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PAGE 1 1 ELECTRON TRANSPORT NEAR THE ANDERSON TRANSITION By ANDREW DOUGLAS 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 2009 PAGE 2 2 2009 Andrew Douglas PAGE 3 3 To my family PAGE 4 4 ACKNOWLEDGMENTS First I would like to thank my advisor, Khandker Muttalib, for giving me the opportunity to continue my studies here at UF. I would like to thank him for his insight, and for instilling the habit of looking at the physics behind the math. I would also like to thank him for his patience and good will Secondly, I would like to thank my family and friends for their encouragement and good humor. And lastly I would like to thank the Newberry Starbucks b ar istas for all the fre e coffee PAGE 5 5 TABLE OF CONTENTS ACKNOWLEDGMENTS .................................................................................................................... 4 page LIST OF FIGURES .............................................................................................................................. 7 ABSTRACT ........................................................................................................................................ 10 CHAPTER 1 ANDERSON TRANSITION ..................................................................................................... 11 1.1 Electron Localization in Mesoscopic Systems ................................................................ 11 1.2 Characteristics of Localization in 1 and 2 Dimensions .................................................. 13 1.3 Characteristics of Localization in 3 Dimensions ............................................................. 15 1.4 Single Parameter Scaling Theory ..................................................................................... 16 1.5 Role of Quantum Fluctuations ......................................................................................... 18 1.6 Distribution in the Metallic Regime ................................................................................. 19 1.7 Distribution in the Insulating Regime .............................................................................. 20 1.8 Inadequ ................................................................... 21 1.9 P(g) Scaling Equation Setup ............................................................................................. 2 2 1.10 The GDMPK Equation ..................................................................................................... 25 1.11 Mapping the GDMPK Equation to Schrdinger Equation ............................................. 31 1.12 Distribution in Q1D ........................................................................................................... 33 1.13 Distribution in 3D .............................................................................................................. 35 2 DIAGRAMMATIC EXPANSION OF THE PROPAGATOR ................................................ 42 2.1 Introduction ....................................................................................................................... 42 2.2 Explicit Expression for GN in 1st Quantized Notation .................................................... 43 2.3 Explicit Expression for GN in Second Quantized Notation ............................................ 44 2.4 General Features of the Diagrammatic Expansion .......................................................... 48 2.4.1 Equivalence to the Time Ordered Greens F unction.......................................... 48 2.4.2 Absence of Self Energy D iagrams ...................................................................... 53 2.4.3 Absence of Vertex C orrections ............................................................................ 55 2.4.4 Absence of C rosse d D iagrams ............................................................................. 55 2.4.5 General Form of the Diagrammatic E xpansion .................................................. 56 2.4.6 Extension of Temporal Integration to the Entire R eal L ine ............................... 57 3 PERTURBATIVE SOLUTION TO THE GDMPK EQUATION ........................................... 58 3.1 Introduction ....................................................................................................................... 58 3.2 Diagrammatic Expansion of GN ....................................................................................... 60 3.3 Exponential Expansion of Perturbative Series ................................................................ 61 3.4 First Order Correc tion to 0 NG 12 Limit ......................................................... 64 PAGE 6 6 3.5 First Order Correction to 0 NG in Small (2 12 Limit .................................... 69 3.6 Extracting the Singular Behavior from the Diagrammatic Expansion .......................... 72 3.7 Determining G2 for the GDMPK Interaction Potential ................................................... 78 3.8 Determining G1 for the GDMPK Single Particle Potential ............................................ 81 3.9 Zeroth Order Expression for p(x,t) in the Insulating State ............................................. 82 3.10 Testing the Validity of the MeanField Approximation ................................................. 83 4 STATISTICAL PROPERTIES OF GDMPK SOLUTION IN INSULATING STATE ........ 84 4.1 Construction of P(g) .......................................................................................................... 84 4.2 Evaluation of the Density ................................................................................................. 87 4.3 Calculation of F ................................................................................................................. 90 4.4 Analysis of P(g) Distribution ............................................................................................ 91 APPENDIX: CONTOUR ORDERED GREENS FUNCTIONS ................................................... 98 BIOGRAPHICAL SKETCH ........................................................................................................... 104 PAGE 7 7 LIST OF FIGURES Figure page 1 1 ................................................................................. 14 1 2 1D and 2D localization lengths plotted vs. disorder. ........................................................... 14 1 3 Typical density of stat es in 1D or 2D ................................................................................... 15 1 4 3D localization lengths plotted vs. disorder ......................................................................... 15 1 5 Typical density of states in 3D .............................................................................................. 16 1 6 Scaling of lng with sample length ......................................................................................... 18 1 7 Gaussian conductance distribution in metallic regime ........................................................ 20 1 8 Log normal conductance distribution in insulating regime ................................................. 21 1 9 Scattering amplitudes in metallic leads connecting to sample ............................................ 22 1 10 Perturbative set up of DMPK equation ................................................................................. 25 1 11 Transition amplitudes in delocalized, localized systems ..................................................... 28 1 12 12 functions as an order parameter [9] ................................................ 31 1 13 Evolution of P(g) with length in Q1D [21] ........................................................................... 35 1 14 Comparison between P(g) in Q1D and 3D in insulating regime [19] ................................. 36 1 15 Comparison between P(g) numerically calculated via Anderson tight binding mo del and P(g) calculated analytically. The squares are data points taken from the tight binding Anderson model. The red curve is the conductance distribution calculated from the present analytic theory at the same PAGE 8 8 2 4 The 4 Greens functions involved in the diagrammatic expansion ..................................... 47 2 5 A first order diagram of external potential ........................................................................... 48 2 6 The other first order diagram of the external potential ........................................................ 49 2 7 Diagram equivalent to the two first order external potential diagrams .............................. 49 2 8 A first order interaction diagram ........................................................................................... 50 2 9 The other first order interaction diagram .............................................................................. 50 2 10 Diagram equivalent to the two first order interaction diagrams .......................................... 51 2 11 Equivalent diagrammatic expansion of G> for t > 0. ........................................................... 52 2 12 Potential well, and interaction vertices ................................................................................. 52 2 13 Greens function involved in expansion of G> ..................................................................... 53 2 14 Hartree self energy diagram .................................................................................................. 53 2 15 Fock self energy diagram ....................................................................................................... 54 2 1 6 Second order self energy diagram ........................................................................................ 54 2 17 Another second order self energy diagram ........................................................................... 54 2 18 Dyson expansion of the propaga tor ....................................................................................... 55 2 19 A typical vertex correction .................................................................................................... 55 2 20 A typical crossed diagram ..................................................................................................... 56 2 21 Typical diagram in expansion of G> (illustrated for N = 3) ................................................ 56 2 22 Dyson expansion of G> (illustrated for N = 3) ..................................................................... 57 3 1 Bare propagator in diagrammatic expansion for G> ............................................................ 60 3 2 External potential line appearing in diagrammatic expansion for G> ................................. 60 3 3 Interaction lines appearing in diagrammatic expansion for G> ........................................... 60 3 4 First order diagram in external potential, 1S ........................................................................ 64 3 5 First order diagram in difference interaction 2(diff)T ............................................................ 65 3 6 First order diagram in sum interaction 2(sum)T ...................................................................... 65 PAGE 9 9 3 7 Plot of erfc ................................. 66 3 8 Mean field approximation ..................................................................................................... 73 3 9 Self consistent mean field approximation ............................................................................ 74 3 10 Typical interaction diagram ................................................................................................... 75 4 1 Plot of eigenvalues density and its small x approximation Red solid curve is numerical solution of Eq. 4 26. Blue dotted curve is small 4 12 .................................................................. 90 4 2 Comparison between P(g) numerically calculated via Anderson tight binding model and P(g) calculated analyti cally. The red curve is that of Fig. 1 16. The blue curve is calculated from Eq. 4 12 ............................ 92 4 3 Comparison between P(g) numerically calculated vi a Anderson tight binding model and P(g) calculated analytically. The red curve is that of Fig. 1 15. The blue curve is calculated from Eq. 4 12 ...................................................... 93 4 4 Comparis on between var(lng) behavior in Q1D and 3D. Black line is linear best fit curve through the data points generated by Eq. 4 32. .......................................................... 94 4 5 Plot showing the severe behavior around g = 1 (lng = 0) The distribution is plotted 12 ............................................................................................................. 95 A 1 Contour used for calculation of G> ....................................................................................... 99 PAGE 10 10 Abstract of Dissertation Presented to the Graduate Sc hool of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ELECTRON TRANSPORT NEAR THE ANDERSON TRANSITION By Andrew Douglas May 2009 Chair: Khandker Muttalib Major: Physics In t his work we examined the probability distribution of conductances in three dimensional metals for signatures of the Anderson transition. To that end, we examined the generalized DMPK equation in the insulating state and developed a formalism to perturbati vely calculate the solution. From the solution we calculated the probability distribution of the conductance (g), and found excellent agreement with numerical simulations using the Anderson model. We also verified that the variance of lng varies with ave rage of lng to the two fifths power, in congruence with earlier numerical studies. Though not definitive, calculations give no evidence of a s ingularity in the distribution near g = 1 in contrast to the behavior in quasi one dimension. PAGE 11 11 CHAPTER 1 A NDERSON TRANSITION 1.1 Electron Localization in Mesoscopic Systems Mesoscopic systems are characterized by the phase coherence of their transport electrons. S ystems are mesoscopic when the system length L is much less than the phase breaking length, o f any present inelastic scattering mechanism: phonons, electron electron interactions B field, spin waves, for example. We will confine our attention to the case where these interactions are absent; so that the only relevant phase breaking length is that of the sample length itself. Phase coherent particles have a proclivity towards localization. We can use a path integral approach to demonstrate qualitatively how the localization arises [1]. Consider the propagation of an electron from a point r to a nother point r The probability of this occurring is: 2 2 *(,) 2Re()cos()ii ijij i i ijPrrAAaa (1 1) where i denotes a particular path through the sample starting from r and ending at r and ii iiAae is its amplitude. The interference terms in the sum average out to zero so that the probability of propagation from r to r is approximately just the sum of the probabilities of traversing each of the paths from r to r For propagation from r to r the situation is almost the same; there are paths wh ich interfere constructively, however. A path and its time reversed pair will have the same amplitude (assuming time reversal symmetry), and will therefore interfere constructively so th at we have for the probability of t hese two paths: 2 2(,) 4TR ii i iiPrrAAA (1 2) PAGE 12 12 This result is twice the probability of each of the separate paths, and so we see that the probability to return to the same point r is twice as high as we would ha ve guessed classically. W e can think of this back scattering as constru ctive interference of the electron wave with itself similar in this respect to the semiclassical picture of e lectron orbits. We can obtain an idea of the salience of this interference on the forward propagation of electrons. If the probability of circling back changes According to Land a u er [2 ], the conductivity is proportional to the probability of an electron tunneling through the sample. 0(1 T he relative correction to the conductivity will then be equal to the relative change in probability of the electron circling back on itself. At any time, t, the probability that it will intersect itself is roughly the volume it will traverse in time dt divided by the volume it has traversed already: 2 3/2/() dtDt where is the wavelength of the electron (roughly its width ), is the velocity, and D is the diffusion constant, which is 2(1/3)(1/3) Dl Integrating from the mean free time (this is the mi nimum time scale for which we can consider the electrons to be diffusing) to the ph ase breaking time, we have 2 3d ddt bDt (1 3) where lD and b is the sample width. The purpose of b is to isolate the part of the electron cloud volume that is in the sample. The corrections are therefore: 2 2 01D: ~ l bl (1 4) 2 02D: ~ln l lbl (1 5) PAGE 13 13 2 03D: ~ l (1 6) These corrections may well be smaller than th e zeroth order term given typical phase breaking mechanisms, but a different picture emerges if we turn these off. Then = L, the length of the sample, and in the thermodynamic limit the corrections will nullify the conductivity completely in one and tw o dimensions regardless of the imp urity concentration, while in three dimensions the relative correction will depend upon the impurity concentration. We also see that the correction is more severe in lower dimensions. This is because in lower dimensions there is greater chance for the electron to overlap with itself. Finally, localizations efficacy is inversely proportional to the energy of the electron. Elaborating, we can observe that when the Hamiltonian is spin dependent, the conductivity will b e enhanced ( anti localization ) since the spin operator will negate a TR path, causing the two pairs to cancel out of the sum above. This general picture also explains why a pplication of a magnetic field suppresses the weak localization effect; self inters ecting paths will acquire a phase difference proportional to the field strength. 1.2 Characteristics of L ocalization in 1 and 2 D imensions In one and two dimensions we have delineated how in mesoscopic systems the electron wavefunction constructively i nterferes with itself to the extent that it localizes into some f inite region of space. To incorporate this into the familiar Bloch picture of electrons in a crystal lattice, we may suppose the electron wavefunction ()r to be the usual Bloch wave, modulated by an exponential envelope, exp(/) r ( where is the localization length ), illustrated below: PAGE 14 14 Figure 1 1. Bloch wavefunction local We can estimate the localization which 00 In quasi one dimension ( Q1D)1 2In two dimensions 2 is given by: 2/ 2 blle and we would conclude, that likewise, in two dimensions all states are localized, though possibly over a length much larger than the actual sample length. We see that the localization length only depends on the dimensions of the sample, impurity co ncentratio n, and the wavelength (energy) of the electron. Generally speaking, dimension, decreases with disorder, and increases with electron energy. If we plot these lengths vs. disorder (W ~ 1/ ), we would obtain something like, 0 2 4 6 8 10 0 1 2 3 1w ( ) 2w ( ) w Figure 1 2 1D and 2D localization lengths plotted vs. disorder If we assume that the Bloch model can still be applied to these metals, namely that the eigen functions and energy levels are the usual Bloch states (now modulated by the exponential PAGE 15 15 envelope), and energy bands of the pure state we may sketc h a typical density of states for these systems. And we may say that all of these states will be localized symbolized by the dark filling. Figure 1 3 Typical density of states in 1D or 2D 1.3 Characteristics of L ocalizatio n in 3 D imensions In contrast to 1 D and 2 D in 3 D the electron wavefunctions do not localize until the disorder ( W ~ 1/ Therefore the localization length is infinite until this critical disorder is reach [3 ], depicted below, for a critical disorder of Wc = 6 (for illustration purposes). 0 2 4 6 8 10 0 5 10 15 3w ( ) w Figure 1 4 3D localization lengths plotted vs. disorder PAGE 16 16 The critical impurity concentration which localizes the electron wavefunctions depends on the wavelength of the electrons in question. In 3D the first states to be localized wi ll be the ones with the least kinet ic energy Such states are located near the top/bottom of the energy bands n(k), and at the edges of the density of states function (E) At some minimum disorder, localization will begin to trap these outlier states, and as disorder increases, the so called mobility edge will move inward, trapping higher and higher energy states [3 ]. At some critical disorder, WCFF is the Fermi wavelength), all states in the band will be localized. Figure 1 5 Typical density of states in 3D 1.4 Single Parameter Scaling Theory We may make the observation that we are dealing with a quantum critical phenomenon. D = 2 is the lower critical dimension. Making contact with the more familiar temperature driven phase transitions, an effective field theory can be constructed for this phase transit ion. In this case, the effective field, F, which goes into the retarded Greens function, is disorder averaged, the short wavelength degrees of freedom integrated out, and a new field Feff is obtained which describes the co upling between long wavelength degrees of freedom [4 ]. The conductance, g, emerges naturally as the scaling variable analogous to temperature in a finite temperature phase transition. 2 finite in the thermodynamic limit in the metallic state, PAGE 17 17 while zero in the insulating state. However the identification should not be pushed too far, as there is no nonat the critical disorder The scaling equation for conductance, ln () ln g g L (1 7) can be calculated perturbatively from the effective action using renormalized perturbation theory, or at least interpolated using the known values of g when g is large and small. T wo length scales define the conductance properties of the sample, l and When L << l we enter the ballistic regime When l << L << the electron wavefunction is extended over the length of the sample. And so propagation is diffusive This is the metallic regime, and within this regime, the conductance will scale as Ld2. When L ~ localization effects b ecome prominent. This is the crossover regime. And finally when << L, then is localized within the sample, and electron propagation is characterized by hopping, which is exponentially damped, by This is the localized regime. And the conductance will scale as 0exp(/) gL So generally speaking, we can say Ballistic Metallic Crossover~ Insulating Ll lL L L But it is important to remember that 1D, 2D systems are qualitatively different than 3D. If w e [5 ], PAGE 18 18 Figure 1 6. Scaling of lng with sample length We note that our previous observations are confirmed in the new language. D = 1, 2 always corresponds to insulating behavior, since as we increase L, ln g will always decrease since th ree dimensions the sample will scale towards the insulating or metallic state depending upon the impurity concentration, reflected by gc. 1.5 Role of Quantum Fluctuations We have been able to give a qualitative picture of why the localization phenomeno n occurs, and how it causes the phase transition in 3D. And we can calculate in princip l e the critical exponents from the single parameter scaling theory [3 ]. But our picture i s not completely satisfactory because the scaling variable, g, which appears a bove i s somewhat ambiguous. The same phase coherence that gives rise to the localization of the es also causes the absence of their self averaging because the electrons are correlated over the entire length scale of the sample. So g does not self ave rage in mesoscopic systems, and therefore PAGE 19 19 on t he particular samples impurity arrangements, since the phase transition itself will not depend on this. 1.6 Distribution in the Metallic R egime The Boltzman equation gives us the impurity avera ged conductance in 3D. But any given samples conductance w ill fluctuate around this average value, because different impurity arrangements will change the interference pattern the transport electron sets up with the impurities. Analogously we find fluctuations in g when we change the magnetic field, or the gate voltage, since this changes the phase/wavelength of the electrons thereby changing the interference pattern. But a question would be, how broad is the conductance distribution pertaining to different impurity arrangements, and these other effects?. In a typical (non mesoscopic) material, you can divide the sample up into pieces, and measure the conductance for each piece. Then the conductivity of the entire sample will be the average of the conductivities of each little piece. Assuming such, we would as 1/ In other words, we should expect: 0// VV where V0 is the volume over wh ich the conductivity is uncorrelated, typically about ( )3 where is the de phasing length, again. For ordinary materials, then, the relative fluctuations decrease as V increases. But this assumes that the observable is not correlated over the length of the sample, and that we can indeed split it up into these pieces. And this is where the argument fails for mesoscopic samples, since the electrons are correlated over t he entire length of the sample as the phase breaking length is L itself. And in fact we will find that the variance is unaffected by scaling the sample. Using similar path integral arguments, we can determine that 2var()(/)(2/15) ge independent of the impurity concentration [6 ]. The variance is only a function of the sample s PAGE 20 20 quantum mechanical symmetry. Thre if only TRS is present. These are the so called universal conductance fluctuations. Note that this var iance is typically much smaller than the samples actual conductance, g, at least for macroscopically sized samples. And this g, in the conducting regime, goes as L, so the relative fluctuations would be expected to decrease as a function of L. In this c onducting regime or s tate 0 5 10 15 20 25 0 0.2 0.4 P g g Figure 1 7. G a u ssian conductance distribution in metallic regime 1.7 Distribution in the Insulating R egime What does the probability distribution look like in the insulating regime? We can argue in the following way. In th e insulating regime, conduction is accomplished via tunneling, and we have that the probability of transmission from one side to the other goes as the modulus of the wavefunction, and so 0~exp(/) ggL We can split the conductor into n regions roughly on the size of the localization length. From the Landauer formula, we can say that 12/ // 12~......nL LL ngeeeggg (1 8) since the transmission probabilities would be multiplicati ve. In this case, PAGE 21 21 ln~ lni i iiL gg (1 9) So lng would be the sum of n independent random variables. And therefore we would expect that lng wou ld have a Gaussian distribution with mean: ln~lni i iiL L gg (1 10) and varia nce, var(ln)~varlnvar(ln)~ii iL ggng (1 11) So we have that the expectation and variance of lng would be proportional to each other. We wi ll note that the variance is on the order of the expectation of lng, and both of these increase as we go furth er in to the insulating regime. The distribution P(g) will typically look like, 0 0.2 0.4 0.6 0.8 1 0 10 20 P g g Figure 1 8. L og normal conductance distribution in insulating regime Clearly, in the insulating regime, there is no truly representative value for the conduct ance. 1.8 g ) for Mesoscopic S ystems In the metallic regime, g has a Gaussian distribution with a universal variance on the order of 1 /10 [1]. This universality stems from the fact that in the diffusive regime, the e lectron scatters from impurity to impurity and as such rather globally samples the material, washing out PAGE 22 22 the particular details of H. And given the universal variance, only 1 parameter, PAGE 23 23 mn(x,y)exp(ikz) that the electrons can occupy. At T = 0, only the electrons at the Fermi surface contribute to the current. Requiring that 2222 xyzFkkkk will fix the allowed (m,n,k) triads. We will find that there are 2/FNkA ( where A is the crosssectional area of the wire ) such propagating modes Since kF is roughly a lattice spacing, we see that the number of propagating modes is roughly the number of lattice sites along th e cross section. Given this, the wavefunction in the left and right leads will look like, 1(,,)(,)( )nnN ikzikz L nnn nxyzxyaebe (1 12) 1(,,)(,)( )nnN ikzikz R nnn nxyzxycede (1 13) respectively. The wavefunctions are normalized typically so that the current in each incoming channel is unity. From this, we can in principle calculate the wavefunction in the left and right leads, by solving Schrdingers equation inside the material and demanding continuity/differentiability of the wavefunction across the bound aries. Thus we can in principle determine the B, C coefficients. We can formally relate L to R in the following way. The S matrix will relate the incoming flux to the outgoing flux. CA BD S (1 14) We can define an S matrix as: tr rt S (1 15) where t+ is NN matrix, whose elements (modulus squared) give the fraction of current in the i th channel on the left which transmits through into the j t h channel on the right. tis the analogous NN matrix from right to left. r+ is the reflection matrix whose elements (squared) PAGE 24 24 give the fraction of current in the i th channel on the left which reflects into the j th channel on the left. And ris the a nalogous reflection matrix on the right side. If we define Ti by, 2N i ij jTt (1 16) then it will represent the fraction of current that starts in channel i and transmits to the other side of the sample. Then the conductance can be calculated using Landauers formula [2 ], 00 0Tr() /i iggTgtt ge (1 17) Well note that from the equation above, we can equate the transmission coefficients Tn with the eigenvalues of the matrix tt. In the metallic regime/state, we can determine that, Tn ~ z (where Lz is the longitudinal length across which the leads are connected) by comparing the Drude conductivity formula to the Landauer expression above. In the insulating regime, the conductance goes as exp( Lz ld expect that transmission eigenvalues would scale like this as well. We will want to determine the probability distribution function for these values Ti, as a function of length of the sample. From this we can calculate the probability distribution of the conductance g. To facilitate writing an equation for the Ti, we go back to the original setup. It is L R via: M AC BD (1 18) A useful feature of M is that it obeys a composition law. We can write it terms of the transmissi on/reflection matrices: // /1/ trrtrt M rtt (1 19) And M has the following polar decom position. PAGE 25 25 01 0 00 1 VU M VU (1 20) where U, U nn = (1 Tn)/Tn. where Tn are the eigenvalues of tt [8 ]. We have already interpreted the transmission eigenvalues Tn as the eigenvalues of the matrix tt. The matrices U, V can also be interpreted as roughly the matrix of eigenvectors of tt. 1.10 The G DMPK E qua tion In Q1D, the scaling equation for the probability distribution of eigenvalues, Ti, is known [8 n = (1 Tn)/Tn: ()2 1 (1)()  2aa ij a ij aap JpJ tJ N (1 21) where N is the number of transmission eigenvalues. We will now review the derivation of the generalized DMPK (GDMPK) equ ation describing the evolution of the distribution of transmission eigenvalues in arbitrary dimension, see [ 9 ]. Restricting ourselves to the case of SRS and TRS ( ) symmetry case, we can consider the following set up. Suppose we have a sample with le ngth L0 to which we attach a small piece L1. Figure 1 10. Perturbative set up of DMPK equation The transfer matrix, M, relating the wavefunction on the left to the one on the right, obeys the composition law. PAGE 26 26 10()LR LRMMM (1 22) As such we can write a convolution equation for M relating the probability distribution for the transfer matrices pL(M), at length L 1 11001()()()()LL LLpMdMpMpMMMM (1 23) which is to say t hat: 11 1()()LL L MpMpMM (1 24) To obtain the probability distribution of transmission eigenvalues we will need to perform a polar decomposition of the transfer matrices, M. So we note the following decomposition **01 0 00 1 u M UV u (1 25) Generally speaking the decomposition looks like, ()()() i ij i ijdMJddud J (1 26) So we can write, 111111000(,,)()()()(,,)(,,)LL L LpUVddUdVpUVpUV (1 27) Now we perform the ensemble average over M11, U1, and V1. This will provide us with the probability distribution for p distribution of p will have to integrate proceeding rotation to obtain, 111100(,)()(,,)(,)LL L LpVdMpUVpV (1 28) PAGE 27 27 Now we do a Taylor series expansion of pL out to second order integrated over the Us, and so wed have, 1111 11222 22 22(,) (,)(,) (,) (,) (,) (,) 11 () ()...22LL LL L MMM LLL M MMp pp p ppp (1 29) 11. We need to determine what these perturbative corrections are. The matrix X 1 12 4zzQQI X QMMQQ (1 30) So now we form this matrix out of 1 01MMM and we obtain, XXXw (1 31) where, **** 11111111111 ** ** 111 1111 11111 1(1) (1)1 w uu uu (1 33) ... ...abba aaa ba ab mn an am mn nmww w w (1 33) Before proceeding with the disorder average over M1 we note the following facts. First, if were in Q1D, d efined by the transverse length L PAGE 28 28 populated. This would be expected because in this situation, if we send in an ewith m omentum k, by the time it arrives at the other end of the metal, it will have scattered so many times, that its final momentum, k, will have an equal probability of being anywhere on the Fermi surface. Therefore adding M1 will not change this fact, and w to M1 perhaps, we will expect tha closed, and some open. But we do expect that this situation will not be changed by increasing length, at a certain point. The 3D situation is illustrated below; the LHS represents weak d isorder and the RHS strong disorder. In the strong disorder case, there is at best a single path which will allow the eto tunnel out of the sample. Figure 1 11. Transition amplitudes in delocalized, localized systems And so we will still make the as independent of M1. This assumption has been explicitly tested, numerically, and is seen to hold for Lz > L [10]. This assumption will then truncate our Taylor expansion to: 11 12 2 2(,) (,)(,) (,) 1 ()... 2L LL L MM L Mp pp p (1 34) PAGE 29 29 Now we are ready to disorder average over M1. First we note that M1 (the diagonal unit matrix) 11uI (1 35) 111 ** 111111 mmm m L mnmamnn mn L (1 36) T hese averages define how one takes the disorder average over M1 [9 ]. And thus we come to the 1 122 42 (12)  2(1)abab a a ab M ba ab ababaaa M (1 37) 22 2 4 2() ()()(12)2 (,) () (,) 2(1)() 2L LLL a a a abab L ambm ba m ab m L aa amam mp ppp d p d (1 38) T he matrix distribution is strongly peaked about its average, which wi ll legitimize pull ing the matrices out of the integrals [10]. We will note that in Q1D the me an field approximation is exact. PAGE 30 30 22 44 22 22 22(,)() () ()(,) (,)() () ()(,)LL am L am mm mm LL ambm L ambm mm mmpp d dp pp d dp (1 39) we then define the eigenvalue correlations, 22 4()(,) ()(,)L ambmab m L amaa mdp K dp K (1 40) whic h will have to be parameterized numerically. This brings us to the following equation, ()1 () (1) 2abaaaa a aa ab ab ab ab aapp KJ tJ K J K (1 41) In Q1D, we have, 1 1 1ab ab abKL N (1 42) and the original DMPK equation is recovered. Studies of the matrix K [10] suggest a simple two parameter model in the strong disorder insulating regime (W >> Wc), namely that in 3D, 11 12~/ ~/2aa abKKl LL (1 43) Even though the matrix elements Kaaab deviate from K11 12 somewhat substantially, in the ins ulating state, it is the interaction between the first eigenvalues which dominates the properties of the material namely the conductance, etc. And so it is most important to model these correctly. That this simplified model can capture the full behavio r of the Anderson transition is demonstrated in [11]12 serves as an effective order parameter, as evinced by the PAGE 31 31 following plot (Fig. 1 12). In the large L thermodynamic limit it scales to 0, or 1 in the insulating, critical, and conducting regimes respectively. Figure 1 12. Demonstration that 12 functions as an order parameter [10] Therefore we can study the metallic, critical, and insulating regimes by tuning the values 12 and K11. 1.11 Mapping the GDMPK E quation to Schrdinger Equation In order to solve this partial dif ferential equation for the probability distribution of the transmission eigenvalues it is convenient to make the aforementioned approximation that 11 12,aa abKK and define K11 z/ switching variables to 2sinhiix maps our GDMPK equation to: 1 22 12()1 () 2 lnsinhsinhlnsinh2i iii jii ij ip p txxx xxx x x (1 44) This partial differential equation must have initial and boundary conditions to enable a unique solution The initial conditions would describe transmission at length 0, and thus we should have ballistic initial conditions the probability of transmission Ti is equal to 1 (xi = 0) for each channel. PAGE 32 32 (,0)()i ipxx (1 45) The boundary conditions can be determined from probability conservation. We rewrite the GDMPK equation as a conservation equation. 1 2i iip p t Fp p xx F (1 46) If we integrate over dNx in, say, a Gaussian cylinder perforating the xk = 0 face with a height, h (half the height in 0kx area and half in the 0kx area), t hen taking the 0 h (for t will give us 0lim ()0 lim()0i ix ii xp xx p x x (1 47) Note that these boundary conditions are not so much as imposed on the G DMPK equation as they are derived from it. Then, following [ 12], we map the GDMPK equation to a Schrdinger equation We can do this through the mapping 1(,)()(,)() 1 exp() 2 ptGt xyxxyy x (1 48) The G DMPK equation transforms into a complex time Schrdinger equation for G so that G HG t (1 49) 2 22 1212 2211(12)1 2 2sinh2 (2) 11 4sinh()sinh()ii ii ij ijijH xx xxxx (1 50) PAGE 33 33 If we assert that p( x ,t y ) obeys symmetric initial conditions, 12 12 1122 ()(,,...,;0,,...,) 1 ()()()...() !NN S NNpxxxtyyy xyxyxy N yy (1 51) (which reduce to the usual aforementioned initial conditions when the small y limit is taken) then we can show that G obeys the initial conditions, 12 12 ,1122 ()(,,...,;0,,...,) 1 ()()()...() !NN SAS NNGxxxtyyy xyxyxy N yy (1 52) where G variables. Under the small y limit, we will recover the proper initial conditions on p. The boundary conditions on G transform to: 1 (,)0 sinh2 lim(,)0iii xGt xx Gt xy xy (1 53) 1.12 Distribution in Q1D The DMPK equation can be solved exactly in all three symmetry cases In the conducting regime, defined by LZ [6 ]: 2 2/222/2 2()sinhsinhexp()sinh2jiji iii ij ipxxxx xxx x (1 54) defined to be z. From this, the conductance distribution can be obtained [12]. 21/3 1 ()~exp 22/15 g Pg (1 55) PAGE 34 34 The expectation of g is identical to the Drude + weak localization corrections result. The in accordance with the discussion above. A more careful analysis of the distribution shows severe properties near g = 1. It has been demonstrated [12] that the distribution possesses an e xponential cutoff at g = 1 + ewhose severity diminishes smo othly as a function of disorder. When Lz In this regime, the exact solution reduces t o [ 13]: 2 2/222 2()sinhsinh()exp()sinh2jiji iii ij ip xxxx xxx x (1 56) Since the eigenvalues, xj, are exponentially separated, i t follows from this that the conductance distribution will look like [14]: 21 ln~expln(/4) 4 Pg g (1 57) And so we have in the insulating regime  PAGE 35 35 Figure 1 13. Evolution of P(g) with length in Q1D [15] 1.13 D istribution in 3D For the case of TRS and SRS 12 we will expect to find a Gaussian distribution with average conductance consistent with the Drude formula and weak localization corrections derive d above. And we also expect a The distribution will be Gaussian as before, but it is hypothesized that there is no singularity near g = 1 in the 3D case, and that this marks the dif ference between the Q1D and 3D situations, namely that in Q1D, the wavefunctions are always localized, while in 3D, the wavefunctions in the metallic state are extended rather. However, in the insulati 12 << 1) we find a qualitatively different distribution than the one found in Q1D. In the insulating state we find that does vary directly with the PAGE 36 36 PAGE 37 37 1 2222 12 2()exp(,)() (,)lnsinhsinhln 2 1 ()lnsinh2ln 2NN iji ij i ij jiji ii iip uxxx uxx xxxx xxxx x (1 58) From this the conductance distribution can be calculated by forming the integral: 2 1 1 (,) 1()...()sech ...N Nn n fg NPgdxdxpgx dxdxe xx (1 59) In order to avoid calculating N integrals, one can calculate the first two integrals and approxima te the remaining N 2 as a functional integral over a continuous density of eigenvalues. 12 12() ()exp(,,,) Pgdxdxxfgxx D (1 60) Using 22/ 2 01 ()limxxe (1 61) and separating out the first two eigenvalues, we may write f as 2 22 21212 12 2 22 2 12 2()()(,) ()()(,)(,)()()(,) 1 sechsech()sechx x xx xfxxuxx dxxxuxxuxxdxxdyyuxy gxxdxxx (1 62) For simplicity, one can make the approximation : 12() (,) ()lnsinh2ln () uxxy uxy ux xx uyyx (1 63) PAGE 38 38 This simplifies f to the form: 22 21212 2 22 2 12 2()()(,)()()2()()() 1 sechsech()sechx xx xfxxuxxdxxxuxuxdyy gxxdxxx (1 64) A saddle point approximation can be made to evaluate the functional integral. Therefore we must minimize f with respect to negative and normalized to: 3()2xdxxN (1 65) The first constraint can be implemented by restricted the range over which x2 may vary. The second can be implemented using Lagr ange mult ipliers Taking the functional derivative we have, 2()20xfdxxN (1 66) Defining, 2()()2() h()sech Vxxux xx (1 67) The solution of this equation i s 12 1 2()() () () ()SPVxhx Vx ux hx (1 68) It is observed that 2(,)0SPxx for certain values of 2xx if 2 x is allowed to range over all real values. Thus to enforce the non negativity of the density we require 22minxx Taking account of all the constraints yi elds the following expression (for g < 1): PAGE 39 39 2 2 122s2 212 2min() () exp((,),;(,) h()h((,))p xx Pgdx Fxgxxxx xxgx (1 69) 22h() ()2 h() ()xx xdxx ux Plotting P(lng) vs. lng yields the following typical result s in the deep insulating state 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 30 25 20 15 10 5 0 ln(g) P(lng) F igure 1 15. Comparison between P(g) numerically calculated via Anderson tight binding model and P(g) calculated analytically The squares are data points taken from the tight binding Anderson model. The red curve is the conductance distrib ution calculated from the present an alytic theory at the same PAGE 40 40 0 0.02 0.04 0.06 0.08 0.1 0.12 60 50 40 30 20 10 ln(g) P(lng) Figure 1 16. Comparison between P(g) numerically calcul ated via Anderson tight binding model and P(g) calculated analytically The squares are dat a points taken from the tight binding Anderson model. The red curve is the conductance distribution calculated from the present analytic theory at the same PAGE 41 41 0 10 20 30 40 50 60 70 1 2 3 4 5 6 <lng>2/5 var(lng) Figure 1 17. P lot of var(lng) vs. < lng>2/5 As aforementioned, according to [16 ] we expect linear be havior. Separating out more eigenvalues fails to improve agreement, and actually exacerbates the disagreement. In spite of appearances to the contrary however the GDMPK equation does accurately describ e the behavior of 3D metals as indicated in [ 11]. The problem with the analysis above is that it is 12 ] 12 solution. Doing so violates the requisite mathematical symmetry the actual solution possesses. Furthermore in hindsight, the saddle point approximation to the N dimensi onal integral leaves out the contribution of the entropy to the free energy, f which must be included in order to obtain an accurate eigenvalue density In the following chapters we will design a perturbative solution to the GDMPK equation which preserv es the symmetry of the exact solution, as well as incorporate s the interaction terms neglected above. Thereupon, we will improve the methodology used to approximate the N dimensional integral through which P(g) is obtained. PAGE 42 42 CHAPTER 2 DIAGRAMMATIC EXP ANSION OF THE PROPAGATOR 2.1 Introduction In the previous chapter the GDMPK equation was reduced to a complex time Schrdinger equation. Let us consider this equation in generality. So we suppose we have N particles in a potential field, subject to singl e particle boundary conditions, and mutually interacting. We will consider the function 11(,)(,...,;,...,)NNGtGxxtyy xy which will satisfy the Schrdinger equation in the variable x 2 2 111 ()(,) 2NNN n mn n n mn nG VxxxG tmx (2 1) and the coordinates will be assumed to range between a and b conditions, from which any other desired initial condition can be easily obtained. Well be interested in two cases symmetric initial conditions, and anti symmetric initial conditions. 12 12 1122 ()(,,...,;0,,...,) 1 (())()()...() !NN NNGxxxtyyy xyxyxy N yy (2 2) y ) signifies all permutations of the order of the coordinates y1, y2, yN, and (())y gives 1 for all permutations of y if were interested in the symmetric case, or gives 1 for even permutations of y and 1 for odd permutations if were interested in the anti symmetric case. We will also consider that it obeys single particle boundary conditions of the form, 12()(,,.,,..,)0j jNBxGxxxx (2 3) for each coordinate xj. These conditions may t ake the form of enforcing periodicity in each coordinate, or that the Greens function vanish es at the endpoints of each coordinates interval, etc. PAGE 43 43 2.2 Explicit E xpression for GN in 1st Quantized N otation Let j enumerate singl e particle states that satisfy the boundary conditions. Then let, 12 12 ()1 ... (())... !NNjjj jjj Njjj (2 4) enumerate symmetrized (or anti symmetrized) N particle states constructed out of these. And let, 12 ()1 (())... !Nxxx Nxxx (2 5) be symmetri zed (or anti symmetrized) N particle position eigenkets. Then the solution to the Schrdinger equation is merely the N particle propagator in position space, expanded in the basis that satisfies the single particle boundary conditions. ,1 (;) 1 (;)Ht HtGte N e N Gt jj jjxyyx yjjjjx yjjjjx (2 6) We see that this equation does satisfy the Schrdinger equation, for we have, 111 () !!!Ht Ht Hte HeHe tN N N yxyxxyx (2 7) This expression also satisfies the initial conditions as can be seen by setting t = 0, and evaluating the result usin g the definition of  x > and  y >. The boundary conditions are satisfied as well, as long as one uses the representation above in terms of the complete basis of single particle states which satisfy those boundary conditions. This formulation of the solution lends itself perhaps to path integral methods of calculation, in addition to perturbative Feynman diagram expansions. PAGE 44 44 Well opt for the latter. The first quantized form of GN us to segue into a second quantized formulation. 2.3 Explicit E xpression fo r GN i n Second Quantized N otation In order to develop a Feynman diagram expansion of this expression, well like to go to second quantized notation. Let 0 denote the N particle vacuum state. And let () j create a particle in the state j which again is a single particle state satisfying the boundary conditions. we want anti symmetric ini write (;) Gt xy as: 11 111 (;) ()() 1 (,)() !NN Ht mn mn NN mn mnGt xey N xty N xy0 0 00 (2 8) In the second line we use the fact that H, given below, annihilates the vacuum: 2 211 () ()() ()()(,)()() 22b bb a aaHdxx Vxxdxdxxxxxxx mx (2 9 ) This expression for G satisfies the partial differential equation, as can be demonstrated by explicitly evaluating its time derivative, and using the partial differential equation satisfied by the position space annihilation operator. 0(,)(,)(,)(,)(,) xtHxidyxyytxt t (2 10) where, (,)(,)(,) xtxtxt (2 11) and PAGE 45 45 2 0 21 (,) () 2 Hxi Vx mx (2 12) It can be verified that the initial conditions are sa And finally, the boundary conditions are satisfied if we use the eigenbasis which satisfies the boundary conditions to expand the position space field operators As is typical, it is advantageous to switch to the interaction picture. So we introduce, 22 11 21 21(,)exp()(,)exp()tt II ttSttTdtVtSttTdtVt (2 13) Since S also annihilates the vacuum, we can insert S and S r to come to the following expression, 111 (;)(0,)(,)(,)(,)(,0)(,0) !NN mn mnGtStxtStStSty N xy0 0 (2 14) which we can notate as, 111 (;)(0)(,)(,0) !NN Cmn mnGtS xty N xy0 0 (2 15) where SC is the time evolution operator which contour orders the operators from t = 0, to t = and then back to t = 0. Instead of evolving from 0 to infinity and back, we may evolve from 0 to any finite time and back, but the calculations simplify if we extend the time evolution to infinity. Our evolution contour is illustrated below. PAGE 46 46 Figure 2 1. Temporal contour used for Greens function These contour ordered Greens functions are discussed in the appendix. The perturbative calculation of this quantity follows from standard perturbation theory [18 ]. We assign N incoming propagators at the coordinates xm, t, and N outgoing propagators at the coordinates yn, 0. Figure 2 2. General expansion of G> And we connect these propagators together via the interaction vertices: Figure 2 3. External potential and interaction vertices PAGE 47 47 These diagrams will then involve four different Greens functions. Figure 2 4. The 4 Greens functions involved in the diagrammatic expansion And they are given in Eq. 2 16. Assuming that the eigenfunctions (satisfying the boundary conditions) of the unperturbed k(x), the 4 Greens functions would explicitly be: () (,;,)(,)(,)()()(,;,) (,;,)(,)(,)0 (,;,)(,)(,)(,;,)() (,;,)(,)(,)(,;,)(ktt kk kGxtytxtyt xyeGxtyt Gxtytytxt GxtytTxtytGxtyttt GxtytTxtytGxtytt 00 00 00 00 ) t (2 16) And we would integrate over all internal positions from a to b, and all internal times from 0 to > isnt d efined for tt (because the sum diverges), unlike in the real time case. So G> is only partially defined. This same issue makes G++ completely undefined. G is the only well defined greens function. Nonetheless, we will formally work with these functions, and find that in the end completely finite results will emerge, completely analogous to the results we would find in the real time case. PAGE 48 48 2.4 General Features of the Diagrammatic E xpansion The diagrammatic expansion delineated above simplifies considerably. And we will consider these consequences below. First we will find that the expansion reduces to that of the time ordered Greens function for all positive times. Secondly, we will find that a large class of diagrams typi cally associated with such diagrammatic expansions do not appear. That is, there will be no self energy diagrams, no vertex corrections, and no crossed diagrams. Self energy diagrams will be absent because the self energy is the interaction of the partic le with the background field. But our background field is the vacuum In a similar vein, the interaction itself will not experience any corrections because there will be no background to mediate it Crossed diagrams can be thought of as propagation back wards in time, but this will be disallowed as we will see All propagators must proceed forwards in time. 2.4 .1 Equivalen ce to the TimeOrdered Greens F unction When we work out the diagrammatic expansion of G> we will find that the infinities inherent in the expression cancel, and the finite parts reduce to the expansion of the time ordered Greens function. To illustrate, well look at the first order diagrams in the single particle potential and interaction. To first order in the external potential, there are only two diagrams. Lets consider one of them. Figure 2 5. A first order diagram of external potential This diagram is given by the expression, PAGE 49 49 111 11 11)(,;,)(,;,)((,;,)tdxGxtxtdtGxtxtVxGxtxt (2 17) Its sister diagram is the following: Figure 2 6. Th e other first order diagram of the external potential which is given by, 111 11 11)(,;,)(,;,)((,;,)tdxGxtxtdtGxtxtVxGxtxt (2 18) This diagram is infinite too. But adding them together gives, 111 11 11)(,;,)(,;,)((,;,)t tdxGxtxtdtGxtxtVxGxtxt (2 19) which is completely finite. So well note that the two structurally identical diagrams add to give one result which can be written as 111 11 11 0)(,;,)(,;,)((,;,)dxGxtxtdtGxtxtVxGxtxt (2 20) i.e, Figure 2 7. Diagram equivalent to the two first order external potential diagrams PAGE 50 50 Let us consider the first order term in the interaction potential. Again there are two contributing diagrams. One is: Figure 2 8. A first order interaction diagram Without loss of generality we ll set 0 t Then well have, 112 1111122222(,;,;) (,;,)(,;)() (,;,)(,;)bb taaGxtxtyydtdxdxGxtxtGxtyxx GxtxtGxty (2 21) And its sister diagram: Figure 2 9. The other first order interaction diagram is given by, 112 111112 0 2121(,;,;) (,;,)(,;)() (,;,)(,;)bb aaGxtxtyydtdxdxGxtxtGxtyxx GxtxtGxty (2 22) Each is infinite, but a dding them together gives PAGE 51 51 1 11 11121112 0 22(,;,;) (,;,)(,;)() (,;,)(,;)tbb aaGxtxtyydtdxdxGxtxtGxtyxx GxtxtGxty (2 23) which is finite. Again well no te that the two structurally identical diagrams add to give one result which can be written as 1122 1111 00 1212 2222(,;,;) (,;,)(,;) ()() (,;,)(,;)bb aaGxtxtyydtdxdtdxGxtxtGxty ttxx GxtxtGxty (2 24) which is, Figure 2 10. Diagram equivalent to the two first order interaction diagrams It has been verified out t o secon d order that this reduction continues. Therefore it seems plausible that the a priori divergent calculation of G> reduces to the completely finite calculation of G. This is not surprising since we always consider positive times (since time is a length) and for t > 0, 111 (;) (,)(,0) !NN mn mnGt xty N xy0 0 (2 25) is identical to: 111 (;) (,)(,0) !NN mn mnGtTxty N xy0 0 (2 26) PAGE 52 52 Therefore the diagrammatic expansions should be equivalent. But we will note that there is an ( x ,t y ,0). Therefore it will satisfy a trivially different PDE, 1122 ()()()()...()NNGHGtxyxyxy t x (2 27) which for times greater than 0, reduces to what we are interested in. Therefore, we may replace our rules with the following. Calculate all diagrams o f the form, Figure 2 11. Equivalent diagrammatic expansion of G> for t > 0. And we connect these propagators together via the interaction vertices: Figure 2 12. Potential well, and interaction vertices which involves only the single greens func tion, PAGE 53 53 Figure 2 13. Greens function involved in expansion of G> And we would integrate over all internal positions, and times from 0 to From the theta function in the definition of Gwe see that in the diagrammatic expansion, particles only propa gate forwards in time. The arrow points towards later times. As a general rule, any diagram which would force the particle to propagate backwards in time will be 0. This eliminates a large class of diagrams from the expansion. For the following well neglect any single particle potential present in the Hamiltonian. Its presence does not affect the substance of the following observations. 2.4.2 Absence of S elf Energy D iagrams We will generally find that the bare propagator receives no self energy corr ections. Consider the Hartree self energy term. From now on, well leave off the extraneous ( ) signs on the vertices. Figure 2 14. Hartree self energy diagram The loop propagator is interpreted as the expectation of the particle density. But this is 0 since we are taking the expectation with respect to the vacuum. Next, consider the Fock diagram. PAGE 54 54 Figure 2 15. Fock self energy diagram This diagram is 0 for the same reason middle propagator to be interpreted as the particle density. Next let us consider the three second Figure 2 16. Second order self energy diagram This diagram is 0 because the pr opagator at the top of the bubble requires later times to proceed from left to right, while the one at the bottom of the bubble requires later times to proceed from right to left. Therefore, the two propagators cannot be both be non zero at the same time. Another is: Figure 2 17. Another second order self energy diagram This is also 0 because the righ t side interaction line operates at time t1, and the left side interaction line operates at time t2. If t1 > t2, then the second from right propagator is 0. If t1 < t2, (Fig. 2 17) is 0 for the same reason the Fock diagram is 0. A general argument for the absence of any self energy term can be PAGE 55 55 made. Consider the N = 1 case. This Greens function, G1, would have a general diagrammatic expansion of the form, Figure 2 18. Dyson expansion of the propagator But the differential equation satisfied by G1 is: 2 11 21 () 2 G VxG tmx (2 28) which is the same as that satisfied by the bare propagator. Thus G1 is simply the bare propagator. And therefore there can be no self energy corrections 2.4.3 Absence of V ertex C orrections Consider the following prototypical vertex correction. Figure 2 19. A typical vertex correction The upper interaction line operates at t1 and the lower one operators at t2, say. Again, if t1 > t2, then the propagator second from right is 0. If t1 < t2, then the propagator second from left is 0. Making similar arguments, we should find that the interaction receives no corrections in general. 2.4.4 Absence of C rossed D iagrams Consider a typical crossed diagram. PAGE 56 56 Figure 2 20. A typical crossed diagram This diagram too is 0. Let the back slash interaction line be t1 and the forward slash int eraction line be t2. Depending on which time is greater, either the middle top or middle bottom propagator will be 0. 2.4.5 General F orm of the D iagrammatic E xpansion Given the arguments above, we can observe that the general form of a term in the diagr ammatic expansion will be like the following, illustrated for N = 3. We have N horizontal lines for N particles. And all the interactions take place along the vertical, connecting any two particle lines. External potential lines can be inserted anywher e. Figure 2 21. Typical diagram in expansion of G> (illustrated for N = 3) We can prove t his supposi tion by the following argument. Taking N = 3 for illustrative purposes again (the argument for general N is completely analogous), supposing the above is correct, G3 can be constructed via the following recursive equation, PAGE 57 57 Fig ure 2 22. Dyson expansion of G> (illustrated for N = 3) This would be written in 4 vector notation as: 3123123 0 3123123 0 1233123123 121132233 3123123(,,,,) (,,,,) (,,,,,) ()()()()()() (,,,,,) Gxxxyyy Gxxxyyy dtdxdxdxGxxxttxxx xxVxxxVxxxVx Gxxxtyyy (2 29) If we operate on this equation with th e single particle differential operator, H0, (where H0 includes t he single particle potential), t hen we obtain, 3 0 3 123123()()(,,,,)ij ijHxxGtxxxyyy t (2 30) which is indeed our differential equation, for times greater than 0. Therefore this is the corr ect diagrammatic expansion. 2.4. 6 Extension of Temporal I ntegration to the Entire R eal Line Instead of integrating over the internal times from 0 to between nal time points on the vertices are 0 for any negative time. This extension puts the diagrams in a temporally translationally invariant form so that we may use examine the diagrams in frequency space. PAGE 58 58 CHAPTER 3 PERTURBATIVE SOLUTION TO THE GDMPK E QUATION 3.1 Introduction Let us return to the problem of calculating the probability distribution of the eigenvalues from the GDMPK equation, 1 22 121 2 lnsinhsinhlnsinh2i iii jii ij ip p txxx xxx (3 1) Since 12 is small, we will proceed with a perturbative expansion of the Greens function in terms of the interaction. But since it is prohibitively difficult to use the exact single particle eigenfunctions, we will also have to perturbatively incorporate the s ingle particle potential as well. To facilitate this, let us generalize the problem above by writing, 1 22 121 2 lnsinhsinhlnsinh2i iii ji i ij ip p txxx xxx (3 2) Then making the customary mapping, 1(,)()(,)() 1 exp() 2 ptGt xyxxyy x (3 3) transforms our equation into: G HG t (3 4) 2 22 1212 221(2)1 2 2sinh2 (2) 11 4sinh()sinh()ii ii ij ijijH xx xxxx (3 5) PAGE 59 59 with initial conditions, 12 12 ,1122 ()(,,...,;0,,...,) 1 ()()()...() !NN SAS NNGxxxtyyy xyxyxy N yy (3 6) symmetric (S) or anti symmetric ( AS ) initial conditions depending upon In the small y limit, we will recover the proper initial transform to: (,)0 sinh2 lim(,)0iii xGt xx Gt xy xy (3 7) Before proceeding we will make the following observation. In order to recover p( x ,t) from x ,t y ) we have to take the small y limit in: 12 121 0 /2 2 2 /2 /2 0 2 2 /2(,)lim()(,)() sinhsinhsinh2 lim (,) sinhsinhsinh2jii ij i jii ij iptGt xxx Gt yyy y yxxxyy xy (3 8) The denominator is singular in this limit, and so in order for this limit to exist, we must have that, 12/2 2 2 /2(,)~sinhsinhsinh2 1jiim ij iGt yyyy xy (3 9) But we will find that the individual terms in the diagrammatic expansion do not conform to this behavior. Therefore the customary perturbative expansion will not be sufficient to re cover p( x ,t). To recover finite results from our perturbative approach we will have to re organize the diagrammatic series to conform to this behavior. We will find that reorganizing the series exponentially will give well defined results. PAGE 60 60 3.2 Diagrammatic E xpansion of GN W e will start with th e free particle Fourier basis. And so we have the general FD rules, Figure 3 1. Bare propagator in diagrammatic expansion for G> Figure 3 2. External potential line appearing in diagrammatic expansion for G> and the two intera ction terms have the Fourier representation : Figure 3 3 Interaction lines appearing in diagrammatic expansion for G> where the dashed line interaction represents the csch2(x y) potential, while the solid line interaction represent s the csch2(x+y) potential. Both diagrams conserve momentum at each vertex. In the first, momentum flows down the interaction line in one direction or another, while in the second, momentum flows from the center of the interaction line outwards into the vertices. This can be interpreted as conserving total momentum in the first case, and relative momentum in the second. The free particle Greens function looks like, PAGE 61 61 2() 0 2 1(,) 2xy tGxtye t (3 10) which has the Fourier transform, 02 11 (,) /2q qGqi q i (3 11) The Fourier transform of the external potential, and interaction technically doesnt exist since the integrals diverge but a suitable expression can be found nonetheless. W e have that, cothcoth 2iqzq dzezi (3 12) which can be obtained from inserting the asymptotic serie perform a dubious integrate by parts, then we may suppose that, 2coth sinh 2iqxeq dx q x (3 13) This result is consistent with known identities and so we may take the right hand term as a good representatio n of the left Therefore the Fourier transform of the single particle potential and interaction can be supposed to be: 1212(2) () coth 24 (2) () coth 42 q Vq q q qq (3 14) These expressions give sensible results as well see below. 3.3 Exponential Expansion of P ert urbative S eries We will repeatedly find it convenient to organize the perturbative seri es exponentially [19 ]. For instance, 00 11()()()()exp()nn nn nnGtGtWtGtFt (3 15) PAGE 62 62 Then the first few terms, Fn, are given by: 11 2 221 3 33121 22 4 442121311 2! 1 3 11 24 Fw Fww Fwwww Fwwwwwww (3 16) where wn = Wn/G(0). We have two different perturbations, and so we will perform an exponential expansion in both perturbative parameters. We can expand G in powers of the two perturbations, 0 ()()()exp()ab NN VUab abGtGtFt (3 17) V refers to the coef U the coefficient in front of the interaction. F( ab ) is the abth order term in the exponential series. By comparison, the Taylor series expansion is, 0 ()()() ()ab NN VUab abGtGtWt (3 18) We can list a few of these terms, Fab in terms of wab. 1010 0101 11111001 2 202010 2 0202011 2 12 Fw Fw Fwww Fww Fww (3 19) One of the advantages of this approach is that truncation of the exponential series at any order will preserve the positivity of G, unlike the Taylor series expa nsion which can become negative should the perturbation become too large. Additionally, convergence to the exact result for G PAGE 63 63 (tested in simple cases) seems to be faster when an ex ponential expansion is used. Additionally, to facilit ate the calculation o f P(g) we will want p( x ) to be in the form p( x ) = exp( f ( x )). Primarily though, the motivation is provided by the following. The Taylor expansion of GN looks something of the form, 0 12()()(2)NNGGsymmSsymmTO (3 20) where 1S is the first order diagram in the external potential (see Fig. 3 4) 2T is the first orde r d iagram in the interaction (see Fig. 3 5 and Fig. 3 6) and symm indicates that we are symmetrizing each diagram appropriately. The first order diagrams are typically ln divergent for small y and x And in this limit we will shortly see that 0 1212 1 1212ln() ln ~(2) (2) 44 ln (2) (2) 4NN j ij NN j ij N ij ijy yy GG yy O (3 21) To recover p( x ,t), we must y ) and then take the small y limit. So then w e have, 12 12/2 2 2 /2 /2 0 2 2 /2 0 1212 1212 1(,)~ sinhsinhsinh2 lim sinhsinhsinh2 (2) (2) (2) ln() ln ln 444jii ij i jii ij i NNN N j ij ij j ij ijpt xxx yyy G y yy yy yx (3 2 2 ) But c learly the small y limit does not exist. But if we re organize the perturbative series exponentially, t hen we will have PAGE 64 64 12 12/2 2 2 /2 /2 0 2 2 /2 0 1212 1212 1(,)~ sinhsinhsinh2 lim sinhsinhsinh2 (2) (2) (2) exp ln() ln ln 444jii ij i jii ij i NNN N j ij ij j ij ijpt xxx yyy G y yy yy yx (3 2 3 ) Now to first 12, the singularities cancel, and the small y limit is finite So the first order term in the exponential series cancels the singularity to first order 12. It is suspected, but not verified, that including the second order term in the exponential series ( y ) to second order 12, and so on. So it would seem that the exponential expansion is the natural way to expand such Greens function s But to start, well only concern ourselves with the first order term. 3.4 First Order C orrection to 0 NGin S12 Limit Let 1S be Figure 3 4 First order diagram in external potential 1S and 2(diff)T be, PAGE 65 65 Figure 3 5 First order diagram in difference interact ion 2(diff)T and 2(sum)T be: Figure 3 6 First order diagram in sum interaction 2(sum)T s o that 22(diff)2(sum)TTT Then the first order result, W1(x ,t  y ) would be, 0 1 1 1()() ()[] 0 2 2(,)(,) ()[,](,)()(,)(,) ()(,,,)(,)mmNmm m mnmnNmnmn mnWt SxtyGt TxxtyyGt y yxyy xy y xy (3 2 4 ) where [,] mn means a sum over distinct pairs, and x(m,n) means the set of coordinates x excluding xm and xn. We have at 0th 12 = 0), the symmetrized, small y limit of the GF to be: 2000 2 11 1(,)(,)(,) 2N x t Nn nGtGxtGxte t x (3 2 5 ) Now lets consider 1S again. Well calculate this diagram in position space, 00 1 11(,) (,)(,)ikxikyit kk kkSxtyeeeGkVGk (3 2 6 ) PAGE 66 66 The s ** 1(,)()()()() kktt kk kk kk Sxtyexyeyx (3 2 7 ) ()() () (2) coth 24 82kk kk k kk izkxV xx ez dz zk (3 2 8 ) () 11 ~ coth ()1 (2)482 22 42 1 ~ln () 1 42 42ikx ikx k ikx ikxx ieiek kx k exek x (3 29) Taking the large x, small y l imit in S1, we obtain, 0 112/ ln1 (,)(2) erfc (,) 42 2/ xt y Sxty Gxt t (3 3 0 ) defining, 2erfc()erfc() 2xxex (3 3 1 ) plotted below: 0 1 2 3 4 5 0 0.5 1 erfcp x ( ) x Figure 3 7 Plot of erfc Well note that erfc PAGE 67 67 Now we turn to the calculation of 2(diff)T It is best to perform the calculation in frequency momentum space, in the center of mass reference frame. In that case the center of mass integrals are easy to perform, and the relative momentum integrals r educe to a calculation similar to the one performed above. We obtain, 2(diff)1212 0 12 12 1212 21212(,,,) / ln()1 (2) erfc (,,,) 42 2/ Txxtyy xxt yy Gxxtyy t (3 3 2 ) where 0 21212(,,,) Gxxtyy is the un symmetrized two particle GF. Next we want to calculate, 2(sum)T Again, we switch to frequency momentum space, and the center of mass reference frame. In this case, it is the relative momentum which is free, and the total momentum which carries the interaction. We obtain, 2(sum)1212 0 12 12 1212 21212(,,,) / ln()1 (2) erfc (,,,) 42 2/ Txxtyy xxt yy Gxxtyy t (3 3 3 ) Now we may construct F1. Sup pose that we consider bosonic symmetrization. Then the N particle symmetrized G0 would be, in the small y limit, 00 1 1(,)(,)N Nn nGtGxtx (3 3 4 ) The N particle symmetrized S1 would be, in the small y limit, 0 1 1()() ()[] 0 []1 ()(,)(,) 2/ ln 1 (,)(2) erfc 42 2/N mmNmm m m m N mS SxtyGt N xt y Gt t yy xy x (3 3 5 ) Dividing by 0(,)NGt x we obtain, PAGE 68 68 0 []2/ ln 1 (2) erfc 42 2/m Nm m Nxt Sy G t (3 3 6 ) Now lets consider the two particle functions. The N particle symmetrized T2 diagram would be, in the small y limit, 0 2 2(,)(,) ()[,] 0 1212 [,] 0 1212()(,,,)(,) / ln() 1 (2) erfc (,) 42 2/ / ln() 1 (2) erfc ( 42 2/N mnmnNmnmn mn mn mn N mn mn mn NT TxxtyyGt xxt yy Gt t xxt yy G t yy xy x x[,],)mnt (3 3 7 ) And now, dividi ng by 0(,)NGt x we obtain, 1212 0 [,] 1212 [,]/ ln() 1 (2) erfc 42 2/ / ln() 1 (2) erfc 42 2/mn N mn mn N mn mn mnxxt T yy G t xxt yy t (3 3 8 ) Therefore, 1 1 00 [] 1212 [,] 12122/ ln 1 (2) erfc 42 2/ / ln() 1 (2) erfc 42 2/ / ln() 1 (2) erfc 42 2/NN NN m m m mn mn mn mn mnST W F GG xt y t xxt yy t xxt yy t [,] mn (3 39) And so up to this order, and tak 12 limits, 0 1()()exp()NNGtGtFt will look like, PAGE 69 69 120 22 22 12 122/ 2 ()()experfc 2 2/ / / experfc experfc 22 //m NNmmn mmn m mn mn mnxt GtGtyyy t xxt xxt tt (3 4 0 ) As aforementioned, we see that the exponential resummation sums the diagrams that preserve the required symmetry of G to first order in 12 ). And now we can fill this into our expression for p, and ta ke the small y limit. If we define 11111 2Zzl tKtKLL (3 4 1 ) then well have the main result of this section. 2 22 12 12 12()explnsinh2erfc 2 lnsinhsinh 2 exp 2 2 erfc erfc 22 22iii i ij mn mn mnpxxx xx xx xx x (3 4 2 ) We can see that the single par ticle potential pushes the distribution away from the origin, as one would expect. Additionally, as we increase length and so we can see that the particles are diffusing away from the origin. It is also apparent that the interaction term serves to increase the separation of the particles as well, and this separation is se en to increase with length. 3.5 First Order C orrection to 0 NG in Small (2 mall 12 Limit Our result is somewhat deficient can calculate the exact greens function for t he single particle case, and see that asymptotically (in fact this is the case already for x = 2, the wavefunction is exactly sin(kx). This is the case because even though the single particle PAGE 70 70 pote 2 limits dictates that the wavefunction go as exp(ikx), sin(kx) in those limits. This can also be seen from the derived boundary conditions. Since therefore, t 12 limit. Note that we will still be using bosonic symmetry, but with a sin(kx) single parti cle eigenbasis. We have at 0th 12 = 0), the symmetrized, small y limit of the GF to be: 20 0 0 /2 11 0 12 (,)(,)(,) sin()sin()N kt Nn k nGtGxtGxyekxky x (3 4 3 ) In the small y limit we have, 02 1(,)4exp 22 Gxy xxy tt (3 4 4 ) Now let us consider 1S again. Performing a calculation analogous to what we did above, and t aking the large x, small y limit in S1, we obtain, neglecting other terms which are small, 0 11ln (,)(2)(,) 2 y Sxty Gxty (3 4 5 ) The interaction diagrams can be calculated in the sin(kx) basis by writing sin(kx) ~ eikx eikx, and relating each piece to the Fourier basis interaction term we calculated ab ove. We find that the new T2 is obtained from the old via: 21212 21212(,,,)()(,,,)oldTxxtyyPTxxtyy (3 4 6 ) where () P is ( 1)n and n is the number of negative signs in the permutation The result is, in the small y1, y2 limits, PAGE 71 71 (0) 1212 2 12122 12 (0)1212 2 12(2) ln()ln()() 4 / 1 erfc 2/ (2) () 2 / 1 erfc 2 / T yyyyG xxt t G xxt t xy xy (3.4 7 ) There are other terms as well, which contribute to an effective two particle/ one particle potential, but they are minor contributions compared to these. Now we may construct F1 as before, so that up to this order, the N particle GF, will look like, in the small 2 12 limit, 122 0 22 22 12 12()() / / experfc experfc 22 // NNmmn m mn mn mn mnGtGtyyy xxt xxt tt (3 4 8 ) Now lets observe that in the small y limit, we have that: 02(,)4exp 22N nnn nGxy xxy tt (3 49) If we fill this in, then we have, 122 22 22 12 12()exp  2 / / experfc experfc 22 //Nn nmmn n mmn mn mn mnGtxxyyy t xxt xxt tt (3 5 0 ) So well obs erve again, that in the perturbative 12 y ). And now we can fill this into our expression for p. We obtain the main result for this section. PAGE 72 72 2 22 12 12 12()explnsinh2ln 2 lnsinhsinh 2 exp 2 2 erfc erfc 22 22i ii i ij mn mn mnpxxx xx xx xx x (3 5 1 ) Comparing this result to the previous one, we see that the major effect as been simply to make the replacement, erfclniixx which further adds to the repulsion of the eigenvalues from the origin. There are extra interacti on terms in the exponent, but as with before, the contributions they make are much smaller than the ones shown above. 3.6 Extracting the S ingular Behavior from the D iagrammatic Expansion Now well note that even using the almost exact single particle bas is, has not had a significant effect on the two particle interaction term vs. what the free single particle basis produced. Therefore it would seem that the single particle potential contribution to G is somewhat independent from the two particle contribu tion. We may suppose that this is because the single particle potential does not distinguish between positions (except for very close to the origin) and so it does not influence the expectation of the interaction energy, except close to the origin. There fore we ought to be able to separate the effects of the single particle potential from the effects of the interaction especially in the insulating state where only large values for xn are relevant This will also have the advantage of separating the ter ms which are responsible for satisfying the symmetry requirements of G, from the remainder. We have in mind the following kind of approximation of the diagrams, PAGE 73 73 Figure 3 8. M ean field approximation where we decouple the external potential and interacti on terms. L ets go back to Eq. 3 17; separate out the terms in the exponential expansion which dont couple V (the single particle potential) and U (the two particle interaction ) These are the Fa0 and F0b terms. Then we could write the exponential expa nsion as, 0 00 0,0 0 00 0,0()()exp ()expexpexpn n ab NN VnUn VUab n nab n n ab N Vn Un VUab n n abGtGtFFF GtFF F (3 5 2 ) Now use the fact that by definition, 0 0 0 0()()exp ()()exp Vn NN Vn n Un NN Un nGtGtF GtGtF (3 5 3 ) where ()V NGt is the exact N particle Greens functions, exclusive of the interaction, and ()U NGt is the exact N particle Greens function, exclusive of the single particle poten tial. This allows us to write 0 00 0,0()() ()()exp ()()VU ab NN NN VUab ab NNGtGt GtGt F GtGt (3 5 4 ) PAGE 74 74 Now, 0 00()() ()() ()()VU NN NN NNGtGt GtGt GtGt (3 55) is what we get when we sum all diagrams under the approximation illustrated in Fig. 3 8, while exp[ ] contains all of the corrections to this approximation. Note ()V NGt is exactly determinable, since we can solve for the GF in the single particle case, by exactly solving for the s ingle particle eigenfunctions. The next question is how we may calculate the N particle ()U NGt ? Whichever approximation we use, we would like to reproduce the known small y form of GU, namely that, 12/2 22(,)~sinhsinh 1U N jim ijGtyyy xy (3 5 5 ) so as to preserve the mathematical symmetry of the exact solution. In the same way as above, we will approximate the diagrams in such a way as to separate the singularity canceling behavior from the rest of the behavior. The singularity canceling part we can determine exactly, and the rest we will treat as corrections What we have in mind is something of the sort, Figure 3 9 Self consistent mean field approximation PAGE 75 75 To that end, instead of the usual perturbative grouping of diagrams, where we g roup them according to the number of times we bring down the vertex from the time development operator (i.e., the total number of interactions) let us instead group them according to how many times each particle interacts with another. It seems advantage ous to do this because as we see above in Eq. 3 55, 2 particle correlations dominate over 3 particle correlations in some sense. Now each diagram will have some number, n, of interactions between particle s a and b, where both a and b can run between 1 thr ough N assuming N particles. For instance, the diagram below (Fig. 3 10), Figure 3 10. Typical interaction diagram has 3 interactions between particles 1 and 2, 2 interactions between particles 1 and 3, and 1 interactions between particles 2 and 3. Let j label all possible inter particle interactions, 12, 13, 14, 23, 24, 25, 34, 35, 36, etc., and let nj label the number of such interactions. j will run between 1 and N!/2 therefore, and nj can take on any value. Well label all diagrams the refore with the following notation: 12(...)N jnnn nWW (3 5 6 ) and call !/2 12!/2 12!/2 1...j j NN n n n nn jN j (3 5 7 ) PAGE 76 76 the order of the diagram. For example, the diagram above would be labeled, 321W since there are 3 in teractions between particle pair (12), 2 between pair (13), and 1 between pair (23) respectively. And this diagram would be of order 321 321 123 In this notation, the usual Taylor series expansion of the GF would look like, 0()() ()j j jn U NN n nGtGtWt (3 5 8 ) We would like to reorganize the series according to: 0()()exp()j j jn U NN n nGtGtft (3 59) The first few terms would be for the special case of N =3 : 100100010010 001001 2 200200100 2 020020010 2 002002001 110110100010 101101100001 0110110100011 2 1 2 1 2 fw fw fw fww fww fww fwww fwww fwww (3 6 0 ) where 0/jjN nnwWG Again, we can separate the sum into diagrams which involve only the same particles and those which cross particles. So similar to before, we can write, PAGE 77 77 !/2 12 12 1 2 !/2000... 00...0 00...0 0 00...0 00...0 000...!/2() ()exp ()exp ()...exp () exp ()N N j j jU N n nn Nnn N nnn n n nGt Gt ft ft ft ft (3 6 1 ) where the {nj to sets that mix particle interacti ons for example the term f101 above Now we recognize that analogous to before, 00....0 0 00....0()exp ()j jj j jn U nn n nGtG ft (3 6 2 ) i s the two particle Greens function connecting the njth pair of particles. So we may write, 120 !/2()()...expj j jn U UUU NNnnN n nGtGtGGG f (3 6 3 ) So now using a more conventional notation where 2(,)UGab is the exact 2 particle Greens function between particles a and b, we may write out the perturbative expansion for U NG to be, 0 2 0 2(,) ()() exp (,)j j jU n U NN n ij nGijt GtGt f Gijt (3 6 4 ) Now 0 2 0 2(,)() (,)U U NN ijGijt GGt Gijt (3 65) is what we would have if we summed all interaction diagrams under the approximation illustrated in Fig. 3 9. And the exp{ } term contains all of the corrections to this approximation. So al together, we would have, PAGE 78 78 0 2 00 0,0 2 0 2 00 2() (,) ()() exp exp ()(,) () (,) () ()(,)j j jV U n ab N NN VUab n ab ij N n V U N N ij NGt Gijt GtGt F f GtGijt Gt Gijt Gt GtGijt (3 6 6 ) We can approximate GN by leaving off the exponential corrections. Since G1 and G2 must have the same singular behavior as GN, at a reduced dimensionality of course, we can clearly see that th is approximation possesses the requisite singular behavior of GN. We should therefore, in the interests of preserving the mathematical symmetry of the actual solution, take this as our true 0th order approximation, and it would seem that this is the natur al way to perturbatively calculate the N particle propagator The zeroth order a pproximation is already quite accurate as we ll see. And well also note its utility, as it only requires calculating 1 particle, and 2 particle Greens functions, which can often be done exactly. Additionally, it doesnt require that the strength of the interaction, or external potential be small, only that two particle correlations predominate. In principle therefore, this expression could be an adequate starting point for the analysis of the insulating and critical states all the way up to the metallic state, provided the parameters K11 and K12 are suitably modified. Finally this approximation is well controlled so that corrections can be straightforwardly included. S imilar mean field approximations are seen elsewhere for instance in decoupling the 3 particle correlation function equation s describing a strongly interacting gas, or liquid [20 ]. 3. 7 Determining G2 for the GD MPK Interaction P otential The interaction problem defining G2 is: 2 2 2121211220 (,;,)()()U UUG HG Gxxtyyxyxy t (3 69) where PAGE 79 79 22 1212 22 22 12 12 12(2) 11 11 22 4sinh()sinh() H xx xxxx (3 70) So now we change variables to: z1 = x1 x2, z2 = x1 + x2, and G2(z1, z2) = g(z1)g(z2). Then the differential equation separates, and, looking for the e igenfunctions of the differential equation we have, 2 22sinh() zz (3 71) 1212 2(2) 4 Ek (3 72) We make the substitution 1/2sinh(cosh) zfz which maps the equation to the associated Legendre differential eq uation (with complex coefficients) : 2 2 2(1)2(1) 0 1 xfxf f x (3 73) where, 22 12 121/4(1)/4 1/2/2 (1)1/4 1/2 ik (3 74) Evidently, the (un n ormalized) solutions to this equation are (cosh)(cosh) fzPz [21 ]. P has an asymptotic representation [21 ] 12(1)/2 121 (cosh)~ 1 2 ()~ Re 1 (/2) 2coshikzPz z z ik ez ik z (3 75) PAGE 80 80 In order to make some progress we need an a If we restrict ourselves to the insulating regime, where 12 rite, 12 12() /2 ~ (/2) ikik ik ik (3 76) Note that this is the only point in the analysis where we have assumed that 12 And mathematically speaking, this is where our present solution will differ from the Filling this in, we obtain the result, 2 1 12 1221(/2) (cosh)~ sintan(2/) 2cosh k Pz kzk z (3 77) which makes the ( approximately 1/2 1 12()~tanhsintan(2/)kzzkzk (3 78) Keeping in mind that the small z behavior of the wavefunction ought to be 12/2z we can somewhat clumsily modify this to: 12/2 1 12()~tanhsintan(2/)kzzkzk (3 79) which is, numerically speaking, a very good approximation to the actual wavefunction for small z and large z when 12 is small. But in any event, the 12/2tanh z term will not play a role in the discussion to follow y ). Well note that this wavefunction (sans that tanh part) is the exact wavefunction for a repulsive delta function potential of streng th 12. This suggests that if were not interested in the small z behavior we may make the replacement: PAGE 81 81 12 2() sinh() z z (3 80) which might be a useful expedient for future perturbative calculations of GN extending beyond zeroth order. It is reassuring to note that our proposed Fourier transform of the interaction potential matches the Fourier transform of this delta function in the small q limit. Now if we calculate the GF we have, 22 12 11 22 12 12 12 1221212// 111222 2 00 /2 /2 /2 /2 1122 1122(,;,;) 1 ()()()()(,)(,)tanhtanhtanhtanhkt kt kk kkGzzzzt dkezzdkezz JzzJzzzzzz (3 81) and filling in the form o 2/1 1 12 12 0 2212 12(,) 1 sintan(2/)sintan(2/) exp cosh()22 2()2 erfc 2 22ks zzJzz dkekzkkzk zz zzezz (3 8 2 ) 12 12/212 12()1 tanh 22 erfc 2 22Tx xx (3 8 3 ) and taking the small z limit brings our expression for G2 to, 12 12 12 12/2 /2 21212 1 21 21 21 2 0 21212(,,,) ()()tanh()tanh() (,,,) Gxxtyy TxxTxxyyyy Gxxtyy (3 8 4 ) 3.8 Determining G1 for the GDMPK Single Particle P otential The single particle part of the problem is: 2 221(2)1 22sinh2 H zz (3 85) PAGE 82 82 In the same vein as above, we can solve for the eigenfunctions, and then construct the Greens function. In the large x, small y limit, we obtain, /2 10 1(,) ~tanh2 (,) Gxty xyGxty (3 86) 3.9 Zeroth O rder Expression for p(x ,t ) in the I nsulating State Inserting our expressions for G1 and G2 into our equation, our zeroth order calculation for GN(t) comes to: 12 12 12 122 /2 /2 /2()exp tanh ()()tanh()tanh()N iii i i ijij ij ij ijGtxxy TxxTxxyyyy (3 87) And consequently, our expression for p( x 12 12 12 12 12 12/2 2 2 1/2 /2 2 2 1/2 2 1/2 /2 /2(,) sinhsinhsinh2 sinhsinhsinh2 exp tanh2 ()()tanh()tanh()jii ij i jii ij i iii i i i ji ji ji j ijpt xxx yyy xxy TxxTxxyyyy xy (3 88) Taking the small y limit, we find that the singularities are exactly cancelled, as was our design, and our final result valid in the small 12 (insulating state) limit is : 12 12 122 /2 221 (,)explnlnsinh2 2 sinhsinh()()ii i i jiijij ijptxxx xxTxxTxx x (3 89) PAGE 83 83 3.10 Testing the V alidity of the Mean Field A pproximation First well n 12, this expression not surprisingly reduces to the earlier one (Eq. 3 51) obtain ed from evaluating the two first order interaction diagrams. Using the scheme above, we can estimate the exact N particle Greens function for bosons interacting c to be the following. 2 (,)exp 1 22 erfc 2 22ij Ni i ijxx Gtxcc x (3 90) We do of course have that when c = 0 the result reduces to the free particle GF. We should also have that when c = particle GF, since an infinitely bosons to vanish when they meet effectively making them fermions. So expanding the erfc 1/2z (since c is large), we find 2 21 11 22 erfc 2 22zzz O ccccccz (3 91) K eeping only the first order term in the large c limit, and inserting this approximation in to the expression above, GN would reduce to: 2(,)exp iji N i ijGtxxx x (3 92) which is indeed the correct expression for the N fermion free Greens function. This result is interesting since the large c limit i s a distinctly non perturbative result. Additionally we would like to note that the DMPK equation has been solved for the three cases insulating and metallic regimes, these complicated solutions simplify to explicit expressions and each agree s with the result that would be obtain ed via this approximation. This is also a distinctly nonperturbative result, since when PAGE 84 84 CHAPTER 4 STATISTICAL PROPERTIES OF GDMPK SOLUTION IN INSULATING STATE 4.1 Construction of P(g) It turns out that in the insulating stat 12 << 1), the two particle interaction T is quite a bit smal ler than the sinh2x interaction, so we can approximate our p( x ) result as: 1 22 12 2()exp() ()()(,) (,)lnsinhsinh 2 1 ()lnlnsinh2 2NN iij i ijpF FVxuxx uxy xy Vxxxx xxx (4 1 ) It is somewhat amusing to note that this is the expression one would obtain if one simply neglected the interaction entirely, obviating the need f or Chapters 2 and 3, and assumed the particles possessed bosonic symmetry, rather than fermionic symmetry as was done previously. So the distribution of the eigenvalues in the insulating stat e approximately assumes the form of a fractional random matrix m odel. Of course, T becomes more important (and changes form) the closer we come to the critical point, and this will change the form of the distribution accordingly in these states. Now we turn towards calculating the probability distribution of the conductance. I n the insulating stat e, the conductance is dominated by the first few eigenvalues and so it suffices to calculate the distribution of the first one or two eigenvalues to examine P(g). For example, let us consider the probability distributi on of the first eigenvalue. We can write, switching variables 2x 11 111() () 1 23 121()~ ...exp()VV NNPedddFeZ (4 2 ) where, PAGE 85 85 11 22 1212()()(,)(,) (())lnsinh2(()) () ((),()) (,) lnN i ij i ijFVuu Vx x V uxx u (4 3 ) and in the middle W e now have to consider how to calculate ZN. ZN is a classic al partition function of stationary particles. The procedure used heretofore (in Chapter 1) was to evaluate ZN by using a saddle point approximation. But we will have to be still more careful in our analysis as this approximation is inadequate in 3D So instead we proceed as follows. The standard thermodynamic definition of the free energy gives, in the large N limit, 12 1211 lnNFZUS (4 4) 12 serve s as an inverse temperature. T he expectation of the energy can be obtained via : 1 11(2) 11 ()(,)() (,)(,) 2 UdVu ddu (4 5 ) and (2) are the particle density and two particle pair density functions respectively and are respectively normalized to N and N(N 1) [20] The entropy can be obtained from the diagonal elements of the density matrix. 11 ()ln () Sd (4 6) Once calculated we would exponentiate to obtain 12expNZUS (4 7 ) PAGE 86 86 What remains now is to approximate and (2). If the interaction describes a dilute gas, then typically and (2) are expanded in powers of the density. This is the familiar cluster expansion of the f ree energy of a weakly interacting gas. Our situation requires a different approach h owever, since we have a confining external potential which aggregates the particles, and even more, we have a long ranged interaction which crystallizes the particles into a lattice like arrangement. The mathematical effect of this is that the cluster exp ansion of ZN will diverge with the particle number, N which is inconvenient because we want the large N limit of P(g) Another method must be used therefore to calculate ZN for large N. O ur approach will be to examine the integral equations satisfied by and (2) and approximately solve these thereby incorporating all orders of the density to within some approximation. The densities can be written as: ()()i i (4 8 ) (2)(,)()()()()()()(,)ij ijxK (4 9 ) where K is the pair co rrelation function. Following Beenakkers generalization of Dysons equation for non logarithmic interactions [17 ], the density function satisfies the following equation in the large N limit. 11 1212 ()(,)1ln()()(,) 2 du Vu (4 10) T he arbitrary constant acts as an effective chemical potential, allowing us to adjust the particle n umber T he two po int correlation function satisfies [17] 1121 (,)(,)() dyuK (4 11) PAGE 87 87 Now we can change variables back to x and obtain, 11 111() () 1 23 121()~ ...exp()Vx Vx NN xxxPxedxdxdxFeZ x (4 12) 11 22 22 12 12()()(,)(,) () (,) () (,) lnsinhsinhN i ij i ijFVxuxxuxx Vx uxx Vx uxx xx x (4 13) The density equation comes to: 112 12 1 12()(,)()(,)lnsinh21ln() 22xdxxuxxVxuxx x x (4 14) The two particle correlation function to, 1(,)(,)()xdxuxxKxxxx (4 15) and the free energy, is: 1 1112 12 1 12 121 ()()(,)lnsinh21ln() 2 22 1 (,)(,) 2x xxFUTS dxxVxuxx x x dxdxuxxKxx (4 16) To make further progress we have to evaluate the densities. 4.2 Evaluation of the D ensity It makes no difference to the solution (in the insulating limit) to neglect the 12 terms in the density equation and so we will. We write, 122 1 12()lnsinhsinh()(,)ln()xdxxxxVxuxxx (4 17) PAGE 88 88 In Q1D, in the insulating regime the eigenvalues are approximately independently distributed. However in 3D, while the beginning of the continuum (x1) eigenvalues is roughly constant and as such, the behavior of the first eigenvalue is very muc h influenced by its neighbors [7]. In order to make progress on solving the density equation we will make the approximation that sinh2x y 2max(x,y). This turns the integral equation into a Volterra like integral equation. For consistencys sake, w e will also make this approximation on r ight hand side, which will eliminate x1 from the expression. Furthermore, we make the saddle point approximation on the single particle potential. 2 1 12 2()(,)()() 111 ln ln2 422 2 1 1 2spsp sp sp spVxuxxVxVxx V x (4 18) These approximation s are qui te good in the insulating state and wil l bring the equation to: 112()()()()()ln()x sp xxdxxuxdxxuxVxx (4 19) where u(x) = 12x Setting x = x1 1 independent it will be of no concern to us. Taking a derivative we have, 1()()()lnx sp xd uxdxxVx dx (4 20) which implies an initial condition. () 1close to Vxexx (4 21) If we take yet another derivative then well obtain the non linear differential equation: PAGE 89 89 2 2 12() ()2()()0 () x xxx x (4 22) 12. Now since this equation is missing its independent variable, we may be the dependent variable, which reduces the equation to a nonlinear first order differential equation: 2 122 p p p (4 23) which is Bernoullis equation. B y making the substitution p = his will map ped into a linear differential equation. 2 122 24 y y (4 24) This can be solved and inverted to obtain, 1224ln d c dx (4 25) which can be integrated (in principle) to obtain the implicit solution: 0 1224ln ds xc sssc (4 26) The arbitrary constant c can be determined fro 12 for large x. Going back to the different ial equation, setting the derivative to 0 and solving for c gives c = 12). Now to determine more precisely the form of the density for x close to x1, we lns term in the equation above (Eq. 4 26) Completing the integral and inverting gives, 2() 1 1221 aroundxcexx e (4 27) PAGE 90 90 In order for the initial conditions to be satisfied we must set c x sp. And so doing gives us: () 1 122 aroundspVxe xx e (4 28) The density is plotted in Figure 4 1. Figure 4 1. Plot of eigenvalues density and its small x approximation Red solid curve is numerical solution of Eq. 4 26. Blue dotted curve is small 4 28. Both are plot 12 Putting the numerical solution into the original un approximated integral equation (Eq. 4 17) yields almost precise agreement. So we see that for x close to x1, the external potential determines the form of the density, whi le for larger x, the interaction dominates and crystallizes the eigenvalues onto a lattice. W ith regard to (2), w e can easily calculate the two particle correlation function by inverting its integral equation, but its contribution to F will be i ndependent of x1, and so will only affect the overall normalization. 4.3 Calculation of F Returning to F, we will make the same saddle point approximation of the external potential that we did for the density, and for consistencys sake neglect the 12 t erms that we s equation It is easy to include them, but they result in imperceptible changes in the insulating state. So we have, PAGE 91 91 1121 ()()ln() 2sp xFdxxVxx (4 29) In order to obtain the x1 dependence of this integral it suffices t o insert the xclose to x1 form of the density, which carries the major contribution to F. Doing so yields, 11 12 12 122 lnerfc() erfc() 2 8sp sp spFV xx xx ee e (4 30) As a result of not inserting the exact expression for the density into F, F(x1) will not have the proper depende nce for larger x1 > x sp 12, we see that F ought to go as 3 1x B ut the contribution to P(x1), and consequently P(lng) from these values of x1 is negligible and so the approximation remains valid. 4.4 An alysis of P(g) D istribution As noted, the analytical model of P(g) contains two parameters 12, is a measure of the disorder of the system, while 12 = L/Lz is a measure of the systems geometry. For definite comparison with n umerical results obtained from the tight binding model, well use, unless otherwise noted, the approximation 12/2 appropriate for cubic systems. We will note however that in the insulating state the form of the distribution little depends on the exact value of 12 In any event, the probability distribution for x1 becomes, 11 11 1 12()exp() ()()erfc() 8spPxfx fxVx xx e (4 31) x1s contribution to the conductance is sech2x1. Therefore g sech2x1. Changing variables, we have the following approximate conductance distribution. 14 (ln)expln 2 Pgf g (4 32) PAGE 92 92 where f is as defined above in Eq. 4 31. Typical results are shown below. A comparison with previous results is a lso made. In each analytical plot, the value of has been adjusted to match the expectation of lng characterizi ng the nu merical distribution, as before. In the plot below (Fig. 4 2) 0 0.02 0.04 0.06 0.08 0.1 0.12 60 50 40 30 20 ln(g) P(lng) Figure 4 2 C omparison between P(g) numerically calculated via Anderson tight binding model and P(g) calculated analytically The red curve is that of Fig. 1 16. The blue curve is calculated from Eq. 4 32. We used 12 Adding a little more order, we have PAGE 93 93 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 30 25 20 15 10 5 0 ln(g) P(lng) Figure 4 3 Comparison between P(g) numerically calculated via Anderson tight binding model and P(g) calculated analytically The red curve is that of Fig. 1 15. The blue curve is calculated from Eq. 4 12 Improvement in agreement with numerical simulations is quite evident. At the cost of simplicity, even better agreement can be obtained by separating out more eigenvalu es, and using the methodology above to calculate P(x1,x2), P(x1,x2,x3), etc., and from this calculating t he (see Eq. 1 59) into the N dimensional integral (Eq. 4 2) to calculate P(g) direc tly can be done in principal but would significantly complicate the eigenvalue density equation and so this was not pursued Another characteristic of the distribution was the peculiar dependence of var(lng) with < lng>. Using the formula above to ge nerate data points, we find behavior consistent with the observation that var(lng) varies directly with < lng>(2/5). PAGE 94 94 0 10 20 30 40 50 60 70 1 2 3 4 5 6 <lng>2/5 var(lng) Figure 4 4 Comparison between var(lng) behavior in Q1D and 3D. Black line is linear best fit curve throu gh the data points generated by Eq. 4 32. It is also desired to examine the behavior of P(g) near the point g = 1. In Q1D it was confirmed that the distribution possessed a slope discontinuity near g = 1. We can examine the distribution in the present case as well. Separating out 3 eigenvalues, and constructing P(g) yields, 3max 2max3 3min 2min3()(,) 3 2 123 2 11 ()(,)1 () exp(,,) sechtanhxgxgx xgxgxPgdxdx fxxx xx (4 33) where, 123123121323 3 12 12(,,)()()()(,)(,)(,) 2 lnerfc() 2sp spfxxxVxVxVxuxxuxxuxx V xx ee (4 34) 1 1 22 231 cosh sech()sech() x gxx (4 35) and, PAGE 95 95 1 2min3 2 32 (,)cosh sech xgx gx (4 36) 2 33 12 2max3 3 2 312sech 1 (,) cosh 12sech sech1 x gx xgx gx gx (4 37) 1 3min3 ()cosh xg g (4 38) 1 3max2 1 () cosh 2 2 g xg g g (4 39) The distribution has been directly differentiated in the vicinity of g = 1, and in contrast to the Q1D situation, no s ingularity has been found 10 8 6 4 2 0 0 0.1 0.2 P lng ( ) lng Fig. 4 5. Plot showing the severe behavior around g = 1 (lng = 0). The distribution is plotted for 12 This is possibly a consequenc e of the approximation of the free energy. The singularity present in the Q1D distribution occur s when x1 ~ 0, x2 ~ xsp (the minimum of the single particle potential well), and x3 ~ of not being able to evaluate the density for all PAGE 96 96 values of x > x1, our approximation of F in this rare configuration is lacking, as noted above. Therefore we will not expect this result to be definitive. PAGE 97 97 CHAPTER 5 CONCLUSIONS In the first part of this work we discussed localization delocalization phase transition which occurs in mesoscopic materials, and we reviewed the single parameter scaling theory which provided a framework within which the role of the dimensionality and quantum mechanical symmetry of the system played in determinin g whether or not the material underwent the phase transition could be understood. It was considered that the formulation of this scaling theory was not entir ely adequate because the scaling variable, g, was somewhat undefined. This motivated a consideration of the probability distribution of the conductance, P(g) and an analysis of this distribution for signatures of the transition. To wit, the generalized DMPK equation was presented. This equation, capable of describing conductors in all dimensions, was analyzed to determine P(g) in the insulating state. In order to accomplish this the formalism necessary to solve the many body Schrdinger equation was ex plicated, and suitable approximation scheme was introduced. This enabled us to proceed with the construction of P(g) by relating it the partition function of a dense liquid. Upon obtaining this partition function in the large N limit, P(lng) was calculat ed. Excellent agreement with numerical simulations confirmed that the generalized DM PK equation does indeed describe 3D conductors in the insulating state at the least Additionally the analytically calculated distribution was shown to satisfy the known var(lng) vs. PAGE 98 98 APPENDIX CONTOUR ORDERED GREENS FUNCTIONS Contour ordered Greens functions are a generalization of the more familiar time ordered Greens functions. The general definition is as follows. Suppose that we have a contour, such as the one illustrated in Figure A 1 perhaps. Call C1 the upper line and C2 the lower line. Then we have, in real time, 0120 12 120120 0210 21()() for (,)()() ()() forC CC Cinttn tt GttinTttn inttntt (A 1) 1 for fermions/bosons. The >C sign refers to t1 appearing after or before t2 in the direction of the contour. There are 4 generic possibilities. Both t1 and t2 can be on the top contour, or both can be on the bottom contour; otherwise one can be on the top, and one on the bottom. In these four cases, GC breaks down into: 120120121 120120 1221 12 120210 2211 120120212(,)()()for, (,)()()for, (,) (,)()()for, (,)()()for,CGttinTttnttC GttinttntCtC Gtt GttinttntCtC GttinTttnttC (A 2) Observe how in each case the operators are ordered by latest on the left, where latest is assessed in the contour sense. The time development operator can be written as: (0)exp()CC CSTdtsHs (A 3) where H is the perturba tion in the interaction picture. t runs along the contour, C, from 0 to and then from have, (;)(0)(,)(,0)CGxtySxty 00 (A 4) PAGE 99 99 defined, again by the contour: A 1. Contour used for calculation of G> We expand the time development operator to first order, (;) exp()(,)(,0) exp()()()(,)(,0) (,)(,0)()()()(,)(,0)C C C C CC CGxty TidsVsxty TidsVsssxty TxtyidsVsTssxty 00 00 000 0 (A 5) Now, t is located on the bottom contour, and 0 on the top so the first order term simply gives u s the greater Greens function, (,;) Gxty We can still use Wicks theorem as usual to break the second term into twoparticle correlations. So, suppressing the position index, we have, ()()()()(0) ()()()(0) () ()()()(0)C C CC C CCdsVsTsst TssTt dsVs TstTs 00 0000 000 0 (A 6 ) Now we lo ok to the integration around the contour. First the s variable runs from 0 to the top of C. Then s runs along the bottom of the contour from PAGE 100 100 0 0 ()()()()(0) ()()()(0) () ()()()(0) ()()()(0) () ()()()(0)C C CC CC CC CCdsVsTsst TssTt dsVs TstTs TssTt dsVs TstTs 00 0000 000 0 0000 000 0 (A 7 ) This can be written as: 0 0()()()()(0) ()()()(0) () ()()()(0) ()()()(0) () ()()()(0)C C CC CC CC CCdsVsTsst TssTt dsVs TtsTs TssTt dsVs TtsTs 00 0000 000 0 0000 000 0 ( A 8 ) Now keeping in mind that s is along the top of the contour in the first term, t is on the bottom segment of the contour, and 0 is on the top of the contour, we can break down the correlations according to the general rules above to obtain, 0 0()()()()(0) (,)(,0) () (,)(,0) (,)(,0) () (,)(,0)C CdsVsTsst iGssiGt dsVs iGtsiGs iGssiGGt dsVs iGtsiGs 00 (A 9 ) The causal, anti causal Greens functions are identical when evaluated at the same time, and so we have, 0()()()()(0) ()(,)(,0)(,)(,0)C CdsVsTsst dsVsiGtsiGsiGtsiGs 00 (A 10) PAGE 101 101 This is equivalent to the first order diagrams in Figures 2 5 and 26. 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Lett. 89, 246403 (2002) PAGE 103 103 [16 ] Conductance fluctuations in the localized regime: Numerical study in disordered noninteracting systems A. M Somoza, J. Prior, and M. Ortu o, Phys. Rev. B 73, 184201 (2006), DOI:10.1103/PhysRevB.73.184201 [17] C. W. J. B eenakker R ev. Mod. Phys. 69, 731 (1997) [18 ] E.M. Lifshitz, and L.P. Pitaevskii Physical Kinetics (Pergamon, New York, 1981) [19] Gerald D. Mahan, Many Particle Physics 3rd ed. (Kluwer Adacemic/Plenum Publishers, New York) [20] Donald A. McQuarrie, Statistical Mechanics (Harper & Row, New York, 1976) [21 ] I.S. Gradshteyn, and I.M. Ryzhik, Table of Integrals, Series, and Products 6th ed. (Academic Press, London, 2000) PAGE 104 104 BIOGRAPHICAL SKETCH Andrew Douglas was born in Jacksonville, Fl in January 1979. Except for a three year sojourn in Orlando, he spent all of his formative years in the sprawling metropolis of Orange Park. After realizing he was too slow to make a living being a professional runner, and too short (5 He is happy he did so. He graduated from the University of North Florida obtaining his B.S. in physics and mathematics, and obtained his Ph.D. from the Univers ity of Florida in 2009. 