmitriiMaslovonvariousproblemsinstronglycorrelatedelectronsystems.HereceivedhisPh.D.inAugust2006. 174