INTERACTING FERMIONS IN TWO DIMENSIONS:
EFFECTIVE MASS, SPECIFIC HEAT, AND SINGULARITIES IN THE
PERTURBATION THEORY
By
SUHAS GANGADHARAIAH
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 2005
Copyright 2005
by
Suhas Gangadharaiah
To my parents.
ACKNOWLEDGMENTS
First and foremost I would like to thank my research advisor Professor Dmitrii
Maslov, for his constant encouragement and guidance throughout the entire course of my research work. His enthusiasm, dedication, and optimism toward physics research has been extremely infectious. The countless hours I spent discussing research problems with him were highly productive scientifically and intellectually.
I would also like to thank Professor Kevin Ingersent, Professor Pradeep Kumar, and Professor Richard Woodard who were always willing and open to discuss any researchrelated issues. I am honored and grateful to Professor Russell Bowers, Professor Alan Dorsey, Professor Kevin Ingersent, and Professor Khandker Muttalib for serving on my supervisory commitee.
My thanks go to the Physics Department secretaries, Ms. Balkcom and Ms. Lattimer; and to my friends Aditi, Gregg, Hyunchang, Lingyin, Tamara, Sudarshan, and Yongke for their valuable help and support. My special thanks go to Aparna, Evren, and Ronojoy for their support, friendship and encouragement.
I would like to thank my parents for the unconditional love, support, and encouragement they provide through the years.
iv
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS .................... ........... iv
LIST OF FIGURES ..................................... vii
ABSTRACT ........................................ ix
CHAPTER
1 INTRODUCTION .................... ............ 1
1.1 Survey ...................... .............. 1
1.2 Deviation from FermiLiquid Universality ......... .......... 2
1.3 MetalInsulatorTransition (MIT) in 2D .......... ........... 7
1.4 Quantifying Fermiliquid Parameters .................... 14
1.4.1 Applicability of the LifshitzKosevich Formula in 2D ....... 15 1.4.2 Absence of Polarization Dependent Effective Mass ......... 16
1.4.3 Effective mass and Landeg Factor Near the Critical Densities . . . 18
2 INTERACTING FERMIONS IN TWO DIMENSIONS . ............. 19
2.1 Introduction ............. ..... . .............. 19
2.2 Scattering Processes ................... ........ . 20
2.3 SelfEnergy: Nonanalyticity and Singularity . ............... 22
2.3.1 SelfEnergy at the Second Order . .................. 25
2.3.2 HigherOrder ForwardScattering Contributions . ......... . 30
2.3.3 Resummation of ForwardScattering Contributions . ....... 31
2.3.4 Renormalized Imaginary Part of the SelfEnergy ... . ... . . . . 37
2.3.5 Renormalized Real Part of the SelfEnergy ..... ........ . 44
2.3.6 Spectral Function ........... ........ ....... . . 49
2.3.7 Coulomb Potential ............. ..... ......... 51
2.3.8 Corrections to the Tunneling Density of States ....... . . . . 52
2.4 Conclusion ................................... 53
3 SPECIFIC HEAT OF A 2D FERMI LIQUID . .................. 55
3.1 Introduction . .................. . . . . . . . . 55
3.2 Specific Heat Calculation From the SelfEnergy . ............. 56
3.2.1 ZeroSound Mode Contribution to the Real Part of the SelfEnergy 58
3.2.2 Zero Sound Mode Contribution to the Imaginary Part of the SelfEnergy ....... .......................... 58
3.2.3 Finite Range Potential . .......... ... . ........... . 60
3.3 Specific Heat Calculation from the LuttingerWard Formalism ...... 60
3.3.1 RealFrequency Approach ................... . . . . 62
3.3.2 Matsubara Formalism .................. ........ . 66
v
3.3.3 Contact Interaction: Beyond Second Order . ............ 68
3.3.4 Generic Interaction ................... ....... 70
3.4 Specific Heat for the Coulomb Potential ................... 72
3.5 Conclusion ................... ............. .. 76
4 ANOMALOUS EFFECTIVE MASS . . . ........ ........ ...... 77
4.1 SpinPolarized Effective Mass ................... ..... 79
4.1.1 Landau's Phenomenological Approach ...... .......... 79
4.1.2 WeakCoupling Approach ................... .... 85
4.2 MultiValley System ................... ........ . 89
4.2.1 Effective Mass ................... ........ . 89
4.2.2 spin susceptibility ................... ........ 97
4.3 Conclusion ...... .......... .. ................ 99
5 CONCLUSIONS ................... ............. .. 101
APPENDIX ..... . ............... . .......... ..... 103
A SPECIFIC HEAT CONTRIBUTION: COULOMB POTENTIAL . ....... 103 B SPINDEPENDENT EFFECTIVE MASS ................... .. 106
C CORRECTIONS TO THE SPIN SUSCEPTIBILITY . .............. 112
REFERENCES ....... ............ ............... . 117
BIOGRAPHICAL SKETCH ................... ........... 122
vi
LIST OF FIGURES
Figure page
21 Scattering processes responsible for divergent or nonanalytic selfenergy contributions in 2D. A) Parallelmoving quasiparticles scatter at each order
by exchange of a small momentum. B) The quasiparticles are moving
opposite to each other, initial and final momenta remain essentially unchanged. C) Each of the oppositely moving quasiparticles undergo momentum change close to 2kF. ................... ...... 21
22 Selfenergy diagrams with explicit and implicit polarization bubbles...... 24
23 Second and third order Vertex diagrams with maximum number of particlehole bubbles. Additional diagrams, obtained from those in the second column by a permutation a  , 7  E, are not shown. . ............ 32
24 Scaling functions F, and GI as a function of x. Note the strong asymmetry
of Fl about x = 0. ................... ....... 40
25 Scaling function FR. Note the strong asymmetry of FR about x = 0. .... 47
26 The logplot of spectral function A(e, k) in units of 1/7r2u2EF. A(e, k) is plotted as a function of x = 2(Ek)/u2E for log(EF/u2jel) = 2 and e/27rE, = 0.025. A kink at x = 1 is due to the interaction of fermions with the zerosound mode. Inset: part of the spectral function Ai(E, k) for e/27rEF = 0.25. A maximum in A1 at x = 1 gives rise to a kink in total A (main
panel). . ............... ... ............ ....... 50
31 Example of a secondorderskeleton diagram. Notice the fully interacting
Green's function. . . . . . ................... .. ..... . ... 61
32 Diagrams through the third order that contribute to the thermodynamic potential. .................... . .............. 61
33 Contour for summing up the Log. The thick line represents the branch cut
and the small circles represent the discrete Matsubara poles. ........ . 63
34 Plot of u~ (u*) vs u* . . . . . ................. . . .. . . . 69
35 Contour for summing up the discrete sum of the Matsubara thermodynamical potential. A) Matsubara poles at the Matsubara frequencies and at the plasmon positions wp, and wpi (for the integration variable eo ); the
branch cut from Wk to Wk represents the particlehole region. B) Particlehole spectrum for 2D. ....................... 74
41 Effective mass vs the degree of spinpolarization. The electron densities in
units of 10"cm2 are: 1.32 (dots), 1.47 (squares), 2.07(diamonds), and
2.67 (triangles). . . . ........ . . . ... . ..... . ....... .78
vii
42 Distribution function in the primed frame and lab frame are represented by
dashed and full circle respectively. . ................. . . . . 83
43 SelfEnergy of spinup and spindown electrons. . ................ 86
44 Effective mass for a spinup and a spindown electron at polarization ( = 0.3. 88 45 Particlehole spectrum for a multicomponent and a 2component system. 91 46 Diagrammatic corrections. A) The RPA potential. B) Corrections to the
polarization bubble due to the interaction terms. C) Vertex corrections to
the selfenergy. ................... ............. 93
47 Percentage change in effective mass between a fully polarized and an unpolarized system as a function of rs. ................... ... 95
48 Plot of the effective mass formula obtained from the weakscreening case
and from the strong screening case. . .................. ... 96
49 Effective mass as a function of degree of polarization for different values of
rs, starting from the lower one r, = 2 to r, = 3, 4 and 5. . ........ . 97
viii
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
INTERACTING FERMIONS IN TWO DIMENSIONS:
EFFECTIVE MASS, SPECIFIC HEAT, AND SINGULARITIES IN THE PERTURBATION THEORY
By
Suhas Gangadharaiah
May 2005
Chair: Dmitrii Maslov
Major Department: Physics
In this work, we analyzed the anomalous behavior of a two dimensional interacting Fermi system. The first anomaly that has been investigated is related to the lowenergy property of Fermi liquids in two dimensions. The forward scattering processes, in the second order, lead to a log divergence in the imaginary part of the selfenergy on the mass shell. We found that this divergence becomes even more singular at higher orders and takes the form of a powerlaw, potentially leading to the breakdown of Fermiliquid theory. We deal with the divergence by resumming the maximally divergent terms at each order of interaction, with the result that the renormalized imaginary part of selfenergy is well defined on the mass shell. We also investigated in detail the contribution of the collective modes: zerosound mode (for a contact potential) and the plasmon mode (for a Coulomb potential) to the nonanalytic temperature (T2) dependent term in the specific heat. We found that the zerosound mode does not make any contribution to the nonanalytic T2dependent term in the specific heat, whereas the plasmons do make a nonanalytic contribution.
The second anomaly is the lack of experimental observation of a polarizationdependent effective mass as a function of the applied parallel magnetic field in a SiMOSFET sample. We investigated this experimental observation in the framework of an ix
Ncomponent model, where N is an additional degree of freedom (which in our case is the valley degree of freedom). This model gives the change in the effective mass between an unpolarized and a fully polarized state to be not more than 1 to 3%, a result consistent with the experiment.
x
CHAPTER 1
INTRODUCTION
1.1 Survey
The physics of twodimensional (2D) systems has been the subject of intense research both experimentally and theoretically, over the last few decades. Exciting physics such as Wigner crystallization of electrons at low electron densities [1] and the scaling theory of localization [2], were first predicted and later confirmed by experiments. On the other hand, novel phenomena such as the existence of a metaltoinsulator (MIT) transition [3], the quantum hall effect [4], the fractionalquantum hall effect [5], and radiationinduced zeroresistance states [6, 7] were discovered first and later given theoretical explanation. Many 2D systems have been described successfully in the framework of Fermiliquid theory. However, the validity of Fermiliquid theory in 2D has been a contentious issue. Fermiliquid theory is known to break down in ID. Anderson conjectured that even in 2D, Fermi liquid is destroyed for arbitrarily weak interactions [8].
The basic postulates of the Landau Fermiliquid theory state that, low energy
properties of a interacting fermionic system can be described similar to a noninteracting Fermi system [9]. This phenomenological theory makes specific predictions for the lowenergy dependence of thermodynamic and transport properties on temperature, frequency and other small energy or momentum scales. In particular, the predictions for specific heat and spin susceptibility are as follows. Specific heat, C(T) scales linearly with temperature T, for T * 0, whereas the spin susceptibility approaches a constant as T + 0. Over the past five decades, this theory has been enormously successful in describing the lowenergy properties of many fermionic systems, particularly 3He and also a large number of metals. The common features of a metallic Fermiliquid system include, the existence of a Fermi surface, a specific heat proportional to temperature, a constant spin susceptibility, a finite density of lowenergy single particle excitations and a conductivity which increases with decrease in temperature. This theory, though, has failed to account for the behavior of
1
quite a few heavy fermion systems and highT superconductors. The deviation from the simple Fermiliquid picture for the electrical conductivity and other features such as the thermal conductivity, the nuclear relaxation rate etc., in the normal state of highT, superconductors is quite perplexing given the fact that many of these materials have large Fermi surfaces [10, 11].
Issues related with the validity of FermiLiquid theory in systems which exhibit
metalinsulatortransition have also come under intense scrutiny. Twodimensional systems can exhibit insulating and metallic behavior depending on the concentration of impurities and the density of charged carriers. Among the many different theoretical approaches to understand 2D systems in either the metallic or the insulating regime, those based on the framework of Fermiliquid theory have been more successful, compared to other theories, in describing the transport and thermodynamic properties. Nevertheless, deviations from the Fermiliquid universality have been observed, such as the absence of spinsplit mass in a spinpolarized system and nonanalytic terms in the thermodynamic and transport quantities. Section 1.2 discusses nonanalyticities in the thermodynamic and transport properties, and addresses the important question regarding the validity of Fermiliquid theory in 2D. Section 1.3 discusses the metalinsulatortransition in 2D. The issues related to anomalous effective mass behavior have been taken up in Section 1.4.2.
1.2 Deviation from FermiLiquid Universality
Recent studies on the low energy behavior of Fermiliquid systems away from the critical point, reveal deviations from the Fermiliquid universality due to the presence of softmodes which are responsible for the nonanalytic terms in thermodynamic quantities like specific heat and spin susceptibility. These nonanalytic terms survive near the quantum critical point and are responsible for the breakdown of the HertzMillisMoriya (HMM) theory [12], leading to either a firstorder or incommensurate ordering (or both) [13]. Thus a study of the lowenergy physics of Fermi systems, which has its own inherent merit, acquires extra significance due to its implications near the quantum critical point [13, 14, 15].
Approaches to understand a system of weakly interacting fermions in 3D through perturbative techniques on a tractable microscopic model of shortrange interacting
fermions have added to a regular T term in the leading specific heat term and a constant term for the spin susceptibility as T  0, in agreement with the results of a Fermi gas [16, 17]. These results, when extended to arbitrary dimensionality D > 1, yield similar results provided the interaction falls off rapidly. The similarity in behavior of the Fermigas result and the Fermiliquid result obtained via perturbative calculations, stop at the leading order corrections. Perturbative calculations beyond the leading term in 3D obtain a nonanalytic T3 In T term [18, 19, 20] and in 2D a T2 term [21, 22, 23] for the specific heat, different from the regular T3 term as obtained for the Fermi gas. The nonuniform susceptibility, in terms of the momentum Q, gets a nonanalytic Q2 In Q term for 3D [15] and for 2D max(Q, T) [13, 15, 22, 24] instead of a analytic max(Q2, T2) as the nexttoleading term. Some of these predictions, especially on the specific heat behavior, have been confirmed by experiments. Specificheat measurements on bulk 3He have revealed beyond the linearinT term an additional T3 log T term [25], a similar T3 log T behavior was also observed in heavyfermion UPt3 [26]. The origin of the nonanalytic terms were attributed to nonperturbative spinfluctuation processes. Heat capacity measurements on monolayers of 3He adsorbed on atomically flat graphite yield a T2 correction to the specific heat in agreement with theoretical predictions [27].
For D = 1 the perturbation theory is singular and Fermiliquid theory is destroyed. This breakdown shows up already at second order in the interaction [28]. Apart from the low dimensionality there are other sources which can lead to the breakdown of Fermiliquid theory. Using weak coupling analysis one can show that for certain special shapes of the Fermi surface, for example, a Fermi surface satisfying the 'nesting' property (i.e., (p = (+g) the Fermi liquid is unstable. A 'nesting' Fermi surface can lead to the breakdown of translational invariance by the formation of a charge/spin density wave with a typical ordering vector Q. There have been suggestions that the nonFermi liquid behavior of highTc materials is due to the shape of the Fermi surface [29]. A dispersion relation in momentum space which has saddle points on the Fermi surface can also affect the low energy properties of fermions (the velocity is zero at the Fermi surface). A secondorder perturbation analysis reveals that the quasiparticle decay rate depends
4
linearly on energy or, in other words, is of the same order as the real part of the selfenergy [29], signaling a nonFermi liquid behavior. The validity of Fermiliquid theory in two dimensions has been a source of intense research since Anderson suggested that in two dimensions the Fermi liquid is unstable even for weak coupling [8]. His argument was that scattering of two interacting particles at the same point of the Fermi surface will cause a finite phase shift. This means that even in the limit of the lowest energies, two particles with opposite spin cannot occupy the same quantummechanical state, as is assumed in the quasiparticle picture of Landau's Fermiliquid theory. A second argument was that the presence of an antibound state (pole in the particleparticle channel) will lead to an instability of the Fermi liquid. The effect of phasespace shifts and the antibound state has been analyzed in detail with the conclusion that one cannot infer the breakdown of the Fermi liquid from these effects [30].
In this work we have considered a 2D fermion with a weak shortrange interaction potential. A perturbative analysis of such a system with a linearized singleparticle spectrum reveals a singularity in the imaginary part of the selfenergy at second order in the interaction [22, 31, 32, 33]. This singularity is present on the mass shell (e = (p) and is logarithmic in nature. The divergence is even more singular at higher orders and starting from the third order onward goes as a power law. An important point is that these divergences are present even for an arbitrarily weak interaction. A similar divergence on the mass shell is observed in 1D. This divergence is due to what is known as the infrared catastrophe, where an onshell fermion can emit an infinite number of soft bosons consisting of spin and charge fluctuations, leading to a powerlaw divergence on the mass shell [28]. At each order in the interaction, the divergences can be cutoff by invoking the finite curvature of the singleparticle spectrum, yet the series itself remain nonconvergent at low energies [34]. These result seem to justify Anderson's conjecture about the invalidity of the Fermiliquid picture in 2D. This, however, is not the complete analysis.
The divergence in the selfenergy which at each order in the interaction is due to the interaction of fermions with the upper edge of the particlehole continuum, when summed to all orders corresponds to the interaction of fermions with the zerosound mode. A
careful selection of diagrams that at each order are the most divergent, and resumming to all orders in the interaction removes these divergences, thus restoring the Fermiliquid picture. The resulting renormalized selfenergy with the divergences removed acquire features which indicate a deviation from the standard Fermiliquid behavior. Of these the most prominent is a nonLorentzian shape of the spectral function. The spectral function acquires a shoulder at the threshold for the emission of zerosound bosons and a nonanalytic wlw term in the real part of the selfenergy. In addition to the contribution to the wlwi term from the nonperturbative processes, the second order term in perturbation also contributes to the wlwl nonanalytic term to the real part of the selfenergy. A nonanalytic wiwl term in the real part of the selfenergy, in principle, translates into a nonanalytic T2 term in the specific heat.
The nonanalyticities in the thermodynamic and transport (for a disordered system) properties are due to what are known in a general sense as the Generic Scale Invariance (GSI) of the system [14]. Correlation functions away from the critical point generally decay exponentially and are shortranged with a characteristic length or time scale. On the other hand, near the critical point the correlations become long ranged and the decay is only powerlaw. Correlation functions with longrange order exhibit scale invariance: a change of scale, e.g., in length, represents a change in the correlation function by an overall factor. For cases where longrange order is obeyed, even for regions away from the critical point, the system is said to exhibit "generic scale invariance" (GSI). In simple terms, correlation function that are long ranged and belong to the class of GSI (also called soft modes) have their Fourier components oc 1/k (where a is some positive exponent and k is the momentum) for regions even away from the critical point. Correlation functions which are shortranged have the form oc 1/(ka + m), where m is a constant mass term. One such example is an O(N)symmetric Heisenberg ferromagnet which exhibits, in the ordered phase, nonanalyticity in the magnetization as a function of the magnetic field. The ordered phase contains gapless Goldstone modes and the corresponding correlation functions are longranged and belong to the class of GSI. These are responsible for the nonanalyticity in the magnetization. Other examples include the nonanalytic behavior of conductivity as a function of frequency. In a disordered system the leading term in the
6
conductivity is ao = ne2T/me (Drude result, where n, e, 7 and me are the density, charge, scattering time of electrons with impurities and the mass of electrons respectively) and the leadingorder weaklocalization correction is given by [36]
 0"0 d D 1 (1.1) 7rNF I Dq2 _ i'
where D, q, Q and NF are the diffusion constant, momentum, frequency and density of state respectively. For D = 2 this leads to a logarithmic correction and for 2 < d < 4, this results in a nonanalytic frequency dependence o(Q  0) = 1 + 13O(d2)/2 (1.2)
with 3 > 0.
The thermodynamic properties acquire nonanalytic corrections even for disorderfree, pointlike interacting fermions. Although the bare interaction is shortranged, the effective interaction at finite frequencies is longranged. At secondorder in the interaction U the effective potential is U = U2II( , q), where H(Q, q) is the dynamical polarization bubble, which for the momentum range
IQI/VF < q < kF is given by U= U2aDI , vFq
where aD is a constant and 1/q is the generic scale invariant term [22]. The long range interaction forces, prevent a regular expansion of the selfenergy in powers of w leading to a nonanalytic expansion in frequency. The nonanalyticity that arises in the selfenergy is transferred to the observable quantities thus resulting in nonregular subleading terms (Section 3.2.1). In a system with disorder, fermions interact with Friedel oscillations (these oscillations are due to the presence of a static local perturbation). This interaction falls off as rD with the distance from the impurity and are the source for nonanalyticities in transport properties(these are the interaction corrections, unlike the earlier example of weak localization correction).
Chubukov and Maslov [22] identified the nonanalyticities in the selfenergy for D = 2 with the three scattering processes, which are essentially onedimensional in character. These are the forwardscattering process with the incoming momenta nearly
7
the same, undergoing small transfer of momentum and two backscattering processes involving scattering of nearoppositemomentum quasiparticles, undergoing either a small momentum transfer or large "2kF" transfer. Secondorder perturbative analysis reveals a nonanalytic T2 behavior of C(T) arising from the backscattering processes , whereas the forwardscattering process, for a linearized spectrum, contributes an unexpected log singularity at the mass shell in the imaginary part of the selfenergy. As discussed earlier this divergence is the analog of a stronger powerlawsingularity in 1D and is dealt with by resumming the perturbative series by including the maximally divergent diagrams. An offshoot of this resummation is a collective mode. In the past, the role of collective modes in the nonanalyticity in the thermodynamic potential had been treated by assuming that the collective modes are free. Indeed, by invoking power counting arguments [21], it can be shown that the collective mode contributes to a TD term in the nonanalyticity. However, collective modes are not free excitations. The contribution of the collective mode to C(T) has been analyzed in detail in this work (Secs. 3.2.1 and 3.2.2 ). We show that the collective modes do not contribute to the T2 term in the specific heat in 2D.
In the next Section, we will discuss the validity of Fermiliquid theory in 2D systems which exhibit a metaltoinsulator transition.
1.3 MetalInsulatorTransition (MIT) in 2D
In the late 1970s, an interesting theoretical viewpoint emerged on the conductance of a 2D system. Following the concepts of Thouless [35], Abrahams et al. [2], developed a singleparameter scaling theory of localization (electronelectron interactions were not taken into account). The scaling equation being d In g(L)
dlng(L) /3(g(L)), (1.3) d ln L
where g = G(L)/(e2/2h) is the dimensionless conductance and is the scaling parameter, and L is the typical length along each of the directions. The asymptotic values of 3(g(L)) obtained via physical arguments can qualitatively describe the flow equation. That is, starting from a given conductance G(L) for a system of size L (volume LD, where D is the system dimensionality), the flow equation can in principle predict the conductance (for example, when the linear dimension L is doubled). (Of course it is implicit that the
material property, the density of disorder and the electron density, do not change). In the largeconductance (G > e2/h) regime, the conductance shows Ohm'slaw behavior with G oc LD2, thus 3 = D  2. In the other extreme with the electrons localized, the conductance gets exponentially suppressed with increasing length and / = L/(, where ( is the localization length. It is assumed that 0 varies monotonically between these two asymptotic limits.
The consequence of this theory is that a 2D system with L  oo00 is an insulator at zero magnetic field and zero temperature. For a finitesized sample, the scaling relation implies that an insulating behavior will be observed. If the 2D sample were dirty to begin with, the system would show exponential localization as a function of system size. If the sample were relatively clean to begin with, then the system on further increase in size would show logarithmic localization (in 3D, there is a critical disorder separating localized and metallic behavior ). The form of the 2D conductivity showing logarithmic behavior is given by
ae2 ln(L/Lo)
a(L) = a(Lo)  2 ln(L/Lo) (1.4) 7r2h
The above result obtained for zero temperatures can describe the conductivity at nonzero temperatures with a slight modification. According to Gorkov et al. [36] and Thouless [37], at finite temperatures, inelastic scattering introduces random fluctuations in the time evolution of an electronic state thus limiting the quantum interference necessary for localization. Typical lengths (cutoff lengths) beyond which the phase is not conserved are given by LTh = (DTin)1/2, where D is the diffusion constant and Tin is the inelastic lifetime. ( Altshuler et al. [38, 39] argued that for a collision event with small energy change AE < 7', rTi cannot be taken as the dephasing time; the dephasing time will be To ~ (AEri)2/n. ) The inelastic lifetime itself is temperaturedependent: 7T1 oc TP, with the lifetime decreasing with increasing temperatures. This temperature dependence of the inelastic lifetime is responsible for the temperature dependence of the conductivity. Thus Eqn. (1.5) the conductivity equation (in 2D) ape2 ln(T/To)
a(L) = a(Lo) + 22h (1.5)
9
acquires a nonanalytic log T temperature dependence. This result was almost immediately confirmed by experiments performed on thin metallic films and silicon metaloxidefieldeffecttransistor (SiMOSFET). For relatively clean samples, the resistivity showed the expected logarithmic increase with decreasing temperature and applied electric field [40, 41, 42].
In an alternate theory, Altshuler et al. [3] considered interaction effects in the framework of a dirty Fermi liquid in the regime when kFl > 1 (where kF is the Fermi wave vector, 1 is the electronic meanfree path) and TT < 1 (this is the diffusive regime, with
7 being the scattering time) and obtained similar nonanalytic temperature dependence for the conductivity. The conductivity obtained is expressed in terms of the Fermiliquid parameter F
e2(1  F) ln(T7)
22h = (1.6) where the factor F is small for weak screening. However, Altshuler et al. [3] did not consider localization effects (phenomena that can occur without electronelectron interaction). That is, the maximally crossed impurity lines in the current vertex that give rise to localization were disregarded.
Both correlation effects and localization effects lead to logarithmic temperature dependence of conductivity. The information about these effects can be disentangled by experiments performed in the presence of a magnetic field [43), since the correlation effects and localization effects show significantly different behaviors [44, 45]. With good agreement between theory and experiment, the nature of the conducting state of a 2D system was considered resolved.
Yet theoretical results, particularly those of Finkelstein [46, 47] were contrary to the general viewpoint. By performing fieldtheoretic renormalizationgroup calculations, he showed that the resistance of the system scales to weak coupling leading to a metallic state at zero temperature (his theory explicitly suppressed the maximally crossed diagrams that give rise to logarithms in the localization problem). Altshuler and Aronov [48] then showed that by considering spinorbit or spinflip scattering, the Finkelstein solution
10
to the resistance scales to strong coupling, and shows an insulating behavior [48, 49], thus paving the way for metaltoinsulator transition to be observed in D > 2 systems.
With the advent of better technology, it became possible to fabricate samples with mobilities an order of magnitude higher than the ones developed earlier. The earliest experiments which reported the metallic behavior of resistivity were performed on high mobility SiMOSFET's by Kravchenko et al. [50, 51]. Both metallic and insulating behaviors were observed in the same SiMOSFET sample by just varying the density of electrons. Since then, numerous studies have observed the metallic state in systems other than SiMOSFET's: p and nSi/SiGe [52], pGaAs [53], nGaAs [54], nAlAs [55]. Nearly all of these studies have reported the observation of some key features. Metallic temperature dependence of the resistivity was observed for densities above a sampledependent critical density n, in the ballistic regime TT7 1, showing almost an order of magnitude variation of resistivity with temperature. For densities below n, an insulating behavior was observed. The 2D system shows an unusually sharp response to the application of a parallel magnetic field. Conductivity is strongly suppressed with increased magnetic field, up to the saturation field, whereupon conductivity saturates. Interestingly, the field required for saturation of conductivity depends on the density of the 2D sample and is approximately equal to the field required for complete spinpolarization of the 2D sample [56]. Since the parallel magnetic field only couples to the spin degree of freedom and not with the orbital degree of freedom, the extreme sensitivity to parallel magnetic fields implies that exchange and spinrelated interactions play a crucial role. Despite agreement as to the general behavior of 2D samples on the insulating and metallic side, there are questions on which general consensus has not been reached. First, does the resistivity on the metallic side continue to decrease (dp/dT > 0) down to the lowest temperatures (i.e., no upturn toward insulating behavior at the lowest temperatures)? Prus et al. [57] have observed that the resistance of some curves with metallic behavior show an upturn upon further decrease in temperature, signifying a greater role played both by electronelectron interaction and weak localization; and signaling the possibility that the zerotemperature state of a 2D sample is indeed an
insulator. In contrast, Kravchenko et al. [50, 51], found no resistance upturn for SiMOSFETS with similar mobilities. The lowest temperatures they achieved were an order of magnitude smaller than those achieved by Prus et al. [57], indicating a zerotemperature MIT. The second question which eludes general consensus is whether the crossover from metal to insulator is accompanied by a transition from a paramagnetic to a ferromagnetic phase and if so, what is the order of this phase transition. This is discussed in Section 1.4.
The theoretical approach to understand the MIT has been dependent on underlying assumptions about the ground state of 2D systems. Next we briefly summarize two approaches based on the Fermiliquid picture and one not based on Fermiliquid theory, which we call the nonFermi liquid approach.
The nonFermi liquid approach. Chakravarty et al. [58] assumed that the ground state of disordered electrons is a nonFermi liquid. The singleparticle Green's function, when analytically continued to the lower halfplane, instead of containing a quasiparticle pole consists of a branch point. The spectral function used for the calculation was
1
A(w, k) oc (1.7) ,JW IVFkIla
where a is positive and wc is a highfrequency scale proportional to the inverse of the noninteracting density of states. For a < 1/2 they found that the nonFermi liquid state is localized; whereas for a > 1/2, they found that the disordered system is a perfect conductor, with a stable nonFermi liquid fixed point.
Dobrosavljevic et al. [59] took a phenomenological approach and assumed that at large conductance interactions can modify 0 and make it positive, leading to a metallic state. By assuming the form of the function 3 to be
O(g) = (d  2) + A/g', A > 0 (1.8) at large conductance, the conductance obtained showed a logarithmic decrease with increase in temperature
g  (ln(To/T))/a. (1.9)
12
They argued that this state cannot be a Fermi liquid because of the unusual temperature dependence of conductance and because the system will turn to an Anderson insulator once the interactions are turned off.
Fermiliquid approaches. Among the Fermiliquid approaches to describe the
resistivity of a 2D system, the most prominent are those of Altshuler et al. [3], Finkelstein [46, 47], Castellani et al. [49], and Zala et al. [60]. We have already considered Altshuler and colleagues' solution which predicts localizing behavior in the diffusive regime with kFl > 1. Next, we briefly review Finkelstein's solution in the diffusive regime and Zala and colleagues' approach in the ballistic regime, both of which predict metallic behavior.
* The Diffusive regime:
The range of electrondensities for which the system is diffusive (TT < 1) and
also show metallic behavior is rather limited. For densities such that p _ h/e2 and temperatures T < TF (where TF is the degeneracy temperature), the system lies in a diffusive regime. The temperature dependence of the resistivity in this regime can be described by the renormalization group (RG) equations developed by Finkelstein.
The RG equation relating the resistivity and temperature is
dp = 2n, +1  (4n  1)(1 + 2 +n( Y2)  (1.10)
 = 72 1+
where n, is the valley degeneracy, ( =  ln(TT/h), the dimensionless resistivity p is
in the units of h/e27r and 72 is the interaction parameter [61]. The coupling constant
also gets renormalized and its equation is given by dy2 (17+ 72)2
Sp (1.11) d 2
These RG equations [Eqs. (1.10), (1.11)] can describe nonmonotonic resistivity behavior as a function of temperature. Starting from hightemperature regimes
where the system shows insulating behavior, a decrease in temperature causes the
resistivity and the interaction parameter to increase (the interaction parameter
shows a monotonic behavior) until a critical interaction strength ' y is reached. At this interaction strength the RHS of Eq. (1.10) changes its sign. Further decrease
in temperature causes the resistivity to decrease and hence show metallic behavior.
This result was not taken seriously, since for n, = 1, 72 diverges at low energies.
The situation however is completely different for a twovalley system n, = 2 (SiMOSFET along the (001) plane) where the interaction strengths required are within
the perturbative regime.
13
* The Ballistic regime: Zala et al. [60] showed that insulating behavior in the diffusive
regime and metallic behavior in the ballistic regime are both governed by the same
physics: coherent scattering of electrons by the selfconsistent potential created by all the other electrons (Friedel oscillations). For the ballistic limit they were
able to show with the usual quantummechanical arguments that the scattering of
electrons by impurities and by Friedel oscillations created due to impurities can be coherent, provided the scattering is a backscattering event. As a result, the
amplitude of backscattering processes is enhanced compared to the case with no
Friedel oscillation. The correction to conductivity is linear in the ballistic regime and
is given by
A(T) = 12 + 3F (1.12) rh 2 1 + Fo'
where the sign of the correction to conductivity crucially depends on the Fermiliquid interaction parameter Fo in the triplet channel and near the Stoner instability
the corrections to conductivity is greatly enhanced. This result, essentially the
effect of a single impurity, when extended to include the case of multiple impurities
results in a conductivity correction that is logarithmic in temperature as obtained
by Altshuler et al. [3]. The nonanalytic temperature dependence of the conductivity
observed in both the ballistic and diffusive regimes arise due to the contributions
from long range correlations or the soft modes [14]. The soft modes which are
present in the dirty system have their analogues in a clean system as well and are responsible for nonanalyticities in the thermodynamic quantities like specific heat
and spin susceptibility.
Although there has been considerable progress in understanding the transport
properties of 2D systems in the ballistic limit, the question regarding the nature of the ground state of 2D electrons has not been resolved. This is because the conductivity can show various temperature dependences depending on the strength of disorder and also on the concentration of electrons; this can range from a purely Fermiliquid behavior with large conductivity to a strongly localized behavior (hence a nonFermi liquid) with exponential temperature dependence of conductivity in the dirty limit kFl
14
By fitting the experimentally determined conductivity with Zala et al.'s [60] theory, which implicitly assumes Fermiliquid ground state, Fermiliquid parameters Fo and m*(rs) were extracted [64, 65]. These were then compared with the parameters obtained independently from other experimental techniques, the most prominent being the Shubnikovde Haas oscillation in the conductivity [66, 67]. An excellent match between the Landau parameters extracted by these two independent approaches points to the Fermiliquid nature of the underlying state in the ballistic limit. At the same time there has been a interesting observation which does not fit well with the predictions of the Fermiliquid theory. The theory predicts that the effective mass is renormalized by the density and hence the Fermi energy. A similar renormalization should also be achieved by a parallel magnetic field since the field alters the Fermi energy and the distribution of electrons. The prediction for a field dependent effective mass is for a clean system, still one expects a qualitatively similar dependence to hold for a dirty system which shows Fermiliquid behavior. However no dependence of the effective mass on the field has been observed [68]. The issues related with the extraction of the Fermiliquid parameters from the Shubnikovde Haas data, the anomalous effective mass and the nature of the ground state of 2D electrons are explored in the next Section.
1.4 Quantifying Fermiliquid Parameters
An analysis of quantum magnetooscillation in terms of the LifshitzKosevich (LK) formula yields information regarding the quasiparticle effective mass m*, the Pauli (spin) susceptibility X* and the Lande g* factor. These quasiparticle parameters are related to the harmonics of the Fermiliquid interaction via the following relation:
FO  * 1, (1.13) where FO' is the interaction term in the triplet channel, and m*
Fs = 2(  1), (1.14) mb
F7 is the first harmonic in the singlet channel and mb is the band mass.
This analysis is based on the assumption that the LK formula originally developed to describe the magnetooscillations in 3D Fermi liquid, can be extended to describe the
15
oscillations in 2D systems by a change in the electronic spectrum and in terms of the zero field Fermiliquid parameters of the 2D system. Recently, there have been questions raised about the validity of LK formula in 2D [69, 70]. In the following we will discuss this issue in more detail.
1.4.1 Applicability of the LifshitzKosevich Formula in 2D
In 3D systems, it was shown by Luttinger [71] and also by Bychkov and Gorkov [72] that the presence of a magnetic field does not affect the zero field Fermiliquid parameters to leading order in 1/N , where N > 1 is the number of occupied Landau levels. The correction to the magnetooscillations obtained by including the oscillations in the selfenergy is of the order of N3 which is smaller by a factor of VN compared to the leading oscillatory part. Thus additional oscillations of the selfenergy can be ignored in 3D. The situation in 2D is different: the contribution coming from the oscillations in the selfenergy is 1/N2 smaller than the monotonic part and is of the same order as the leading oscillatory term in the magnetooscillation. Thus, for zero temperature the LK formula in 2D is not applicable for describing the oscillations [69, 70, 73]. On the other hand, experiments performed at finite temperatures seem to agree well with the LK formula, thus the range of validity for the LK formula requires a careful inspection. The amplitude of the kth harmonic of the magnetooscillation is given by k 2T 27rk[E, + iEo(ien, T)]) (1.15) Ak  Wc W
W, = W, ) (
En>O
where E, = 7(2n + 1)T is the Matsubara energy, w, = eB/mc is the cyclotron frequency and E is the selfenergy of electrons. The selfenergy for a generic Fermi liquid in the presence of shortranged impurities is given by iE0(en, T) = ae, + sgn(en)/27. One obtains a simplified form of the amplitude By expressing the effective mass as m* = m(1 + a), one obtains a simplified form of the amplitude
Ak = 4r2kT/w, exp( r2TD) (1.16) sinh(27r2kT/w;) we '
16
where To = 1/2wT is the Dingle temperature. For temperatures and impurity concentrations such that
272(T/w,TDlw) _ 1, (1.17)
the amplitudes of the oscillatory components will be exponentially suppressed, the suppression being stronger for higher amplitude terms [70]. A similar argument holds for the oscillatory part of the selfenergy, and only the first harmonic needs to be retained. The contribution from the first harmonic of the selfenergy oscillation to the magnetooscillations is of the same order as the second harmonic of the quantum magnetooscillation. This implies that the first harmonic of quantum magnetooscillation does not obtain any additional contributions from the oscillations in the selfenergy, hence, as long as only the first harmonic is the major contributor to the quantum magnetooscillation one can still use the LK formula for analyzing the data.
The above analysis was performed for the de Haasvan Alphen effect, however, it is conjectured that many of the arguments, especially, the validity of the first harmonic term also holds for the description of Shubnikovde Haas oscillations in the conductivity for large temperatures. Thus the quasiparticle parameters which get renormalized due to electronelectron interactions (within the Landau's Fermiliquid theory) such as the spin susceptibility X*, effective mass m* and Lande factor g*, can be extracted from the Shubnikovde Haas measurements. In this regard it is worth mentioning that in a recent experiment, an opposite phase of the second harmonic term as compared to the one predicted by LK formula was observed [74]. This can be attributed to the inapplicability of the LK formula beyond the first harmonic.
1.4.2 Absence of Polarization Dependent Effective Mass
The amplitude of the oscillations encodes the effective mass and the Dingle temperature, both of which are related to the amplitude via the following relation ehH 3Pxz
In px =m*(T + TD), (1.18) 2x2k sc 1 p
the slope of ln(pxx/p) vs T yields the effective mass, whereas the intercept gives the Dingle temperature. The amplitude and the oscillatory part are multiplied by a third term which
17
is also called the Zeeman term
Zk = COSTr2h(nlT  n) cos[7rkA* (1.19) eB1 hw*
where A* is the Zeeman energy and hw, is the cyclotron energy. The difference (nT n ) oc x*Bt,t, is directly proportional to the total applied field Btot and for fields with no parallel component Zk is a constant. The presence of nonzero parallel component causes Zk to be field dependent leading to beats. From the position of the nodes in the beat pattern X* is extracted and the ratio X*/m* gives g*. The nodes occur at regular halfinteger values of kA*/w* [66]. Perfect nodes implies that the oscillatory terms due to the upand the downspin electrons have the same amplitude and hence the same effective mass, a result in contradiction with Fermiliquid theory. Indeed, the presence of a magnetic field causes the Fermi surface to split into two, one for the upspin electrons and another smaller one for the downspin electrons. A consequence of this phenomenon will be that the velocities of upspin electrons will be higher compared to those of downspin electrons. Thus within the frame work of Fermiliquid theory it is natural to assume that there are three distinct interaction terms, an interaction between two upspin electrons (fast moving electrons), one between two downspin electrons (slow moving electrons and hence strong interaction) and the interaction between a downspin and an upspin electron (Section 4.1.1). These interaction terms in turn imply different effective masses for the up and downspin electrons. This intuitive expectation is confirmed by a perturbative calculation (Section 4.1.2). These results are in contradiction with the experimental findings on SiMOSFETs with a (001) surface, which report absence of field dependence of the effective mass and also indicate that the effective masses of up and downspin electrons are the same, mT = mi [68].
To resolve the paradox, we have turned to a large Nmodel. We have found the spinvalley degeneracy N plays an important role in the mass renormalization. Each of the N components shows polaronic behavior, with the mass renormalized primarily by the high energy plasmons. For typical r, = 2  6 (where r8 = xv/me2/kF) the percentage change in the effective mass between unpolarized and a fully polarized state is within 1  3%, a result which is consistent with the experiment.
18
1.4.3 Effective mass and Landeg Factor Near the Critical Densities
A number of groups have come up with interesting conclusions regarding the ultimate fate of X* and m* as the densities approach the critical one. Shaskin et al. [75] and Vitkalov et al. [76] have both reported divergence of X* in SiMOSFETS near the critical regimes, with the former arguing that the divergence in X* is due to the divergence in m* whereas the g*factor shows only a nominal increase. Vitkalov et al. however report the divergence of g* as the cause for the divergence in X*. These conflicting results point to different phases of the ground state on approach to the critical density: a divergence in m* points toward Wigner crystallization, whereas the divergence in g* points toward ferromagnetism. The mutually opposing results of the above experimental groups are in further disagreement with the results of a third group. Pudalov et al. [66] obtained an increase in both g* and m* near the critical density, however the increase in m* was by a factor of 3 and only a small increase in the value of g* was seen. Clearly the experimental evidence for the very existence of phase transition and its order is contradictory, nevertheless it remains an interesting point that if the phase transition does exist in 2D, what is the order of this phase transition.
The scenario of mass divergence near the MIT regime is similar to the one proposed by Brinkman and Rice [77], who showed using Gutzwiller's variational method [78] that at halffilling the effective mass diverges at the critical value of the interaction. This divergence is accompanied by the vanishing of the discontinuity in the singleparticle occupation number thus indicating a deviation from the Fermiliquid ground state. Galitski et al. [79] and Chubukov et al. [80] have proposed that at the critical interaction, the quantum critical behavior is associated with a densitywave instability at a finite wave vector q,. At the critical point, the interaction diverges for finite wave vector q, leading to the divergence of the fermion's selfenergy and hence the effective mass. It was further argued by Chubukov et al. [80] that in the near vicinity of the critical point the selfenergy has only frequency dependence, and the divergence of the effective mass right at the critical point and not before it indicates that the renormalization factor Z remains nonzero, thus the Fermiliquid state survives till the critical point.
CHAPTER 2
INTERACTING FERMIONS IN TWO DIMENSIONS
2.1 Introduction
The nature of the lowenergy electronic excitation in a 2D system has been of great interest, particularly since the discovery of high Tc superconductors. It was suggested by Anderson that forward scattering of two electrons leads to vanishing renormalization factor Z, i.e., to the breakdown of the Fermiliquid description [8]. Subsequently, the effect of forward scattering has been investigated by many authors within the framework of the tmatrix approximation, and an e2 log E term as the leading correction to the imaginary part of the selfenergy, ImE, was obtained. This result indicates that the real part of the self energy is still dominant compared to the imaginary part of the selfenergy and hence the quasiparticles are well defined [30, 31, 32].
In a recent work, it was shown that for fermions with a linearized dispersion, ImE
acquires a log singularity on the mass shell even at the second order in the interaction [22, 29, 31, 32]. In this thesis we show that the divergences on the mass shell at higher orders in perturbation theory become powerlaw and are more singular. Although the divergences in the individual terms are cut off due to curvature effects (nonlinear dispersion), the series itself does not converge. This result seems to support the Anderson's conjecture regarding the breakdown of Fermi liquid in 2D. However, this is not the complete picture.
At each order in the interaction, the singularity arises due to the interaction of
quasiparticles with excitations near the upperedge of the particlehole continuum. By resumming the singular perturbative terms to all orders, we were able to obtain a closed form for ImE. This form shows that the singularities are due to the interaction of quasiparticles with the collective excitation of the system, the zerosound mode (ZS). The resummation regularizes the powerlaw divergences whereas remaining log divergences are removed by introducing finite curvature. As a result Fermiliquid theory is restored. The outcome of the resummation process is nonperturbative contributions to the selfenergy, 19
20
one of them being a zerosound contribution to the imaginary part of the selfenergy ImEzs that is nonmonotonic as a function of the distance from the mass shell. The nonmonotonic behavior leads to a nonLorentzian shape of the spectral function.
The contribution from the nonperturbative zerosound mode to the real part of
the selfenergy ReEzs is strongly enhanced near the mass shell and is responsible for a nonanalytic U2e~I contribution. This result suggests that the nonperturbative processes contribute to a nonanalytic T2 term in the specific heat. This however is not the case. We discuss the zerosound contribution to the specific heat in Chapter 3.
This Chapter is organized as follows. We begin in Section 2.2 by introducing the
relevant scattering processes, these are forward and backscattering processes. Section 2.3 is essentially concerned with the selfenergy of fermions interacting with a contact potential. We show in Section 2.3.1 and Section 2.3.2 that forward scattering leads to the divergence of the imaginary part of the selfenergy on the mass shell at second and higher orders in the interaction. The procedure used for the resummation of diagrams at all orders is discussed in Section 2.3.3. This is followed by the calculation of the renormalized imaginary part of the selfenergy in Section 2.3.4 and the real part of the selfenergy in Section 2.3.5. The results from Secs. 2.3.4 and 2.3.5 are used to obtain the spectral function in Section 2.3.6. The spectral function obtained shows features that deviate from the usual Lorentzian function for a quasiparticle. The nonLorentzian feature originates due to the interaction between a quasiparticle and the zerosound mode which results in a nonmonotonic behavior of ImE; the effects on ImE are transferred to the spectral function. The nonperturbative processes for fermions interacting via a Coulomb potential also lead to nonmonotonic behavior of ImE, as discussed in Section 2.3.7. The final part of this Section, Section 2.3.8, discusses nonanalytic corrections to the tunneling density of states. We show that the tunneling density of states does not acquire a nonanalytic correction to order U2. We end the Chapter by concluding in Section 2.4.
2.2 Scattering Processes
Scattering events between two lowenergy quasiparticles of momenta k1 and kc2 can be divided into three categories. (1) Exchange of large momentum of the order 6k = Ik1  k21 oc kF, (2) exchange of small momentum 3k _ IIl/vF
21
k k
2 k' k' k k k
13[2 2 k2 la [ 2
A B C
Figure 21: Scattering processes responsible for divergent or nonanalytic selfenergy contributions in 2D. A) Parallelmoving quasiparticles scatter at each order by exchange of a small momentum. B) The quasiparticles are moving opposite to each other, initial and final momenta remain essentially unchanged. C) Each of the oppositely moving quasiparticles undergo momentum change close to 2kF. exchange of momentum near 2kF such that I6kI  2kFI  IQI/vF < kf. Scattering events of the first kind are responsible for analytic contributions to the selfenergy and thermodynamics(Section 2.3). Scattering events of the second and third kinds are responsible for nonanalytic contributions to the selfenergy and thermodynamics. The effect of the scattering events involving small momentum exchange or large 2kF scattering becomes more pronounced with the lowering of the spatial dimensionalty, since the relative phasespace for these events increases in lower dimensions compared to the higher dimensions.
From an earlier analysis of smallmomentum scattering events [22], it was established in 2D that the kinematics responsible for nonanalyticity in the selfenergy are ID in nature. These ID processes can be subdivided into two kinds: A) those in which the two quasiparticles move almost in the same direction and undergo small momentum transfer so that all four momenta are close to each other k1 k2  k'  k; this process is also called the g4 process and its diagrammatic representation is shown in Fig. 21A, and B) g2 processes which involves quasiparticles with antiparallel momentum undergoing small momentum transfer l  k2  k' E k2. The diagrammatic representation of a g2 process is shown in Fig. 21B. The third kind of scattering event responsible for the nonanalytic term in the selfenergy involves exchange of 2kF momentum and is essentially 1D in character due to the quasiparticle occupancy restriction. The 2kF process is depicted in Fig. 21C.
22
2.3 SelfEnergy: Nonanalyticity and Singularity
In this Section, we will first derive the imaginary part of the selfenergy to second order. We obtain two interesting results: a nonanalytic term in the selfenergy and a massshell singularity. We will consider both of these issues in detail. The Hamiltonian of our system is the same as for the Coulomb potential case, except that the potential is now short ranged.
Before we go into the details of the calculation, we will discuss here through some simple arguments the origin of the nonanalytic term in the selfenergy. An analytic selfenergy forms a regular series in e, the real part of the selfenergy being expressed as ReER(e) = ale + a2E3 + a365 +   , whereas the imaginary part of the selfenergy is expressed through the series ImER(e) = bl,2 + b2e4 + b3E' + .... An example of a nonanalytic term in ImER, which we will often encounter is e2 log I e. Let us consider ImER at T = 0,
ImR (e)  dw J dDqImGR(e  w, k  qImVR(w, q, (2.1) where ImGR and ImVR are the imaginary part of the Green's function and the potential respectively. From this form, it is clear that, if upon a momentum integration we obtain a w term, then one ends up with a regular e2 term after the remaining frequency integration is performed, whereas if the momentum integration yields w log Iwl (as in 2D) or w2 (as in 3D) then the frequency integration yields a nonanalytic E2 log eIj or e13 term respectively.
Now to be a bit more explicit, the potential term V(r, t) is real, thus ImV(q, w) is odd in frequency and we can express it in terms of a characteristic momentum Q of the system (which serves as an upper cutoff) in the following form
ImV(q,w) = F( q Iw), (2.2) Q' QVF,
where F is now a even function of frequency. The angular integration over ImGR yields on the mass shell (e = (k; where (k = k2/2m  k2/2m)
fdOlmG" = Jd 6(  V.+ q/2m) = 2 AD( +2/2m), (2.3) vFq vFq
23
where the subscript D stands for the dimension,
A3(x) = 0(1 IxI)
A2(~:>1The function AD primarily serves to impose a lower cutoff q Iw/vF and we can ignore the specific functional form. We plug the result of Eq. (2.2) and Eq. (2.3) back in Eq. (2.1) to obtain
ImER(E) dqq D2F( q v F. (2.4)
0 qw/vyF Q QUF
Now if the momentum integral is dominated by large momenta of the order Q, then the function F to leading order can be considered to be independent of frequency (since IwI/vFQ ~ e/VFQ < 1, as Q the upper momentum cutoff is large and e is small) and one can replace the lower limit by 0 as well. The momentum and frequency integrations decouple, and one obtains an analytic E2 term. Thus a process which involves large momentum transfers yields analytic terms. Let us consider a case where the decoupling between the momentum and frequency is not possible. Such a scenario takes place at the second order in the interaction for a contact potential. The retarded part of the effective potential is a constant times the polarization bubble,
ImVR(w, q) = U2ImIIR(w, q), (2.5) where InR has the following general structure in an arbitrary dimensionality
ImllR(w, q)= v D( ), (2.6) vFq vFq
where vD is the density of state in D dimensions and the dimensionless function BD imposes a constraint w < vFq. For D = 3, we obtain from Eq. (2.1)
ImE(e) U2 dw dqq2[] , (2.7) where we have neglected the constants, and the functions B3 and A3. The first square bracket represents the result from the angular integration, whereas the second one
24
a)
al) a2)
b)
bl) b2) b3) b4)
b5) j6) b7) b8)
Figure 22: Selfenergy diagrams with explicit and implicit polarization bubbles. represents ImfI. The upper cutoff is of the order of kF. Performing the momentum integration we obtain
ImE(e) ~ U2 dww k F I
0 kF FL FBeyond FL S a E2  bE3, (2.8)
where the first term originates from the large momentum transfer regime and is the Fermiliquid result, the subleading second term originates from the smallmomentumtransfer regime and is nonanalytic. For reduced dimensions, the effective interaction plays a more singular role with the result that the nonanalytic terms become the leading terms in the imaginary part of the selfenergy. In 2D, the nonanalytic term behaves as e2 log Je, a more dominant term than the Fermiliquid e2 term. In the Sections to follow, we will consider the origin of nonanalyticity in greater detail.
25
We will define the Fermion selfenergy (E) via the Dyson equation in terms of the exact (G) and bare (Go) Green's function G1 = Go' + E, (2.9) note that we have defined E with an opposite sign compared to Refs. [16, 17].
2.3.1 SelfEnergy at the Second Order
At second order in the interaction, there are only two nontrivial diagrams (Fig. 22), the rest of the irreducible diagrams in effect renormalize the chemical potential and hence are not important. For a contact interaction, V(r) = US(r), the two second order diagrams in Fig. 22 are related, namely, diagram (al) is 2 times diagram (a2). This is particularly easy to see in momentum space as diagram (al) = 2U2 fQ II(Q)G(P + Q) and diagram (a2) = U2 fQ fQ G(P + Q)G(P + Q + Q1)G(P + Q1) = U2 fQ II(Q)G(P + Q), where fQ is the integration over frequency and momentum and I(Q) is the polarization bubble without including the spin summation. In the retarded formalism, the total contribution at second order in the interaction is given by U2 I 0d ImGoR(ei,Pi)ImIIR(Q, k  pl) 2(e, k) = d'p1 d
(2r)3r o ~ dQ + E1  E  i x tanh T)+ coth T , (2.10) and for T = 0, the imaginary part of the selfenergy acquires the form
Im (, k) =  ( dQa d2qlmG2(E + Q, k + q)ImIR(, q), (2.11) where ImGR(e + Q, k + q) is
ImGR(e + Q, k + q) = r5(Q + e  ek+q), (2.12) with (k = k2/2m  k)/2m. We will be interested in the contributions coming from q QVF
26
whereas for q m 2kF we have
IR( ,q m _ (q  2kF q  2kF2 ii0+)21/2
R(, q)= i (~ F + 2kF 2kFVF (2.14) 2x 2kF 2k, 2kyvy
The above two results for IIR may be obtained by using the result of Stern [81]; alternatively, one can use the result for the polarization bubble in the Matsubara representation m 2.15)
IIH'(in, q)= 2 ~ (2.15)
where x = (qkF)2  q4/4 + m20Q and y = q2mQn, expand for small q
The nonanalyticities in Eq. (2.13) and Eq. (2.14) represent Landau damping and the Kohn anomaly, respectively. For Q < vfq damping of excitations with frequency Q and momentum q can take place only inside the particlehole (PH) continuum, similar to the damping of a collective mode inside the PH continuum. The polarization bubble near 2kF shows different nonanalyticities depending on whether Q < 2kF or Q > 2kF (where Q represents the transferred momentum). For Q < 2kF, the nonanalytic part IR cX iQ/(2kF  Q)1/2 and for Q > 2kF the nonanalytic part IR c< (Q  2kF)1/2. In what follows, we will calculate the forward and backscattering contribution at 2nd order.
Forward and backscattering contribution at second order. In an earlier work
[15, 22], the leading nonanalytic contribution to the selfenergy were identified to originate from processes with small and large 2kF momentum transfers. To logarithmic accuracy, contributions from the Kohn (2kF) anomaly to the nonanalytic term of the selfenergy were found to be identical to the g2 contributons. In the following, we will show that the nonanalyticity which originates from q m 2kF due to the dynamic nonanalytic term in the particlehole response function is responsible for the nonanalyticity in the ImER. The imaginary part of the polarization bubble near Q = 2kF+q < 2kF, is given by the dynamic term mQ/47rVfF lqjkF, thus the contribution to the imaginary part of the selfenergy from the Q 5 2kF region is
/U2 IQf _ Q 1 Im2(e, k) = 22 d d 2QS(  k+) 4, (2.16)
(27)2 o 4vF qIk
27
where for k = kF, (k+Q = (4k2 + kFq + q2)/2m + (2kF + q)kF cos 0/m. Performing the angular integration
Sd(a  bcos) = 20(lbl  lal).
V a2
QdQ = (2kF) dq
and keeping the most dominant terms (with typical q such that IqvF > ), we obtain
ImER2 (E,k) = (2 dQ Jr)IV dq(2kF)[ 4kq/m2 [4rv il U2 W
2 e2 log , (2.17) 8TEF E
where Um/27r = u and W represents the lower cutoff on the momentum integration, which is typically EF. The other nonanalytic piece of ImnlR oc (Q  2kF)1/2 = 4V which originates from Q > 2kF does not contribute to the nonanalytic piece in the selfenergy. The contribution to the nonanalytic term in the second order ImER [Eq. (2.11)] from the q = 0 region (we need to keep the nonanalytic piece of the polarization bubble near q = 0, i.e. m/2 vF/(vq)2  Q2 ) is expressed as
Im2(e, k) = ()2 da d2q6(E   5k+q) [  , (2.18)
(27)2 27V/(Vq)2  2] 2.18) where (k+q = k2/2m + q2/2m + kq cos 0/m  k'F/2m. As before, we perform the angular integration on the delta function, f mG =_ 27r
dI(vtQ)2  (Q + )2
using this result, we find that Eq. (2.18) acquires the following form, U2 2(f qd[ ImR(E, k) = (2r)2 JdJ qdq 2 (  q)
x 27rV(vq)2 _ 2 , (2.19) where (k = k2/2m  k)F/2m and A = e  (k has the interpretation of the distance from the mass shell.
28
For A = 0 and neglecting the curvature (eq = 0) we obtain
U2 eo~ k ImER(e, k) (2')2 dQ dq2@/2[  2j2
(2)2 0 /vF F (VFq)2  2 U2 eo [ m ] fokr 1
2 d  dx[ , (2.20)
i.e., an unexpected log singularity on the mass shell. The origin of this divergence is due to the form of the polarization bubble at small momenta which is square root divergent at 191 = VFQ and due to the divergent form of the Green's function obtained after performing the angular integration. Combining the two contributions and performing the d2q integration one obtains a logarithmic singularity due to the soft modes on the mass shell (A = 0). Higher orders in the interaction have larger number of particlehole bubbles and hence stronger singularities. These case will be considered in the next Section.
By keeping A finite, we obtain the following form mU2 E E2
ImE(e, k) = V(27r)3 IdQQ log IA(2Q A)I
mU2 A 2E E 2
(4r)3 2 og I A + 2e(e + A) + 4E2 log IA 2E mU2 A2 E 2 A2 E
 167r3v 102gK + F4 lo2 4 )log ], (2.21) and near the mass shell
u2 2 E, E,
ImE(e, k) = 8r2EF [log f+ log + k (2.22) It has been established that the two log contributions in Eq. (2.22) have different origins [22]. The first term is due to a process with two incoming particles of almost identical momentum undergoing small angle (q m 0) scattering (in 1D terminology it is the g4 process), whereas the second term is due to a process in which two incoming particles of almost opposite momentum undergo small momentum scattering (a g2 process). The following analysis will convince us that there are only two, q m 0 processes which contribute to the E2 log e term. We will reexamine the contribution to ImER
ImER c dQ qdqd06( + A  kq/m cos 0) Q
29
where k is the momentum of the incoming particle. The imaginary part of the Green's function yields a 1/q term and for kFq > Q the imaginary part of the polarization bubble yields a 1/q term, the 1/q2 term obtained is responsible for the nonanalytic term. The above expression reduces to
0oq kq
S d jVn d6 ( + A)  cos 0 ),
for small Q and A typical angles near ï¿½r/2 will yield the nonanalytic E2 log e term, this is the angle between the incoming momentum k and the transferred momentum . A similar relationship exists between the momentum transfer and the incoming momentum # of the "other" electron. The information about the "other" electron is in the imaginary part of the polarization bubble and is given by
ImlIR(Q, q) oc pdpd{5(Q  eg  pq cos)  6( + q  pqcost)} = P dpd { 6( Eq  cos6)S( E cos4) q Poq pq
oc (2.23)
 Q2
thus for pq > Q, we find ImIlR cc Q/q (the 1/q form of the polarization bubble is crucial for the nonanalytic term), clearly the typical angles involved in this contribution are
0 : ï¿½7r/2. Thus both the fermionic momenta f5 and k are perpendicular to 7, implying two processes, g4 and g2, are responsible for the nonanalytic contribution to the selfenergy from q M 0 regimes. In this work, we will classify g2 as a backscattering contribution since it involves scattering of two incoming momenta with total momentum close to zero. For a contact interaction, the total backscattering contribution which is a sum of g2 and 2kF processes is given by
u2e2 EF
Im2 (e) = E2log (2.24) S 47rEF I (2.24)
Since there is no massshell singularity from the backscattering contribution, we do not need to consider higher order terms and hence we can stop the series at the second order in the interaction U. We will, however, pursue higher order contribution from forward scattering process because of the massshell singularity in this channel.
30
2.3.2 HigherOrder ForwardScattering Contributions
In an earlier analysis, we have established that the singularities from the soft mode (q  0) in the particlehole bubble causes a log singularity of the ImER at the second order. At higher order in the interactions, these singularities proliferate leading to a powerlaw divergence of ImE R. The third order diagram of the selfenergy consists of a maximum of two polarization bubbles, having the same momentum and frequency (Fig. 22),
(II )2 (.2[ 1 + i 2
2(HR)2 ) v(vFq)2  (Q + iO+)2
= ()2 2Q 1
27 (vFq)2  (Q + iO+)2 ((vFq)2  (Q + i0+)2)2 Thus at the third order, the most singular term in Im(IIR)2 is given by
2
Im[IIR]2 =  [a21aQ6(j 2  v2q2)]. The singular term at the third order is given by
Ea 0 d qdq 27r
Im3,F(e, k) = U' 2 (2)2 fT 2 () U(VFq)2 _ (A + )2] x [m2Q 6(Q 2  (qvF)2)] U3m2 O Q I&I 327rvF J 2 (A+ )2 327r3vJo0 (2  A)A S3 E () (2.25)
where in the last step of the above equation the singular 1/VA4 term has been extracted. The additional e function signifies that the expression is zero if A and e are of the opposite sign. At higher orders we obtain imER, V UnEn/2+1
In, oc ,, 8 (2.26)
31
for n > 2. Collecting forwardscattering contributions to all orders in n, we obtain
Im () lo ) C (2.27)
n=1
where C, are the numerical coefficients. This series converges only for (u2 /A) < 1, outside this regime the series will not converge. Thus there is a clear need for resummation of the perturbative series. This is the topic of our next Section.
2.3.3 Resummation of ForwardScattering Contributions
A proper resummation of the selfenergy series involves keeping at each order in the interaction the most singular terms. We find that such terms correspond to diagrams with the maximum number of particlehole bubbles (with small frequency and momentum). A convenient way to resum the contributions will be to first identify the singular fourfermion vertex F and then relate the fermion vertex to the selfenergy E via the Dyson equation.
Fourfermion vertex. The diagrams that form particlehole bubbles are of two kinds (for a short range interaction): the ones which have explicit bubbles and the others with implicit bubbles (those which are not easily identifiable). These explicit and implicit particlehole bubbles also appear in the selfenergy. A typical example is the 2nd order selfenergy where we consider two bubbles, one of which has an explicit bubble Fig. 22(al) and the other an implicit bubble Fig. 22(a2). An explicit bubble comes with an additional factor of 2 from the spin summation. Due to these differences, the procedure of finding an overall prefactor at order v is rather involved, as it requires counting the number of diagrams at the same order in a conventional diagrammatic technique operating with a nonsymmetrized vertex, F. Once we obtain the nonsymmetrized vertex, we antisymmetrize it, and use the result in the Dyson equation to obtain the selfenergy. In the following, we derive an expression for the forwardscattering part of the nonsymmetrized vertex, F, summing up diagrams with the maximum number of polarization bubbles to all orders in the contact interaction, U. The diagrams for a nonsymmetrized vertex up to third order are presented in Fig. 23.
32
a ,a aa E
Figure 23: Second and third order Vertex diagrams with maximum number of particlehole bubbles. Additional diagrams, obtained from those in the second column by a permutation a + /3, + e, are not shown.
33
In the Matsubara technique, we associate a factor of U with each of the interaction lines, and a factor 2 with each of the polarization bubble. There is also an extra factor of 1 for exchange processes in which the two outgoing legs are permuted (the last diagrams of second and third order in Fig. 23). We present here a general recipe for calculating the uth (v > 1) order vertex diagram. The vertex consists of two parts (Fig. 23). The first part comes from the direct interaction and contains a spin factor dba pe. The second part is due to the exchange interaction, and comes with a spin factor dco56. At each order, there is only one exchange diagram whose contribution is (U)"Iv1'6 Sq (At second and third orders, these are the first and third diagrams in the third column of Fig. 23, respectively).
The rest of the diagrams are due to the direct interaction and contain various number of bubbles. At order v (number of interaction lines = v), the number of bubbles in the diagram (both explicit and implicit) will be v  1, thus the number of explicit bubbles or the rings (R) can vary from 0 to v  1. We consider a diagram with R rings and v  1 interaction lines, for this diagram R + 1 interaction lines will be used up in connecting the rings (we are considering connected diagrams) and also in connecting the rings to the two external solid lines. The remaining N = v  R  1 interaction lines can be arranged anywhere either at the two ends of the chain of bubbles (or in other words, dressing up the two interaction lines which connect the solid line) or inside the R ring bubbles. Thus there will be S = R + 2 sites (two ends and inside the R rings) where N interaction lines can be placed. The number of diagrams with R ring diagrams is equal to the number of ways to arrange N lines among S sites:
(S + N  1)! v!
= (2.28)
(S  1)!N! (R + 1)!(v  R  1)!'
Consequently, the contribution to F from diagrams with R bubbles is v!
(U)"(U 2) " 60, (2.29) (R + 1)!(v  R  1)!
The total contribution from all bubble diagrams at the order v is then
S V!(2 )R (1  (1)
(R + 1)!(v  R  1)! 2 U"IIH~6S (2.30)
34
where we have used an identity v!(2)
= ( ) (1  2)" = (1)". (2.31) =R!(v  R)!
R=0
Adding up direct and exchange terms, we obtain the following form for the nonsymmetrized vertex at order v > 1:
r", , = ~(1  ()")U"HP16,~6  (U)"II"1le6. (2.32) The vertex function can now be readily summed to all orders, with the result
ra~,e(PI,P2;P1  q,P2 + q) = raf,,E(q)
= USc6a3 + 5 ao'Yc v=+
U U2I(q)
= 6,6p4, ( 5,6S,  (2.33)
1  (UII)2 1 + UII(q)
where by including the additional factor U6c, , we are able to extend the summation of the vertex function to v = 1. Using an SU(2) identity
560, = (1/2) (,a, , + , (2.34) and introducing the dimensionless spin and charge vertices
1 1
2 1 + UII(q)'
1 1
9p = 1 1 (2.35)
2 1  UII(q)'
we obtain for the nonsymmetrized vertex the following form
fap, E(PI,P2;P1  q, P2 + q) = r(q)
= U [ , + + + a e + e . (2.36)
Finally we antisymmetrize the vertex by the following procedure
Foao;ye(q) = raF;.Y (q)  ra;E (q). (2.37)
35
For a repulsive potential the retarded charge vertex, G, has pole at the zeros of 1 UIR(q, Q) = 0. The pole exists even for an infinitesimally small U, since for I Q > IvFQI, IR is real and can be arbitrarily large as I1  (vFQ)+.
27r n2  (vq)2
/2 = 2 2
+U (1 + (1 + 2u) =
The collective mode which corresponds to the shortrange interaction is called a zerosound mode whose dispersion is Q1 = cQ and the zero sound velocity is
2 )
c = vF 1 +  VF(1 + u2 1 + Um F( 2
where the expansion is valid for u << 1. The charge vertex exhibits singular behavior near the zerosound mode and has the following form
1 1 P'ZS 2(Q  cQ) [1  UII(Q, \ n=CQ + 2(Q + cQ) [1  UII(Q, Q)] O=cQ (u2v Q2)
(Q + iO+)2  C2Q2ï¿½
For small values of u the zerosound mode cQ is just above the upper boundary of the particlehole continuum, given by VFQ. The residue of the charge vertex is not a constant and is proportional to Q2, thus the quanta of the zerosound mode are not free. The spin vertex 0g on the contrary has a pole in the imaginary axis and remains overdamped.
Dyson equation. The selfenergy due to forward scattering is related to the vertex function via the Dyson equation [17]
EF,a (p) = 3 j UG(p  q)  UFyal (p, p' + p"  p; p', p")G(p')G(p")G(p' + p"  p), (2.38)
where
..  T d2k/ (27)2... (2.39) Although the Green's functions in the Dyson equation are exact ones, we can safely ignore the selfenergy insertions into the diagrams which diverge near the mass shell as these insertions do not lead to any additional singularities near the mass shell. As we keep
36
u=(Um/27r) small, regular corrections are thus irrelevant, and we can safely use bare G's instead of the exact ones in Eq. (2.38). Substituting Eq. (2.37) into the Dyson equation (2.38), we obtain for the selfenergy
EO,(p) = S, lUG(p  q)  1 Uya;,OG(p')I(p  p') + j UF,,;,G(p")II(p  p")
= ,, UG(p  q) + a, 1 2(UII)2 + U3 (p')fI(pp')
J, 1 (UII)2 + 1 I
 5aPfUG(pq)ï¿½+ 6 1(UI(q) G(p  q) / U*II(q) Ir
643 G(p  q). (2.40) Rearranging the result, we obtain the selfenergy in the following form EF,a = 860EF where
~~fq[uu2H( 1 U312 (q) 3 U3I2(q) 1
EF (p) = U + U2II(q) + 1 U3II (q) 3 U3 12 (q) G (p  q). (2.41)
2 (1  UI (q)) 2 (1 + UII(q))
We remind the reader that in the perturbation theory ImER diverges upon approaching the mass shell logarithmically to second order in U, and as 1/V  k to third order. It is convenient to rearrange the terms of Eq. (2.41) and decompose EF into three parts making use of the charge and spin vertices, introduced in Eq. (2.35), as
EF (p) = Ep (p) + E, (p) + Eex; (2.42a) Ep,(p) = U 9p (q) G (p  q); (2.42b) E, (p) = 3U jQ9, (q) G (p  q); (2.42c) x = f [2U  U2II(q)] G (p  q). (2.42d) Terms Ep and E, correspond to the interaction in the charge and spin channels, respectively, and are summed to all orders in U. The remainder, Eex, contains extra contributions of the first and second orders in U, which are not included in the spin and charge terms and are required to reproduce the correct expansion of the first and second order term of the selfenergy.
37
A similar expression was obtained for the selfenergy in the paramagnon model (the spinfluctuation model) [20], although the contribution from the remainder term, E~, was not taken into account. The paramagnon model includes only two kinds of diagrams, RPA and ladder diagrams corresponding to the charge channel and the spin channel, respectively. In contrast, our choice of diagrams is based on a broader consideration of including the maximum number of particle hole bubbles at each order which for forward scattering leadto massshell singularity, thus we have included nonRPA and nonladdertype diagrams as well. Including the maximally divergent diagrams also means that the perturbative expansion near the mass shell is in terms of the diverging parameter ((Um/2x)2e/A) and any expansion which involves the regular Um/2ir is neglected. After the complete resummation, it turns out that the expression for the selfenergy is very similar to the one obtained in Ref. [20].
2.3.4 Renormalized Imaginary Part of the SelfEnergy
The imaginary part of the selfenergy acquires nonzero contributions from the
regions in momentum and frequency space where the effective interaction has a nonzero imaginary part. There are only two such regions, one of them at the collective mode I I = cQ where the charge channel !G has a pole, and the other region where ImIn j 0, i.e., where particlehole excitations are permissible. All three termsthe chargevertex part, the spinvertex part and the extra term acquire contributions from the particlehole region, whereas only the chargevertex part has a contribution from the collective mode excitation. We can represent the imaginary part of the selfenergy as follows ImER = ImER + ImER + ImER
= (ImER + ImER + ImER)pH + (ImER)zs, (2.43) where the subscripts PH and ZS stand for the particlehole and zerosound contribution respectively. In the following analysis we will investigate the forward scattering contribution from the particlehole excitation and the zerosound mode, and show that ImE is completely renormalized.
38
Particlehole contribution. The first term of Ee is real [Eq. (2.42d)] and does not contribute to the imaginary part, whereas the second term differs from the second order selfenergy term, ER, by only a sign. Thus we can immediately infer ImER from the forward scattering part of ImER [Eq. (2.22)], imR 2R U2m2 e2 EF
Im = Im = log . (2.44) 32"3 EF I
The imaginary part of EP and E, in the retarded formalism is given by aE d d2
ImE = U  ImGR(e + Q, k + q)Imgp (2.45) ImhE = 3U jImGo(e + Q, k + q)Img, (2.46) where for q < PF, ReIIR = m/27r, thus for weak interaction Um/27 < 1, the terms ImGp and Img, will be equal in the leading order of interaction
1 UImiR 1 UImIIR
2 (1  UReR)2 + (UImIIR)2 2 1 + (UImIIR)2
1 UImlIR 1 UImIR Im9, =
2 (1 + UReIIR)2 + (UImIIR)2 2 1 + (UImIIR)2' Substituting the value for ImGR and ImIIR from Eqs. (2.12) and (2.13) in Eqs. (2.45) and (2.46) we obtain
ImGpH = ImEP + ImE,
2 EF 2A
 47 [log + GI \ (2.47)
where
G,(x) = 2 log 2  1/2 + log Ix  2Re zdz log 1 + z (2.48) x/z 1 1x/z In the limit x2 >> 1, the scaling function behaves as GI(x) , log ix'1 and hence ImER U2 2 EF
4 E log (2.49)
39
In the opposite limit Ixl << 1, function GI(x) vanishes as xl In x . As a result, net ImEpRH remains finite at A = 0, and for Ixi < 1 (i.e., for JAl < u2e) it behaves as ImER = Un EF 2A In jA' (2.50) m 7PH 4 E, u2E UE u2e 2j Comparing the limiting forms of Eq. (2.49) and Eq. (2.50), we see that higher order terms in u simply cut the logarithmic divergence in ImERH for JAl < u2leI. To logarithmic accuracy, one can then approximate ImEpRH by IMR = 2 2 EF
PHIm In (2.51) 47PH 4r EF i w'
where w  max (AI, u2lel).
Zerosound contribution. Unlike the particlehole region, which contributes to all the terms in Eq. (2.43), the zerosound mode makes a contribution only to ImER:
ImERs = U ) ImG R(E + Q, k + q)Im9pzs, (2.52) Im ZS = UJ R (2)2 (2.52) where the subscript p  ZS, stands for the contribution to the charge channel from the zerosound mode region. The collective mode Imgpzs has a pole given by U(1)2v2.Q2 \ U3n2 UImg _zs = Im 2 Q [s(Q  cQ)  6(Q + cQ)] (2.53) ( + i0+)2  c2Q2 87r We will plug the form of interaction given in Eq. (2.53) into Eq. (2.52) and perform the angular integration on GoR to obtain
ImER d _ QdQ 2r
pZS E (2w)2 (.Q)2  (Q +A )2 m2U3Q
8 [(  cQ)  6(Q + cQ)] m2U3 61c Q2dQ (2.54) 163 0 /(vQ)2  (Qc A)2' near the massshell QcA > A2 and c2  v = (Um/27r)2. After some manipulations, on Eq. (2.54), we obtain
i 2U3 e/c QdQ m2U3 e 2a c QdQ
Im m)2 1673 Re (2.55) 167 0 2  16 T Fo 2 Um2 27 F 2
40
4
SF(x) ..2
x
Figure 24: Scaling functions F, and G1 as a function of x. Note the strong asymmetry of FI about x = 0.
The above result was for E > 0. For e < 0 we have
m2U3 fEQdQ/C
Im = 3Rei QQ(2.56) pZS 1673 o  (Um)2 Thus for any E near the mass shell, the contribution to the imaginary part of the selfenergy is given by
ImERzs _ 2U3 Re Ji QdQ
* 1670 R o gn(E)  (u)2 VS QVF 27 rm2U2E2 2 7min{1,V } 4 (2.57) 167 3EF (2.57)
The integral of Eq. (2.57) can be performed using standard Mathematica software, the result is
FI(x) = 2 min y4dy 37r2 2/8, for 0 < z < 1;
ox2 sin_ x + 1+1)] , for > 1.
(2.58)
Here x = 2A/u2e, and for x > 1, Fi(x) . 2r/5v/. From the plot of F(x), Fig. 24, it is clear that ImERs remains finite in the entire range of x. Combining the asymptotic value of Fi(x) and Gi(x) for x > 1, we recover the 3rd and higher order results obtained via the
41
perturbative process,
P H 2 VEF 20E0 E3
[v'u33E2
20EF
It should be noted that the above form of the third order selfenergy is valid only for regions far away from the mass shell.
Unlike the perturbative case, where the regions near the mass shell are inaccessible due to divergence, no such divergence takes place for F(x). At IAI = u2eI/2, Fi(x) has a maximum and on the mass shell (A = 0), ImEfs vanishes. The vanishing of ImEis on the mass shell is due to the Cherenkov type restriction where the quasiparticle with velocity VF < c cannot emit a zerosound boson, hence the zerosound mode cannot decay the quasiparticle. The asymmetry of FI(x), F(z) = 0 for x < 0, can be understood if we refer to Eq. (2.54), for E > 0 the integral will have nonzero contribution as long as A > 0. For a detailed analysis let us consider Eq. 2.52. The delta functions originating from ImER and ImG, impose severe restrictions on the phase space for the emission of bosons. We find the following condition should hold for nonzero contribution to ImEzs or for the emission of Bosons (in the following we will use c m vF(1 + u2/2))
E _k = VFQ COS
1A
1 < cos0 = u2 < 1
2
u29
0 < < A < 2 < 2E.
20<
4e 2  2E  2
This condition clearly suggests that E and A should have the same sign. To study the decay of a quasiparticle, we consider a fermion with energy e such that U21EI/2 < A or x > 1, in this case all Q's satisfy Q/4e < x/2. As x is increased further the condition x/2 < Q/ue imposes constraints on the lowenergy bosons, thus fewer and fewer bosons are able to contribute to the decay processes, and hence FI(x) reduces. For A such that u2 /2 > A, i.e, x < 1 decreasing A (or x + 0) implies the frequency range available to the
42
bosons that contribute in the decay process shrinks, thus Fi(x) decreases. This explains the nonmonotonic variation of ImERs as a function of the parameter x = 2A/U2E.
An important thing to notice is that the contribution from the zerosound region ImERs is small compared to the contribution from the particlehole region, ImZE F,(x)
ImERH log EF IE'
since Fi(x)  1, ImERH is logdominant with respect to ImEZs.
Finitecurvature effects. From the analysis thus far, we conclude that the divergences present from the 3rd order onwards in the perturbative expansion of ImER [Eq. (2.27)] have been regularized by resumming the series. Nevertheless, there still remains a log divergence at second order due to the extra term ImER, which must be cut off due by the finite curvature of the fermionic dispersion. The logsingularity is cut at q2/2m A = E2/EF, and we obtain from Eq. (2.44)
#2 E2 Er
ImEx u2 log E (2.59) 47r EF El "
Thus all the terms of Eq. (2.43) are renormalized and ImE is finite at the mass shell. A natural question that arises is: is it possible to remove the massshell divergence in the perturbative expansion of ImER [Eq. (2.27)] without resorting to resummation and instead removing the divergence by including the cutoff due to finite curvature? Let us insert the cutoff A, = E2/EF in Eq. (2.27). We obtain 22 E_ m u2En/2
Im 4EF [log( 6 ) + EC ) . (2.60) n=1
This series does not converge for 1 < we, where w, = U2EF, thus the finite curvature does not lead to the convergence of the series and the need for resummation remains.
For A > Ac curvature effects can be totally neglected (this is the regime that has been considered in our work) and the results for ImRs and ImpH will remain the same as has been derived by the resummation procedure. For A < Ac the results will change, since Ac can no longer be ignored in the integrations. To obtain the functional form all
43
one has to do is replace A by A, in the functions
F,"' + FI,2 , = F ( ) u26 4 2E GI(  GI ( 2A=, GI (2) U2( 4 62E c W We will use this information to obtain ImEpH and ImCEs on and near the mass shell. Let us consider the limit AI < Ac. Following our earlier analysis we will make the necessary changes ImEH (JAI)  ImEH (IAl), obtaining imE 2E2 EF 2A, = 2E [log + G,21 lJ 4x 4E EF + As a reminder the scaling function: Gi(2EI/wc) = GI(x) behaves as xln xl, for x ï¿½ 1, IEI E, GI(x) = in , for x < 1, l < Ec (2.61) In , for x> 1, Ie > w. From this result we obtain S2IEn T, for e < w,; ImPH = 1 4E2F I fI E I (2.62) 2EF2In le, I There is no such scaling relation for ImrE (A) + ImER (Ac) and the functional form remains the same for either limit, e > w or E < we, ImR 2 U22 EF 47rEF log The zerosound contribution to ImERs obeys a scaling relation governed by the scaling function F(x). For x < 1, FI(z) scales as 372x2/8 and for x > 1 scales as 27/5yv, this then implies
aImEY u22 fo2 i2 Im _ =f;(2.63) 45E >W
44
Adding up contributions from all forward scattering terms, the final form reads R"2 log U2 i, for J wi,;
ImER = ImEH + ImEzs + Im = 4EF(2.64) 2F log , >> e.(264)
Notice since Fi(x)  1 for all ranges of x, the contribution from ImERH and ImE. will be dominant by a large log term compared to ImER and has therefore not been included in Eq. (2.64).
ImER on the mass shell: final result. The leading contribution to the ImER comes from both forward and backscattering contributions. We have successfully renormalized the forwardscattering contribution and as discussed before no such renormalization scheme is required for the backscattering contribution, so the total contribution is
U2 2 EF for eI < .;
ImER = ImER + ImE E In (2.65) 2xrEF Ie 1,  > ei >>
Thus near the mass shell ImE behaves as the usual e2 log e l.
2.3.5 Renormalized Real Part of the SelfEnergy
In the following analysis, we will derive the real part of the selfenergy near the mass shell from the backscattering process by making use of the KramersKronig relationship between the real and the imaginary part of the retarded selfenergy. A detailed calculation of the zerosound contribution to the real part of the selfenergy is also presented, where instead of the KramersKronig relationship we have considered the explicit form of the ReE in the retarded formalism.
Back scattering. In an earlier analysis we obtained ImE.R' and ImE'. A sum of these two backscattering contributions is ImR() U2 e2 W
E) log (2.66) IIEB 4r EF IE(
where W is an upper cutoff typically of the order of EF. We will make use of the KramersKronig relation to obtain the real part of the selfenergy from the imaginary
45
part.
1if ImER(E, k = E) MU262 W W 1 E2
ReR(e) = dE mU22P lim dElog 1 +
S j Ee 87r4v Wo Jo E E2  E2 Focussing only on the cutoff independent and nonanalytic term, we obtain
ReER((E) U2E P lim dE log W (1 + E2
87r4F Woo E E2  E2 mU2 i61 W dx log x 8U24 W 062
(2.67)
8 EF
Forward scattering. The real part of the selfenergy consists of three contributions: from the remainder term (ER), from the particlehole continuum (ERH), and from the collective mode (ERs). The contribution from the remainder term and the particlehole term is extracted similar to the previous approach, that is through the KramersKronig relation, we obtain for the remainder term Re 8 = U2 EI + O(A2 log A) (2.68)
8 EF
and for the particlehole term
ReEPH =u (2.69)
8 EF 4 EF
adding up the contributions from Eqs. (2.68) and (2.69) we obtain
ReER + ReERH  A (2.70) 4EF
Forward scattering: zerosound contribution. In this Section, we will evaluate the
nonanalytic EIEI contribution from the zerosound mode. The explicit form of ReE in the retarded formalism is
ReR(E,P ) = d2l f dw de1 ImGR(el,Pi)ImGR(w,,p  p1)
(2 )2 7o oo 27 w + E  E tanh E + coth ,
T T 2T'
46
where ImGR is given by Eq. (2.52). Following the usual angular integration, we perform the dp1 integration utilizing the delta function term from Im9g. This yields
ReER(e,p) a dw de W2 tanh l + coth ) a .o )/()  (1 + A)2(W + El) 2T 2T
 af0d d.El )2 i A)2(w . ) tanh E& + coth . where a = u2U/47r. For T = 0, the trigonometric functions impose constraints on El and w. Taking the zero temperature limit,
ReER(e,p) = a dw de ( 2 2
o 1 )2  (E + A)2(W El1) l r 2w2 +aj dwj del 2
oo ()2 _ A)2( + E) = ajdu dE 2w2 o0 fE 2 1) d ) )2 _El +N)2((, + El)
Since we are working in the near vicinity of the mass shell the condition e > A holds, thus
ReER(e,p) = af dw del 2w2
o E 2 1 2 1)
= a dw jE del A)2(w+1)
0 0 2  2 +a f dWfe de2
0 ( )2 ( E TA)2(w  E1) = I +12 .
Evaluation of both I, and 12 integrals is similar, we will choose to integrate I2 and simply state the result for 11. We impose an uppercutoff p for the frequency integration (this constraint represents the damping of zerosound modes at very high energies), and a lower cutoff at aC to keep the integral real, contributions from the high frequency regions will be
47
1.6
1.2
X
Crx
LL
0.8
0.4
0
1 0.5 0 0.5 1 1.5 2
X
Figure 25: Scaling function FR. Note the strong asymmetry of FR about x = 0. irrelevant and will not be considered.
f0 _ dw 2w2
2 = E dE1 1 2w2
2o A/ ()2 _ ( + A)2(W  1)
2c e 2 2 log 4
[ E2dCE1 log 1  )2  (p/C)2l 1
o ,/(I )  (El 2 +A)212_(1_ E ,)2 (D E
It can be shown that the term inside the log and the square root in the denominator are purely imaginary, this further simplifies the above expression 2c E2 E2 log 4 6 2 del 12 =   + + rckRe p 4 4 2  2 Similarly we obtain for 1 a2c E E2 log 4 + 7rcRe 61 li=a  + + Re
P 4 Vu2 I221A Adding up I, and 12 we obtain
l1 + 12= raRe e de + oRe e de1
Jo Vu2 ~?  2E1 Jo j /u2e + 2e ,A
48
The first term is more dominant and we will retain it, so the final expression for ReEfs is
ReER = URe f'/VF qdq z 4r JVu2 2A/vFq
 ,FR (2.71)
where the scaling function FR(x) is,
3 3 (2.72) F(x) = Re (1 + X) V'I+  X2 log 1+ (2.72)
2 2I
As shown in Fig. 25, the scaling function is asymmetric with respect to x + x, for x > 1 the function vanishes and for large and negative x,
1
FR(x) 4Re 1.
The real part of the selfenergy away from the mass shell is
ReERs  Reu,
20 EF XA
the ZS contribution away from the mass shell is smaller by a factor of u compared to the u2ee contribution to the selfenergy from the backscattering process. However, right on the mass shell the interaction with the zerosound mode is strongly enhanced, the function FR(x) + 1 and ReEfs = u2E1/8EF, this result is of the same order as the contribution from the PH region. The small x expansion of FR(x) is given by 3x2 1
FR(x) = 1+ x + 3 log  . (2.73)
4 X
Using this expansion in Eq. (2.71) we obtain to linear order in A, the following correction to the selfenergy
ReEzs + 1A (2.74)
8 EF E
Adding up contributions from Eqs. (2.70) and (2.74) we obtain ReEF = ReERH + ReER + ReER Us 2 = (2.75)
8 EF
49
Final result for the real part of the selfenergy. Collecting up all nonanalytic u2 contributions to the ReEF from the forward scattering processes,
ReEF = ReEfH + ReR. + ReEs u2 EE
= (2.76)
8 4EF'
we find the above result from the forwardscattering process differs from the backscattering result by a sign. Thus near the mass shell
ReER = ReR + ReEf = O(u2A2 log A). (2.77) The conclusion from this result is that, for a short range potential the nonanalytic u2e el term is absent.
2.3.6 Spectral Function
Having determined ImER and ReER, we are now in a position to evaluate the spectral function. The spectral function in terms of the Green's function is expressed as A(E, k) = lImGR(E, k)
1 ImER(e, k)
= k(2.78)
7 r [A + ReER(e, k)]2 + [ImR(, k)]2 (2.78) We will plot the spectral function as a function of x = 2A/u2E, keeping e fixed and varying A = E  (k through its momentum dependence. We will also consider a situation where A u21II > Ac (as a reminder Ac is the energy scale at which the curvature effects become important), so that we can neglect curvature effects on the spectral function. Near the mass shell, ReER in the denominator can be neglected compared to A as ReER u2A2 log A < u22 log IwI. Thus the spectral function reduces to
1 ImER(E, k)
A(ek) = (2.79) (r A2 + [ImER(e, k)]2
The variation of ImER as a function of E or (k cannot be neglected, and is responsible for nonLorentzian type features in the spectral function. The imaginary part of the retarded selfenergy is a sum of four different contributions,
ImER = Im + Im ImE + IMm , + ImE s,
50
103
2
102 0
X
LL 2
W 1 0 1 2
Cj 101
100
111
10
2 1.5 1 0.5 0 0.5 1 1.5 2
X
Figure 26: The logplot of spectral function A(e, k) in units of 1/7r2u2EF. A(E, k) is plotted as a function of x = 2(e  (k)/u2e for log(EF/u21I) = 2 and e/27rEF = 0.025. A kink at x = 1 is due to the interaction of fermions with the zerosound mode. Inset: part of the spectral function Ai(e, k) for E/2wEF = 0.25. A maximum in A1 at x = 1 gives rise to a kink in total A (main panel). where the last three terms are due to the forward scattering contribution. Near the mass shell, both ImER and ImE' are independent of A and to leading order the sum ImER + ImE vanishes, thus ImER shows dependence on A through the remaining terms,
U2 2
ZSImE F (x) U2 2 EF]
mE = 0og u21 + GI(x) . (2.80) The spectral function acquires becomes
1 log + Gl(x) + FI(x) A(e,k) = . M + (2.81)
The scaling function Gi(x) exhibits weak dependence on x for x ~ 1. Whereas, FI(x) is strongly dependent on x with a sharp peak at x = 1. The spectral function shows a narrow peak at x = 0 and a wellpronounced kink at x = 1, the kink corresponding to the sharp peak of F(x). A plot of the spectral function A(x) as a function of x is shown in Fig. 26.
51
For typical A 2 u2'E or x ;1 , log(EF/u2e ) > [GI(x) + Fi(x)]. Therefore, we can split the spectralfunction into two parts: a part which describes the regular quasiparticle peak (Ao) and a part with scaling behavior (A1)
A(e,k) = Ao(e,k) +A1(E,k),
1 log(EF/u2kli) 7r2U2EF X2 + log(EF/U22))
Al(e,k) = 2E [GI(x) + F(x)] . (2.82) Z2 + rlog(EF/u2 E )
Near x = 1, Ao(x) is a smooth function whereas Al (x) shows a rapid variation. This variation associated with Al(x) is due to the fact that, for IAI > u21'~ (x > 1), a fermion with frequency e > 0 can emit a zerosound boson of any frequency from 0 < 0 < E whereas for IAI < u21 el, a Cherenkov type restriction makes it impossible to emit and absorb bosons with frequencies above IAI/(1  vF/C) = A/u2.
2.3.7 Coulomb Potential
Next, we will consider the interaction between the fermions and the plasmon modes and show that the kinklike behavior in the spectral function, which was obtained for fermions interacting with the zerosound mode in the case of a shortrange potential, is also present for the case of electrons interacting with plasmon modes. Herein, we will consider the low energy limit: with E l, k < Wp1; where wp, = v/r,EF is the maximum (undamped) plasmon frequency. Near the plasmon mode the potential is given by V(q)
V(w, q) =
1  V(q)II(q,w)
27re2w2/q
(w + iO+)2  Qo(q)2'
2re4mv
re~MV (2.83) (w + i0+)2  0Q(q)2
where Qo(q) = (e2mvi2q)/2 is the plasmon dispersion and we have used the following form for the polarization operator: R wq
52
Using Eq. (2.83) as the potential, we can calculate the imaginary part of the self energy (we will keep E > 0), i.e., the damping due to the interaction of the quasiparticle with the plasmons,
ImR(E, )  j dQ f d2qImGR( + Q  k+q)ImVR(Q, q)
= d d2q(e    VFqcs ) (Q2  Q2 (q)), (2.84) where the first Sfunction is due to the Green's function and the second is due to ImVR. The second delta function forces the transferred boson momentum to be small, i.e., q
= 0 otherwise (2.85) For a fixed value of E, ImER has a kink at (k = 0, also the kink is independent of the coupling constant. A kink in the spectral function could, in principle be detected in photoemission experiments on layered materials or in a momentum conserving tunneling between two parallel 2D gases [82].
2.3.8 Corrections to the Tunneling Density of States
The presence of EA term in the real part of the selfenergy of the particlehole [Eq. (2.70)] and the ZS [Eq. (2.74)] contributions can in principle give rise to a nonanalytic correction to the tunneling density of states. The tunneling density of states is represented as
2 f d~k
N(E)  d(2)2ImG (e, k). (2.86) The term
a k I~ j= E _
EF EF
in the real part of the selfenergy is absorbed by the Zfactor in the Green function, yielding a simple form for the interacting Green's function in terms of the free Green's
53
function
1 1 Z(e)
GR(k, w) = 1Z() (2.87) 1+ E k + i0+ E + iO+
EF
The tunneling density of state for a free system is m/r. Thus we can deduce the tunneling density of states for an interacting system to be
N(E) = Z(e)  1  a E . (2.88) This would be the case if a nonanalytic elA were present in the real part of the selfenergy. However, we see from Eqs. (2.70) and (2.74) that the JeA terms cancel out, moreover, the backscattering term of the ReE does not contain a EA term. Thus the tunneling density of state does not get any nonanalytic correction. The above result, valid for a contact potential, will also hold for a finite range potential, since as before the nonanalytic [etA terms cancel between the different forward scattering contributions, whereas the term is absent in the backscattering contribution.
2.4 Conclusion
We will conclude this Chapter by restating our main results. Secondorder perturbation theory yields two interesting results, namely, a nonanalytic term in the imaginary part of the selfenergy and a logarithmic singularity on the mass shell for fermions with a linearized dispersion. The nonanalytic contributions originate from processes involving two incoming particles of nearly opposite momenta undergoing either a q m 0 or q m 2kF scattering, and also from the scattering process between two incoming particles with nearly identical momentum that undergo q m 0 momentum transfer. The latter process (the g4 process in 1D terminology) is also responsible for a logarithmic singularity on the mass shell, indicating that the quasiparticles are illdefined. We find that at higher orders in the interaction the singularity is enhanced and becomes a power law, causing a formal breakdown of the perturbation theory. These results, which are obtained for fermions with a linear dispersion, are modified by restoring finite curvature of the dispersion. The singularities from the individual terms in the perturbation theory are removed, yet the series itself remain divergent. A resummation, performed by selecting the maximally divergent diagrams, removes these singularities even for a linearized dispersion, with the result that
54
Fermiliquid theory is restored. However, we demonstrate that the interaction of fermions gives rise to a kink in the spectral function at A = u2E/2. The nonLorentzian shape of the spectral function should be amenable to a direct check in photoemission measurements or in momentumconservedtunneling between two parallel 2D electron gases.
CHAPTER 3
SPECIFIC HEAT OF A 2D FERMI LIQUID
3.1 Introduction
Thermodynamic quantities in a Fermi gas form a regular and analytic series as a function of temperature T and momentum q. For example, C(T)/T = 7 + aoT2 + alT4..
xs(T,q=O) = o(O) + boT2 +..
Xs(T=O,q) = Xï¿½(O)+coq2+.. (3.1) where the even powers of temperature T are due to the particlehole symmetry of the Fermi function about the Fermi level and the even powers of q are due to the quadratic dispersion of the free particles. This result, valid for a Fermi gas, does not necessarily hold for interacting fermions. Fermiliquid theory predicts only that
C(T)/T = 7*,
Xs(T, q) = X*(0),
where the constants y* and X*, differ from the free Fermi gas result and in general depend on the interaction parameters. To obtain higher orders in the temperature and momentumexpansion of Cv and X, for an interacting system, the usual approach is to consider microscopic models (for example, fermions with shortrange repulsion) and employ perturbation theory. Corrections to the linearin temperature specificheat term are conveniently obtained by calculating the selfenergy on the mass shell, and relating the selfenergy to the specific heat. A nonanalytic correction to the selfenergy leads to nonanalytic terms in the thermodynamic quantities, for example, in 3D an e3 log ElE term in the real part of the selfenergy leads to a T3 log T term and in 2D an eHEI term in the selfenergy gives a nonanalytic T2 term [19, 20, 21, 22, 23, 24].
55
56
In the last Chapter we investigated the nonanalytic contributions to the selfenergy from perturbative and nonperturbative processes. It was shown that upon resummation, the nonperturbative contributions to ReER become of the order U2 near the mass shell and exactly cancel the second order U2eeI term. This would then mean an absence of nonanalytic T2 terms in the specific heat at second order in the interaction. We investigate the role of contributions of nonperturbative processes to the specific heat and find that the imaginary part of the selfenergy contributes to the nonanalytic term in the specific heat, in such a way that it cancels the contribution from the nonperturbative real part of the selfenergy, thus keeping the secondorder result intact. We confirm our result in three different ways. In Section 3.2 a direct calculation of the specific heat from the selfenergy is performed, and we find that the contributions from the nonperturbative processes cancel out. In Section 3.3 we use the LuttingerWard formula for the thermodynamic potential to show using real frequencies (Section 3.3.1) and imaginary frequencies (Section 3.3.2) that the nonperturbative processes do not contribute to the T2 term in the specific heat through order U2. We will consider the longrange Coulomb potential in Section 3.4 and show that the plasmons (nonperturbative processes) make an interaction independent contribution to the specific heat. We end the Chapter with a conclusion in Section 3.5.
3.2 Specific Heat Calculation From the SelfEnergy
We will use the result of Ref. [16] which relates the thermodynamic potential to the Green's function and establish a relationship between the selfenergy and the specific heat. The thermodynamic potential reads,
Q = 2T J In G(wm, k, T = 0),
where the Green's function is expressed in terms of Matsubara frequencies wmo and has no additional temperature dependence. We can convert the summation over frequencies into a contour integration obtaining
Q= 2 d0J0j [lnG(,k)InGA(w,k)]. (3.2)
57
The thermodynamic potential and the specific heat are related via 02q
C = T (3.3) using Eq. (3.2) and Eq. (3.3) one obtains, 2T a [ d2k " ano Ri C r 2T2 a[ f0dw = arg GR(w, k) . (3.4) SB T T (27)2 oo 9
For the case of a weak interaction such that IE < w, one can expand the Green's function in Eq. (3.4) in powers of E. To the lowest order in E we obtain
C(T) = CFG(T) + JC(T), (3.5) where CFG(T) = mrT/3 is the specific heat for free fermions in 2D and 6C(T) is given by
2 [ f di2k f0 rnoT R. ]\R/ \l 6C(T)/T = 2 [1 2 aw [ER(wk)G (w,k)] (3.6) There are two contributions to 6C(T)one from Re ER and another from ImERwhich are labelled CI(T) and C2(T), respectively:
C (T) = C1 (T) + C2 (T); (3.7a)
2 8 1 d2k " OnoIR R CI(T)/T =7r 5 T (2)2 f 01 dw ImG (u), k)Re'R(w, k)]
81 f d2k 00 8no 1
= 2a (2)2 dW 5(w Ek) RER(w, k) ; (3.7b)
C2(T)/T 2 dwOnP ImE(wk)] (3.7c) Eq. (3.7b) implies that to calculate the specific heat contribution from the real part of the selfenergy, one needs only the massshell form of the real part of the selfenergy. On the other hand, Eq. (3.7c) implies that the contribution from the imaginary part of the selfenergy is zero if the specific heat is independent of momentum k, since the integral
2 2k Jk
vanishes if one assumes the density of states to be constant.
In the discussion to follow we will show that the nonperturbative contribution to the U2T2 term in the specific heat vanishes. Thus contrary to earlier results [21], the U2T2
58
term is not modified by the additional contribution from the nonperturbative process and remains exactly the same as obtained from the second order perturbative calculation [22].
3.2.1 ZeroSound Mode Contribution to the Real Part of the SelfEnergy
The zerosound mode contributes both to the real and imaginary parts of the selfenergy. In this Section, we will calculate nonanalytic contributions from the real part of the selfenergy ReEzs = U2WIWl/8EF to the specific heat. Notice, for calculating the contributions to the specific heat from the real part of the selfenergy we need only the massshell form of the selfenergy. The contribution is,
6Cz(T)/T = 2  d ((k) (u 8EF +  ] OT [T W 2 m 8 8EF 4EF
1 a 2mT' " w (2) 4EF &T r 0 2Tcosh 2()2T 4Nu2T
4Nu 2T (3.8) 7rv2
where
N=[" x3dx 9((3)
Jo cosh2(x) 8
3.2.2 Zero Sound Mode Contribution to the Imaginary Part of the SelfEnergy
For local selfenergy (i.e., independent of momentum) the imaginary part of the
selfenergy does not contribute to the specific heat. In general, the imaginary part of the selfenergy need not be local, this is the case for ImEzs. Near the mass shell Iwl I< kl, thus
pZS Re sn(w) QdQImE 2U3 
2u3 l /ic QdQ
2 Re Q (3.10) o gn(w)
59
The specific heat contribution from the above term can be written as,
JC2 20 [ 1 mdgk W 1 ER
T 7rT T T  2r ood4T cosh2 w
1 [ u3 " d w 1R QdQ
=Re dA d R
7r T 41T2 0o cosh2 2A _ u2
QvF
1O T 2 dA c shd / /2 3 2
10 =3 df Re QdQ 1 (3.11)
7r T 4iT2 coshh2 2 /  2
where we have made the substitution z = 2A/vF. We will first calculate the following integral,
I = fodzRe fjw/cz/2 Q3/2dQ S z o z  U2Q By splitting into regions in the (z, Q) plane where the integrand is real, we obtain
f 2w/(v+2v/U2) dz z/u2 Q3/2dQ
2= 2w / 2 v u d z z/L2 3/2 / /vd
Sdz / /vFz/2d Q3dQ
z  Q  2Q 2u2 f2I/(VF+2VFIU') Z , V  u2Q
16uv +2uv 16uv] 2uv
Using this value back in Eq. (3.11) yields 602 4Nu2T
T  v (3.12) T 7rV2
Adding up the two contributions from the zerosound region, Eqs. (3.8)and (3.12), we obtain
6C2 JC1
S+  = 0, (3.13) thus we find the zero sound makes no contribution to the leading nonanalytic term in the specific heat.
60
3.2.3 Finite Range Potential
The presence of a finiterange potential U(q) does not modify our earlier result regarding the cancellation of the nonperturbative contributions. This is because the nonperturbative terms arise due to forward scattering processes with small momentum transfer hence U(q) can be replaced by U(0). The analysis for the cancellation of nonperturbative contribution to the U2T2 term in the specific remains the same. As a result, the nonanalytic contribution to the U2T2 term in the specific heat is the same as that obtained by the perturbative approach and is given by [22]
6C(T) (3) [ 2[ U2(0) + U2(2kF)  U(0)U(2kF) CFG(3.14)
T x 2 EF
3.3 Specific Heat Calculation from the LuttingerWard Formalism
The approach used in the previous Section to show the absence of a nonperturbative contribution to the T2 term is quite involved, so we will use an alternative method, which is easier to execute, and in the process double check our earlier result. We will use the thermodynamic potential which is expressed in terms of the LuttingerWard formalism to calculate the specific heat. The most commonly cited expression in the literature for the thermodynamic potential of a system of interacting fermions with a potential gV(x) is
Q(g) = Qo +  ,j '  exp(iw,)E(g', , wn)G(g', E, w,), (3.15)
where g is the coupling constant, E and G are the full selfenergy and Green's function respectively [83], and 20 is the thermodynamic constant in the absence of interactions
o =2V log{1 + exp[!3(e  p)]}.
As shown by Luttinger and Ward [84], Eq. (3.15) can be written in a compact form 2V = [log[G(iwn, I)] + E(iw, E)G(in, k)] 2V 1
+ 2m Em(iw, rf)G(iwn, k), (3.16) m in,s
where in the last expression EmC , wn) stands for the mth order selfenergy diagram
61
Figure 31: Example of a secondorderskeleton diagram. Notice the fully interacting Green's function.
k+q
2(b)
1(a) 2(a) k 3(a)
p+q
k+q l+q
3(b) 3(c) 3(d)
k 1
q q
p+q
Figure 32: Diagrams through the third order that contribute to the thermodynamic potential.
determined from the skeleton diagram with fully interacting Green's function. An example of a second order skeleton diagram is shown in Fig. 31.
To calculate the specific heat from the thermodynamic potential we will be interested in those diagrams, that, at each order of interaction, have the maximum number of particlehole bubbles (these are the most divergent diagrams of the thermodynamic series). Typical examples of thirdorder diagrams are shown in Fig. 32(3). It might seem that only one of the diagrams in Fig. 32(3) belong to the category of "maximum number of particlehole bubbles", however a more careful analysis shows that the rest of the diagrams also have equal number of bubbles, though they are 'implicit' ( they are obtained by integrating over the fermionic degrees of freedom). Furthermore, it is easy to show that
62
the diagrams of the kind shown in Fig. 32 (which are "part" of the skeleton diagrams) are obtained from the last term of Eq. (3.16). It turns out that, even for higher orders, the third expression is the only one which is relevant. Following the prescription described above, we obtain the following series for the thermodynamic potential of a short range potential,
1 1
Q2 = Qo+ VT ..,, [_Un , 1l(uII,'2 + 1(UI,)3 _ l(urim)4
+ (UIIm)5  (UIrm)6 + ]. (3.17)
3.3.1 RealFrequency Approach
In an earlier work on nonanalytic contributions to the specific heat [15, 22], it was
shown that the nonanalytic U2T2 contribution to the specific heat arises from two regions in momentum space, from k < ky and k m 2kF. In the following analysis, we resum higherorder terms so as to obtain the nonperturbative contributions from the k < kF regime. We can split the thermodynamic potential into two parts, a part containing contributions coming from the small momentum region (summed to all orders) and a part which has contributions from the largemomentum region (only the secondorder term).
2V [n (Un)2 (Un)3 2(UI)4 + E 2 2 3 4 iw ,Jl~
2V 1 (UII)2 (318) + 0 2 2.18)
iwn,,ljk 2ky
The above expression for the thermodynamic potential can be rearranged to obtain a compact form
spinchannel chargechannel
V (UII)2 3 1
S [2UII  (U2 + (U)2 log(1 + UII) + log(1  UII)
iwn,
nonperturbative Contribution
2V 1 (UII)2
+ 2 2 , (3.19)
iwn, ~2kF
where the terms inside the curly brackets are the nonperturbative contributions. The summations on discrete Matsubara frequencies can be converted to integrals through the
63
Cl
C1
CC
"2
Figure 33: Contour for summing up the Log. The thick line represents the branch cut and the small circles represent the discrete Matsubara poles. usual trick employed in [16]. We will show the procedure of summation for the log terms of Eq. (3.19). By making use of the relation between the Matsubara dielectric function DM(k, iWm) and the retarded and advanced functions DM(k, iw w + iO+) = DR(k, w); DM(k, iw  w + iO) = DA (k, w), the discrete sum can be expressed as an integral with contours, C1 and C2, as shown in Fig. 33. The discrete summation is
aT Ej log(DM(k, iw)) = T log DM(k, O)
iwn
+ J I fB(W) log DR(k, w) OT cs 2Ixri
+ ~T & nB((w) log DA(k, w), (3.20) where riB(w) is the Bose distribution function. The retarded and the advanced function have neither poles nor zeroes in the upper and lower half plane respectively. Also the function, nsB(w)/OT, decays rapidly as w + oo. These properties of the integrand allows us to "open" the contour and express the integrals in terms of contours C'1 and C'2. Notice, the small bumps in the contours C'1 and C'2 cancel the first term of the
64
above equation. So the above equation with contours C'1 and C'2 minus the bumpy part is
OTT 10g(DM(kin))  OT _ 2xi 2)Bas(w)
x [log DR(k, w)  log DA (k, w)] . Next, we will make use of the property of the log and the property of the function on the real axis DR = DA*, to obtain a simplified form. The contributions from the spin and charge channels will be evaluated using,
TE log(D R (k, iws,)) = d2k 1) 2 7;
iwT J(2)2 J sinh2(w/2T) x tan LReDR(, w) (3.21) The contributions that will be considered are those from the spin channel, the charge channel and the II2 term. Out of these only the charge channel will get contributions from the zerosound mode region. Let us first consider the spin channel:
3 f+"dw 3 UImll
2E log(1 + UvI) = E . 2 ep(3w)  tan 1 UReR (3.22)
2O  f . 27r exp(w)  1 1 + URellR where the polarization operator in the particlehole region is, IITR (1 i ).
27 V(kvF)2  w22
The above term does not get any contribution from the zerosound mode region. We will obtain the contributions from the particlehole region under the assumption of weak coupling mU < 1
1 3 3 +" dw 2w kF dk2 T 2 2 7r (2T)2 sinh2(w/2T) , 47r
iw,k "F
x tan' [2 w
[27 m/kwd
65
We are interested in the T2 term, hence we will only be interested in the w2 term from the dk2 integration. The w2 term from the dk2 integration is
dk2  tan W 2U2W2 (3.23) F d 2tan  x/(kvf)2 _92 W2 8 2 Therefore, the T2 contribution is
0 3 9m2U2T2((3) Slog( + ) (3.24) T 20 , 16,3 T2 contributions from the charge channel (particle hole region) can be obtained by utilizing the same approach as for the spin channel. We obtain
0 1 3m2U2T2 (3) E log(1  UI) =  (3.25) T 2 1603 iwn,k
Unlike the spin channel, the charge channel does get a contribution from the zerosound mode region
3 1 S log(1  UII) 2 sinh 2 /2T) arg(Gp), (3.26)
&T 23 2 k i w sinh(w/2T) iwn,k k where
1 1 U2vk2
S= 2 1  UI " (w + i0+)2  c2k2 the integral of Eq. (3.26) is easy to perform as the argument of G,, in the momentum region w/c < k < w/vF is constant. In this region Re(Gp) < 0 and Im(Gp) + 0, giving an argument of 7r. Thus,
8T 1 "o dw '/V kdk w
E log(1  UII) =  (3.27) T ~2 /c 2 (2T)2sinh2(w/2T)R (327) iw,,,k
where c = VF//1  (Um/27r)2 is the zero sound velocity. This gives a 1 3U2m2((3)T2 aT log(1  UII) = (3.28) iw,kF
The contributions coming from the small momentum region to the second order term is
1 n2u2 k dw 2U2W Im(HR)2. (3.29)
T (2)2 M 2 (2T)2 sinh2(w/2T)(32
66
The imaginary part of (HR)2 is
Im(IR)2 = mw m ((kV)2 2 w). (3.30) 272(kF)2 2 42 w)sgn(w).
The first term of the above expression gives contributions to the T and the T3 term, the J function term is responsible for the T2 term
01 2 = dk I dw 2U2w m27rw2 6(kvF  W)
iw, 1 (2 Jo 7 (2T)2 sinh2(w/2T) 42 2w
 3( UM)2((3)T2. (3.31)
7 27rVF
We can now add up Eqs. (3.24), (3.25), (3.28), and (3.31) to obtain the nonperturbative contribution of Eq. (3.19). The result is that the nonperturbative contributions add up to zero, thus there is no nonperturbative contribution to the T2 term in the specific heat, and, only the perturbative second order in the interaction strength, II2 terms of Eq. (3.19), contribute to the U2T2 term in the specific heat. We will show below that the k m 0 and k m 2kF regions give equal contributions to the U2T2 term in the specific heat.
3.3.2 Matsubara Formalism
In this final approach, we will keep Eq. (3.17) as it is and perform the summation
directly on the Matsubara frequencies. A great advantage with this approach is that it is a priori apparent, for weak interaction u < 1, that we do not need to consider terms higher than the second order term for the U2T2 contribution, as none of the terms in the series are divergent and the series converges. On the other hand, a resummation will indeed be required if the interaction is strong, u 2 1, or long ranged. In the following discussion we will consider the two cases: weak potential u < 1 and the strong potential u  1. We will then extend our result to include a potential with an explicit momentum dependence.
Contact interaction: u < 1. For the weak potential it suffices to consider terms
through second order in the interaction. The first order term does not contribute to the nonanalytic temperature dependence since the nonanalyticity arises only from the 2nd order onwards.
We will calculate the contribution from the II2 term which originates from k < kF regions, followed by contributions from the k z 2kF region. The Matsubara polarization
67
operator in the small frequency and momentum region is IIm(wn, k) = 1  ). (3.32) 27 w/ + (kvF)2
Using this form of the polarization operator and performing the momentum integration, we obtain
11 Um2 k' dk2 2 2Iwn
iwk2 2 2 4 w + (kvF)2 w+ (kV)
=(27r X + 4n(n  + lon2g + ,
where N > 1 is the upper cutoff and X = (kFvF/27rT)2 > 1. Notice, the T3 term has been extracted in front. A T3 term in the thermodynamic potential corresponds to a T2 term in the specificheat. Thus, we will be interested only in extracting the term independent of temperature from the square bracket. To this end we will use the EulerMaclaurin formula, which reads
F(n) = F(x)dx + (F(N) + F(1))
n=l
N F(2k1)(N)  F(2k1)(1) + (2k)! B2k, (3.33) k=1
where B2k are the Bernoulli's coefficient and F (2k1) is the 2k  Ith derivative of the function F, and the function F in our case is
2
F(n)= X + 4n(n  + X) + n2 log .n+X From the above expression, it is clear that the only term which can contribute to a cut off and temperature independent term will be the n2 log 1/n2 term. We express the series as
N N
2 (n)2log(n) = 2 x2 logxdx  N2 log N
n=1
N (2 log X)(2k)(N 2 log )(2k)(1)
2 (2k)! B2k k=1 (2k)!
68
thus the constant term is
N1 1 1 ((3)
F(n) =2 n2Lo gn = 2[ +  (3.34)
constant content 9 12 360 272
n=l n=1
and we obtain
1 2 (Urn 2 7T3(3) 2 2u2 2w v' 272 iw,,,k
3((3)u2T2
cv =
7rVF
The remaining U2T2 contribution to the specific heat comes from the 2kF region, to evaluate this contribution we need the asymptotic form of the polarization operator given by
11(wn, k)=  [1 (X + X2 (2k
where x = (k  2kF)/2kF. The square of the second term in the bubble yields another n2 log n term, which as before is responsible for the T2 term in the specific heat. The square of the second term in II yields,
1 E 2U2 1 Umrn)2 kdki2 2/3 20 27 2r
iwn,k
1 U2m2 1 [ (3.35) 2/2 (2 2kF)2 dx x8+xl+ 8knv2(.
where the precise upper limits on x are not important, the logdivergence of x (at x = 0) is cut off at x = Wn/2kFVF. We obtain a similar log term,
1 Un )2wT ((3) I E H2U2 =( 2r )2 (3)' (3.36) iw,,k F
Thus both k < kF and k m 2kf yield identical nonanalytic contributions to the specific heat. The total contribution is twice the contribution from either one of the regions.
3.3.3 Contact Interaction: Beyond Second Order
In this Section we will sum up the T2 contributions from all the terms (Un; n > 2) of the thermodynamic potential Eq. (3.17). We will widen the scope of our result by including selfenergy insertions in to the bare Green's function. Although the general structure of such a Green's function is very complicated, near the Fermi surface, one can
69
25
20
S15
310
5
0 0.2 0.4, 0.6 0.8
U
Figure 34: Plot of ueff (u*) vs u* approximate the exact Green's function by its expression near the quasiparticle pole, G(iw,,, k) =iW0  E*K
where Z is the renormalization factor, Eg = v}(k  kF), V* = kFl/m*, and m* is the renormalized mass. Parameters Z and m* are some functions of the bare interaction u, whose forms, in general, are not known. This amounts to replacing the prefactor and the Fermi velocity in Eq. (3.32) m*z2 W Iml
II(wm, k)  H *(m,k)  1  + v) (3.37) As noted in the previous Section a T3 term in the thermodynamic potential originates from a w2 log w term, which is in turn obtained after the momentum integration of ( 'I )2. In a higher order U*n", the term( i )2 is obtained from the 3rd term of the binomial expansion:
Um*Z2 m IWm
U U m*Z2IWI = ( )+ +1) W . (3.38)
2 ,2n + ( 2 )2 2 V/2+ ()2 Using this result in conjugation with Eq. (3.17), we obtain 6C(T) /T =  2 C),G/E. (3.39)
70
Here CG = 7rm*T/3, EF = kFV*/2, and
, = (u*)2 [1 + 2u* + 6(u*)2 + 4(u*)3 + ..
(u*)2 [  + ,I (3.40) where
u*  Z2m*U/27r. (3.41) This result is valid for 0 < u* < 1. The divergence of u2f at u* = 1, resulting from the spinchannel, signals an instability toward a magneticallyordered state. A plot of u2 (u*) is presented in Fig. 34.
3.3.4 Generic Interaction
In Section 3.3.3, we obtained the specific heat of a contact interaction for arbitrary interaction strengths, this result has a rather limited use as most realistic interactions have a finite range. To obtain a form that is valid for finiterange potential we begin by investigating the structure of vertices.
Let us consider higherorder terms, which can lead to two kinds of corrections; selfenergy insertions and vertex corrections. The effect of selfenergy corrections is reflected in a renormalized effective mass and a Zfactor for the quasiparticle. The second correction is attributed to vertex terms. A typical diagram of nth order consists of n bubbles. To obtain a T2 contribution to the specific heat one needs a product of two Im I/Q terms which are obtained from any two out of the n bubbles. Any extra factor of IQ m/Q will not lead to a nonanalytic term, thus we can set m, = 0 and Q  0 in the rest of the diagram. Thus, even at arbitrary order we are dealing with only two bubbles. The effect of higher order terms can be hidden in the static vertices (these are static because we need to first set Om = 0 and then Q + 0) joining the two bubbles. These vertices are of two types: Fk(k, k; k, k) and rk(k, k; k, k). What we are left with is a "2nd" order diagram, where the wavy lines are replaced by two blocks represented either by a IFk(k, k; k, k) or a Fk(k, k; k, k). In the conventional notation the vertices
71
Pk(k, k; k, k) and Fk(k, k; k, k) are related to Fk(0) via,
k(O) = lim F(kFil, 0; kFi2, OIQ, Q), (3.42)
Il/Q'0
where f(kl,wi; k2, w2 Q, Q) is the vertex for a process (l,wi; k2,w2 *  Q, 1 Q; Ic2 + Q, w2 + Q) and 0 is the angle the two incoming quasiparticles. Since the incoming momenta are nearly antiparallel to each other, we can set the angle to be equal to 0 = 7r. The vertex with the full spin representation is then,
rk6,(7r) = rk(k, k; k, k)ay  rk(k, k; k, k)6, b . (3.43)
We can express the scattering amplitude in terms of the vertex function via the relation,
Aa,6(, ) = Z2kk6(7), (3.44) where the scattering amplitude itself is represented in terms of spin and charge components,
Aai, y6(k, k) = A , y6(7r) = [Ac(7r)6&~y,6 + As(+r)a, ~y~,] m*
 [(A,(7r)  A,(+))6,a + 2A,(7r)66,Jy ]. (3.45) From Eqs. (3.43), (3.44) and (3.45) we obtain the relationship between the spin and charge components and the vertex function,
Z2rk(k, _k; k,  ) = ~[Ac(r)  A,(r)] m*
Z2rk(, k; k, ) = 2A,( (). (3.46) m*
Finally we will substitute Z2Fk(f, k; k, k) for U(O) and Z2 Fk(f, ; i, k) for U(2kF) in Eq. (3.14), to obtain the nonanalytic form of the specific heat in a generic Fermi liquid [85],
6C(T) 9((3) [ 2 [(2rk _k; k _k))2 + (Z2rk(k, ;  ))
T w2 27 ( ))2
_(2rk(k, _; k, _))(Z2F (E, ; _, E)) CFG
EF
C(T)  3((3) [A2 (r) + 3A (r)]T. (3.47)
72
Notice the universal subleading term in the specific heat is not an angular average over all the angles and is different from the leading, Fermiliquid specific heat term (C(T)/T oc constant) which is expressed as an "angularaverage((Fc(O) cos 0))" over the Fermiliquid interaction function.
The result for Cv has been expressed in terms of the scattering amplitude. Alternatively, the specific heat can be expressed in terms of the harmonics of the quasiparticle interaction function F(0). The harmonics of the scattering amplitude and the interaction function are related via
c/(n)
c/s
thus
0F(n)
A,/s(7r)= E(1)n C/S
n=O 1 + ciS
In contrast to the T term in the specific heat which requires only one harmonic, F,, the T2 term requires the full set of harmonics. The parameters of Eq. (3.47) can be extracted from the specific heat measurement on the fluid monolayer of 3He [27]. To a reasonable accuracy, the data can be fitted by a form C/(NTE;) = y(T/E;), where N is the density per unit area in a fluid monolayer, E; = EF(m/m*) and y(x) = a  bx. A fit to the data then yields b = 0.9[A2(7r) + 3As[r]] a 9  14, thus [A2(r) + 3A2[7r]] 11  15.5. In the case of "almost localized fermions" [86], the charge channel is the dominant interaction and F'n) > 1. Thus the scattering amplitude term, A,(7r), is expected to be small due to cancellation between the subsequent terms in the expansion. Neglecting A,(r), the spin part of the amplitude is obtained to be IA,(r) m 1.9  2.3. In addition, if the n = 0 harmonic of F, is the leading contributor to As(r), then FL) a (0.66  0.7). This is consistent with the 3D value Fo") a 0.75 [25].
3.4 Specific Heat for the Coulomb Potential
We have shown earlier that the collective mode for a finite range potential leads to a nonanalytic behavior of the selfenergy near the mass shell. However, this nonanalyticity is not manifested in the contribution to the T2 term in the specific heat. For a long
73
range Coulomb potential, we obtain a T2 correction to the specific heat both from the particlehole and from the plasmon regime. In the temperature range T < KVF < EF, where K = me2 is the screening wavenumber, the correction is independent of the interaction strength. For the Coulomb potential, the singular nature of the interaction for small momentum transfer allows one to consider an effective RPA potential for the selfenergy, and for the thermodynamic potential we need to consider the usual series of ring diagrams. A knowledge of the thermodynamic potential allows us to evaluate the entropy and the specific heat. The finite temperature thermodynamic potential in the ring diagram series is given by
A T f d2k
A 2 (2l log(1  V(k)l(k k, iw,,)), (3.48)
iWn
where the summation is over the bosonic frequencies, HIm is the polarization operator (we include contributions from both up and downspin electrons), and V(k) = 2re2/k the 2D Coulomb potential.
Using Eq. (3.21) to convert the frequency summation into frequency integrals, we obtain
1 d2k I dw _ ImDR(k, w)] S  tan' Rk(3.49)
2 (27)2 7 Sinh2(w/2T) ReDR(k, ) where DR = 1  V(k)IIm(k, iw,). The regions contributing to the entropy are the plasmon region and the particlehole region (Fig. 35). Near the plasmon region, (w + iO+)2 w2
DR = 1  V(q)In(w, q) (W 2 (3.50) thus ImDR  0+ and (1  V(k)ReIHR) < 0 giving a value of 7r for the argument in the integrand. The contribution from the plasmon region is thus Spasmon dw = Fkdk I r sinh2(w/2T)J 2 27
SIpasmon ( T 23((3)
CVpj = _(T)23((3)
74
0)
A B Figure 35: Contour for summing up the discrete sum of the Matsubara thermodynamical potential. A) Matsubara poles at the Matsubara frequencies and at the plasmon positions
wpl and wp; (for the integration variable e ); the branch cut from Wk to Wk represents the particlehole region. B) Particlehole spectrum for 2D. The other contribution to the entropy is from the particlehole region. The argument [Eqn. 3.49] decays as one goes deep into the particlehole region. In the particlehole region the polarization bubble is given by
IR(w,q)=  ( + i ( ) (3.51) thus,
tan (ImDR(w, k) tan1 ( (3.52)
tan=t(3.52) ReD(w, k) 1 + V(k)d2k ky' dw 2me2
Sportiehole  (2 ) [VF tan , (3.53)
(2) o 7 sinh2(w/2T) k + 2me2 We will make the substitutions w/2T  z and vFk/2T + k', and also a change in the order of integration to obtain
1 2T2 dzz T z/Vk02  Z2 Slparticlehole  ()2 dk'k' tan (3.54) 27 vF r sinh2(z) z 2Tk'/lpr +1 where rs = i/kF and p = kFvF/2. It will be convenient to split the integrals into two parts, the first part gives the linear in T correction to the entropy and a T2 term, whereas
75
the second integral gives T2 correction to the entropy.
1 2T dzz2 Slparticlehole = ( )2 z dk'
27r vF 7 sinh2(z) Tk'/(plrs) +
1 2T 2 dzz
2w (VF 7 sinh2(z)
x dk'k' tan'[  1k STk'/(2L2rs) + 1 Tk'/(2r) + 1) the expression in the big bracket is convergent and the main contribution comes from k' 1, hence for T << KVF we can make a further simplification,
1 2T f dzz2 f 1 Slpartidehole = ( )2 dk'
27 lF o 7sinh2(z) I Tk'/(2prs) + 1
1 (2T)2 T dzz
2rvF o 7r sinh2(z)
xj dk'k'(tanl[ ). (3.55)
Finally we have
r, log(rs)T 1 2T 3((3) 1 2T 3((3) + 3((3) Slparticlehoe = + 2( g) (
3m 2V F 27 2x vF 8 2w] r, log(r,)T T 23((3)
3m + ( ) (3.56)
3m VF 47r
So the linear in T contribution to the specific heat from the particlehole region is C = r, log(r,)T/(3m). The T2 term due to the particlehole term is
Sparticlehole = )2 3(3) (3.57)
Adding the contributions from the plasmon and the particlehole region, we get for the nonanalytic correction to the entropy S 23((3) (3.58)
SIparticlehole+plasmon  ) 4 (3.58)
Hence the T2 contribution to the specific heat C = TOS/OTI,,A is
( T 23((3) (3.59) VF 273
76
We notice that in this temperature range the specific heat is independent of the interaction strength. A further discussion is provided in Appendix A where we reconsider the plasmonpole and particlehole contribution to the specific heat from an alternate approach. We obtain the same results as before
3.5 Conclusion
In this Chapter, the important question regarding the role of nonperturbative
interaction effects in the specific heat has been considered. Often while evaluating the contributions to the specific heat from the selfenergy, the contribution from the imaginary part of the selfenergy is neglected. We have found that for a short range potential, the imaginary part of the selfenergy emanating from the zerosound mode region, ImEzs, does contribute to the nonanalytic term in the selfenergy. In fact, this contribution exactly cancels the contribution coming from the real part of the selfenergy (whose contribution also originates from the zerosound region). This result implies that a nonanalytic correction to the selfenergy arises only due to perturbative processes. We have verified this result directly by calculating the specific heat from the thermodynamic potential. To this end use has been made of both the real and Matsubara frequency formalism. An advantage of using the Matsubara formalism is the ease in identifying the terms which gives rise to the nonanalytic contributions to the specific heat. This has helped us in extending our second order result to all orders. This extension to all orders is required for cases when the interaction strengths are not weak. In a further extension of this work, a finiterange potential was considered. We have been able to obtain a closed form solution for the contributions to the specific heat to this problem as well. Finally, we have considered the case of long range Coulomb potential and shown that perturbative process do contribute to the specific heat and, in the lowT limit, this contribution is interactionindependent.
CHAPTER 4
ANOMALOUS EFFECTIVE MASS
The observation of a metalliclike resistivity and an apparent metalinsulator transition in highmobility SiMOSFETs in 1994 [50, 51] challenged the scaling theory of localization and has since led a number of groups to study both the transport and thermodynamic properties of SiMOSFET's and other semiconductor heterostructures. The origin of the anomalous metallic behavior still remains a subject of active research. As discussed in Section 1.3, although models based on the conventional dirty Fermiliquid can account for many observed effects both qualitatively and quantitatively, there are quite a few proposals for the nonFermi liquid origin of the anomalous metallic state. On the experimental side, the main qualitative argument for the Fermiliquid nature of the metallic state is the observation of conventional Shubnikovde Haas oscillations, which implies an existence of well defined quasiparticles. The parameters of the 2DES, the effective mass m* and the Land6 gfactor are obtained by fitting the LifshitzKosevich formula to the Shubnikovde Haas (SdH) oscillation data. Excellent fit to the SdH data using the renormalized parameters and the matching of these parameters with the parameters (X* oc m*g*) obtained via other experimental techniques like saturation of magnetoresistance approach [64, 66, 67], give strong evidence for the Fermiliquid behavior of the 2D system. In particular, studies indicate the spin susceptibility X* shows a strong increase as the density approaches the critical one (insulating regime) with a large fraction of this increase attributed to the increase in the effective mass term than g*, an attribute very similar to the He3 case.
The magnetic field dependence of the effective mass serves as an additional check for Fermiliquid behavior. Experimental observation of such behavior in 2D SiMOSFET's in the presence of a parallel magnetic field shows that, contrary to the expectations, the effective mass shows no dependence on the degree of polarization [68] (Fig. 41). Although the effective mass exhibits strong dependence on the density and hence the 77
78
3
2.5
 2   W  ...   V.. . ....... ......2 1.   E 1.5 0.5
0 0.2 0.4 0.6 0.8 1
spin polarization
Figure 41: Effective mass vs the degree of spinpolarization. The electron densities in units of 10"cm2 are: 1.32 (dots), 1.47 (squares), 2.07(diamonds), and 2.67 (triangles). Fermi energy EF, the lack of dependence on the magneticfield, which alters the Fermi energy, is quite surprising as it does not follow the prediction of Fermiliquid theory. That the masses should depend on polarization can be seen from considering two limiting cases: of zero and 100% polarization. At fixed total density, the Fermi energy is doubled by fully polarizing the 2D system. Thus the ratio of Coulomb to Fermi energy g = e 2 n/EF, which is a measure of interaction strength is halved compared to the unpolarized case. Yet the effective mass remains polarization independent. In this Chapter we analyze this paradox in detail.
The present Chapter is divided into two parts. Section 4.1 deals with the polarization dependence of the effective mass. We will consider two independent approaches, Landau's phenomenological approach in Section 4.1.1 and the weak coupling approach in Section 4.1.2, to show that in a Fermi liquid the effective mass not only acquires a field dependence but also exhibits spinsplitting. In the second part of this Chapter, Section 4.2.1 and Section 4.2.2, we examine the influence of valley degeneracy on the effective mass and on the spin susceptibility, respectively. The importance of valley degeneracy on the transport properties in the (001) plane of SiMOSFET has been emphasized in a recent
79
study by Punnoose and Finkelstein [61]. The diffuson propagator in a spin1/2 system has 4 channels, 1 singlet and 3 triplet, whereas for a system with spin1/2 and 2valley degeneracy (total degeneracy is 4), as in the (001) plane of SiMOSFET, there are 1 singlet and 15 triplet channels. These additional channels have been held responsible for a large (almost an order of magnitude) drop in resistivity near the critical region. Motivated by these findings, we consider a Coulomb gas in the largeN limit (for SiMOSFETs, N = 4). In a system with no valley degeneracy, the effective mass is renormalized by emission of both virtual electronhole pairs and plasmons, in the large Nlimit of the Coulomb gas, the effective mass is renormalized primarily due to a polaronic effect: via emission and absorpbtion of (virtual) high energy plasmons, while electronhole pairs play only a nominal role. As plasmons are classical objects, the quantum degeneracy and, hence the polarization, does not affect the effective mass to the leading order in 1/N. The largeN expansion obtained is rapidly convergent even for nonvalley degenerate system (N = 2) and, as such, it provides a nontrivial way of going beyond the weak coupling limit.
In Section 4.2.2 we investigate the behavior of the spin susceptibility in the largeN limit. We find that the spin susceptibility is renormalized; however this renormalization in leading and subleading order is due to the renormalized effective mass, whereas the Land6g factor remains unrenormalized. This indicates that the enhancement of the spin susceptibility does not reflect the tendency towards ferromagnetic ordering.
4.1 SpinPolarized Effective Mass
In the following two Sections we show how a standard treatment results in a polarization dependent effective mass in the presence of a magnetic field applied parallel to the 2D surface. The parallel magnetic field is assumed to couple only to the spin degree of freedom and not to the orbital degree of freedom, and acts as a knob which can be used to change the population of either kind of electrons, while keeping the total density of electrons fixed.
4.1.1 Landau's Phenomenological Approach
First, we derive an effective mass expression for a partially spinpolarized Fermi liquid system, generalizing the approach used for the single component case [87]. The form of the Landau interaction function for a spinpolarized case is, however, much more complicated
80
due to the breakdown of SU(2) symmetry. We will show that this interaction function effectively reduces to three independent terms which describe the interaction between two upspin electrons, between two downspin electrons, and between an upspin electron and a downspin electron. The strategy to obtain the mass is to compare the change in the quasiparticle energy between a moving frame and a fixed frame via two different ways. In the first way, Galilean invariance is utilized to find the difference in the energy of a quasiparticle between the rest frame and a moving frame. In the second way, one notices that the distribution functions are different in different frames, and thus we can calculate the change in the quasiparticle energy which arises due to a variation in the distribution function.
The general form of the interaction function in a spinpolarized system is given by [88]
fau;,v(j3>i') = F(Thi5')3,,6tv + G(fi')d,.,1, '+ .[X(j,5'),,
+X ( ',)d ,va,,] + Y(, 5 ')[5Y, .] },.], (4.1) where the field is in the z direction, and 8' are the Pauli matrices.
The change in quasiparticle energy due to a change in the quasiparticle distribution function is given by
6 = Trf? ;i', ') ') (27r)2, (4.2)
and the distribution matrix 6i is
0 6n,
where the change in distribution function for the majority (spinup) quasiparticle is given by SnT and for the minority (spindown) quasiparticle by 5n1. We explicitly calculate the product of the density matrix 6n, with the interaction function, to obtain the following
81
terms:
F(if, ')a,~S,,v p,~v F(if, ')S,# [Sn1 + Sn]
G(if,i')0,4.., 'Sn,, = G(if,')[2fn,o  Sap(bnt + bn4)],
X(f, Ji') ,psn,,,h,, = X(pi, #')o,[6nT + 6m] and
EX(f ', p )dy,,6n,,,5,p = X(if ', p)c,vn,,,,p = X(f ', p) [6nt  n~l]b,,
Y(fJ')ahg ,4s,,n , = Y(p,p'),[6 ,1  Snn l]. (4.3) It is easy to deduce that the quasiparticle energy matrix is in a diagonal form since 6,0, oa,, and 6n, ,p are all in a diagonal form. Thus for a spinup quasiparticle we obtain the following relation
JeT(p =  (i#')F(ff,f ') +G(f,f ')+X(ff ')+X(p',p+Y(ff ')
(27)2 [jll if
+6n(if'){F(f, f')  G(if,i') + X(if,i')  X(if',p  Y(i, J')}].
Similarly, for a spindown quasiparticle
6e 4 (p) = 1n (27 (f ') {F(f,E G(f,~  X(if, ') + X(',p  Y(f, ')}
+ng(if'){F(if, ') + G(if,')  X(if,')  X(f', p + Y(if~')}]. To simplify the algebra we will define a new interaction function Fij(i, J'), where i,j =T, I such that
2, F1T 'T ')n ( ')+ F' i(f,')6n (f 2(P ='Fif7 ( ')ny()2') + F4'if,')dn(f ,
82
where
FT'T (p') = F(5,1') + G(j5') + X(5,') + X(5',7) + Y(Yj') FYl"(f,') = F(fi,fl')  G(f,J') +X(k,J')  X(',p  Y(p,')
Fi'1(,J') = F(f, ') + G(, j')  X(f, f')  X(', p) + Y(fp, f').
Thus, we can invoke Galilean invariance to obtain the effective mass [87]. Using Galilean invariance we can relate the quasiparticle energy in a moving frame E' with the quasiparticle energy in a laboratory frame E. Let us assume the movingframe has a velocity ii with respect to the lab frame. Introducing a quasiparticle with momentum fl in the lab frame, we increase the total mass of the system by the bare mass m, since a new particle has been added. The quasiparticle energy in the moving frame is
Ev'( mi) = E(p  ;p +
2
p = p  it,
where f (f ') is the quasiparticle momentum in the lab frame (moving frame) and m is the bare mass of the particle. We thus obtain mu2
Ie(p = e(p+ mI )  (f+ mi). +
The right hand side of the equation can be expanded to linear order in i to obtain the relation
(m  m*)Y. i
'(p3 = A (p3 +
m*
Hence for the particle of the first kind (spinup) we have
e'(p) = e(p + T = 5'(p = ,(4.4) m m
where the bare mass for spinup and spindown quasiparticles is the same and equals m. A second relation is obtained by observing that the quasiparicle energy e'(pj) in the moving frame differs from the quasiparicle energy e(p) in the lab frame because of a
83
n/ 0
L( nl(P)
Figure 42: Distribution function in the primed frame and lab frame are represented by dashed and full circle respectively. different distribution function. The shift in the energy spectrum 5E(p) = e'(p)  E(p) due to a different distribution [the new distribution is shifted by m in momentum space for both up and downspin particles, (Fig. 42)] is 6p = tr f(f ')i (4.5)
(27)2
where tr' is the trace. For spinup
6eT(p = ~F (f, f ')6nT (fi') + FTI (fff ')6n (') (4.6) *T(P7) = ](27)2 [ ï¿½ )]I (4.6) and for spin down
6e (pI =J~ F' (fI4 ')nT(f') + Fl' (fi ,')6n,')] (4.7) where we have used
unt(') = n'(ff')  not( ). (4.8) There is a very simple relationship between the distribution function in the primed frame and that in the lab frame: a particle moving with momentum #' in the primed frame will have momentum f ' + mu in the rest frame, thus the distribution functions in the different frames will obey the relation nj(fl') = n9(ff' + me ), where n' (no) is the distribution in
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the primed (rest) frame. The two distribution functions are shown in Fig. 42. Continuing with Eq. (4.8) we obtain
n',(f ')  no(f ') = no(f ' + mdf)  no(i' ) # ' Ono 0(')
 * m '( t(P*t)  Et( 5)), (4.9) where pit is the Fermi wave vector of the upspin electron. Similarly for the spindown particle, the change in distribution is
n(IV')= m (a (')  m 6(EI(PF)  E(')),
where ffFj is the Fermi wave vector of the downspin electron. For a spinpolarized system the Fermi wave vector of the upspin electron, FT, will be greater than that of the downspin electron, iF!. Substituting the simplified form of 6nT (') and 6n, (f ') in Eq. (4.6) and Eq. (4.7), and using the relations d2ly m* dedO
(27)2 27r 27r
dO'F(fi f ')ii  ' = dO'F(lY 'fl P cos 0') Jil ' cos(O  0')
= Il'lf cos(0) dO'F(I 'p cos ') cos 0', (4.10) we obtain
6ET =  ,u' p 2 cos O'F" (iFT,rFT)
r dO'
m dO
6e   PFM p 2xi cosO'FIf(PF,I FT ') 2xou Pil d2X cos O'Fi'"(fi,itF '), (4.11) where iFT and &IF point in the same direction, and in general FT (ITFT, FI ') # F'(Fo,IjF4 ') / FiT(pfe, pF1 '). Comparing Eqs. (4.4) and (4.11) we obtain for the
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upspin fermions
(~ 1= 1m J 2cos9O 'F ( ,FT ) m) F1 cosO'F (FT,pF ')
similarly for the downspin fermions
(m*)l= 1m pF J cosO'FMI(fi,fI, FT  m ( 2 cosO'F'1(fi,fIjF ')
S(PF 2)2
It is clear from the above equations that the effective mass for the up and downspin electrons is, generally, different. An explicit evaluation of the effective mass requires a knowledge of the F function which, however, is not provided within Landau's phenomenological theory. In the following Section, we will perturbatively evaluate the effective mass of a system of electrons interacting with the long range Coulomb interaction and confirm our prediction regarding the spin splitting of the effective mass.
4.1.2 WeakCoupling Approach
A knowledge of the exact eigen states of a Hamiltonian allows us to calculate relevant physical quantities. In many cases, though, it is simply too difficult to calculate these eigen states. In these situations one resorts to perturbative techniques and obtains the physical quantities as a perturbative expansion in terms of a small parameter of the system. We resort to this approach to calculate the effective mass in terms of the dimensionless constant r, = /2me2/kF (a measure of the interaction strength). The stronger the interaction, the more the deviation of the electron effective mass from the bare electron mass. For a system with different numbers of up and downspin electrons, there will be unequal remormalization of the masses and hence spinsplit masses.
The Hamiltonian of our system is given by
H = f V (r)V (r)  1(r) T (r) dr
2 ( (.12) where the summation on repeated spin indices is implied and U(r  r') is the usual Coulomb interaction potential. The Fermi wave vectors for spinup and spindown electrons are different: k = k/F ï¿½, where E = (ntn)/n is the degree of polarization.
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+   +
SelfEnergy of SpinUp
Electron
+ SelfEnergy of SpinDown
Electron
Figure 43: SelfEnergy of spinup and spindown electrons.
We will consider all interactions present in this system: between two upspin electrons, between two downspin electrons and between an upspin and a downspin electron.
The electrons interact via a Coulomb potential. This potential is screened due to the creation of electronhole pairs. To calculate the selfenergy (required to obtain the effective mass), we choose those diagrams which at each order in the interaction have maximum divergence. Selfenergy diagrams given in Fig. 43 contain all such contributions. The series when summed to all orders [the choice of this particular series is known as the Random Phase Approximation (RPA)] yields an effective potential which is dynamically screened.
The effective mass in terms of the selfenergy is defined as
1 E T I (p,it)
m _ 1+  (4.13)
1  oV_4_I 0 '
where Ell is the selfenergy of up and downspin electrons with momentum p, frequency iDu and quasiparticle energy (p. For p _ k 1, ( = k)(v  vf), where k and v are the Fermi momenta and velocities respectively. Let us consider the selfenergy diagram for
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spinup electrons given in Fig. 43. Because the Fermi momenta of spin up and spin down electrons are different, a probe electron causes different variations in the densities of the two spin orientations. The resulting potential contains the sum of two contributions from both spin orientations. Summing up the RPA series, we obtain the following expression for the selfenergy:
ETI(p, iw) _00 d w kdk V(k) dOG (p + k, iw + iw)
00 2 J (27)2 ( ,wiD f +dw / kdk V2(k)I(k,iw)
 27rJo (27)2 1  V(k)Il(k, iw)
x f dOG (p + k, iw + it), (4.14) where V(k) = 2xre2/k is the bare potential, G is the free singleparticle Green's function and II is the polarization operator. The momentumfrequency integrals in Eq. (4.14) are dominated by the regions k  n < kf, where n  rekF is the screening length and w ~ kvF. Therefore the longwavelength limit of the polarization operator II(k, w) can be used. This form is
1I = rI+ + Il, If (iw, k) = W (1 _ Jl )
2 V(kvF)2+W2
At this point we will summarize the approach that has been taken to calculate the effective mass. In effect, we need to evaluate two quantities, a selfenergy differentiated with respect to the external frequency and a selfenergy differentiated with respect to kWe find that when differentiating Eq. (4.14) with respect to (k, the two individual terms diverge, however the divergences cancel each other. The first term of the selfenergy yields a log singularity which is exactly cancelled by a similar log singularity arising from the second term. Similar care needs to be taken when differentiating Eq. (4.14) with respect to the frequency. Details of the calculation are provided in Appendix B. The result essentially states that the effective mass of spinup and spindown electrons depends on the interaction parameter r, and the degree of polarization, m*il 1+ t + 1) + 0.23r (
nl + 0"23 {,, 1 (4.15) m 1 + r4( + log ) + 0.23 (
7rv22 T r
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1.8
1.7
=0.3
1.5
1.9
0 0.5 r 1 1.5 2
S
Figure 44: Effective mass for a spinup and a spindown electron at polarization ( = 0.3. hence expanding to the lowest order in polarization we obtain m*TI r, 1 log r, r,( S1 + ( + )F (2 + log rs). (4.16)
The general behavior of the effective mass for 0, even the decrease of the effective mass compared to the bare mass at high density, is in agreement with a recent experiment on the 2D electron system in an GaAs/AlGaAs heterostructure [89]. On the other hand the above theoretical predictions regarding the presence of a spinsplit effective mass (Fig. 44) is not consistent with the experimental findings. The LK formula for the case of magnetic field tilted with respect to the 2D electron gas plane reads
Pxx/Po = A cos[273khn,/eBi  r], (4.17)
k,a
where o =T, I is the spin index, and Ak is the amplitude of the kth harmonic which is a function of the electron mass and is assumed to be the same for spinup and spindown electrons. If however, the masses were different then the corresponding amplitudes would differ and the LK formula would change to
P = A,(mT) cos[27r3 shn,/eBï¿½  7] Po ,
 cos[ ~ ]. (4.18)
0
89
The frequency of the oscillation is dependent on the density of the components and for a spinpolarized system a mismatch in frequency results in a beat pattern to be observed in the experimental data. Due to the difference in amplitude term (AA"), the beats will not be complete. Nevertheless the experimental data show full beats, implying that AA, = 0 and that the effective masses for the upspin and downspin electrons are equal [66]. On a similar note, the experimental group of Shashkin et al. [68] found no dependence of the effective mass on the degree of polarization (Fig. 41).
4.2 MultiValley System
The absence of polarization dependence of the effective mass suggests that m* is renormalized via an interaction with some classical degree of freedom which is not affected by the quantum degeneracy of the electron state. As will be shown below, such a mechanism is provided by the renormalization of electrons by the virtual emission and absorption of high energy plasmons. We reanalyze the issue of spin independent effective mass by considering the spinvalley degeneracy in the 2D system. A valley degeneracy provides an additional parameter (N) to the problem. This parameter is 4 for the case of a 2D SiMOSFET (2 spins @& 2 valleys) grown along a (001) plane. The additional parameter N = 4 turns out to be critical in explaining transport data in both the diffusive and ballistic limits. We will consider the effect of valley degeneracy in the context of the effective mass problem and will also calculate the spin susceptibility. To this end our Hamiltonian is defined as follows
H = V1 (r)Vx(r)d r
12m
+ J '1 ((r)'(r')U(r  r')y(r')%'Po(r)d3rd3r', (4.19) where a, / = 1....N, are the spinvalley indices.
4.2.1 Effective Mass
In the longwavelength limit the Matsubara polarization operator at T = 0 for a system with degeneracy N (N components) is mN
IIM(q, iw)  (1 
90
where the Fermi wave vector in terms of the density of electrons n is PF = 2 w/n/N, i.e., the Fermi momentum decreases by a factor of vNH. This decrease in the Fermi momentum is due to the fact that the electrons are distributed in Nvalleys. Within the random phase approximation (RPA), the typical momentum transfer is of the order of K _ e2'mN. The ratio a = K/pF = rN3/2/2, is a parameter describing the crossover between the regimes of weak (a < 1) and strong (a > 1) screening. Both weak and strongscreening regimes are within the reach of perturbation theory (r8 < 1), provided the degeneracy N > 1.
In the weakscreening regime, the scattering processes leading to mass renormalization are essentially elastic and the typical momentum transferred q  K is small. The main contribution to the mass renormalization is via the interaction of electrons with the particleholes [shaded region in Fig. 45A], whereas the emission/absorption of plasmons provides subleading contributions. The effective mass is m* r, vN
 = 1 + ln(rsN3/2) + O(rs).
m 27r
Thus in the weak screening regime, the ratio of the change in effective mass in a fully spinpolarized case (N  N/2) to that in the unpolarized case is rather large. This is similar to the earlier considered case of N = 2.
In the strongscreening regime, rN3/2 > 1, the typical momentum transfers are
much greater than kF. In this regime, as the momentum transferred is large the scattering is almost isotropic (swave). The particlehole continuum contribution to the mass renormalization is greatly reduced for swave scattering. However, the interaction of the quasiparticle with high energy plasmons [shaded region in Fig. 45B] plays the dominant role and leads to the enhancement of m*. As a plasmon is a classical collective mode, it is not affected by a change in N. Consequently, the leading term in the N1 expansion for m* does not depend on N, whereas the nexttoleading term is numerically small.
The Matsubara polarization operator for q > PF is
Ilo(q, i) = N d d d2p 1
S27 (27r)2 (iE + iW 4p.+q)(ie  p)
= 2nEq/(E 2 w2), (4.20)
