Existence of a Phase with Finite Localization Length in the Double Scaling Limit of N-Orbital Models
EExistence of a Phase with Finite Localization Length in the Double Scaling Limit ofN-Orbital Models
Vincent E. Sacksteder IV ∗ Department of Physics, Royal Holloway University of London,Egham Hill, Egham, TW20 0EX, United Kingdom andDepartment of Physics and Astronomy, Rutgers University, NJ, USA (Dated: February 1, 2021)Among the models of disordered conduction and localization, models with N orbitals per siteare attractive both for their mathematical tractability and for their physical realization in coupleddisordered grains. However Wegner proved that there is no Anderson transition and no localizedphase in the N → ∞ limit, if the hopping constant K is kept fixed. Here we show that thelocalized phase is preserved in a different limit where N is taken to infinity and the hopping K issimultaneously adjusted to keep N K constant. We support this conclusion with two arguments.The first is numerical computations of the localization length showing that in the N → ∞ limitthe site-diagonal-disorder model possesses a localized phase if N K is kept constant, but does notpossess that phase if K is fixed. The second argument is a detailed analysis of the energy and lengthscales in a functional integral representation of the gauge invariant model. The analysis shows thatin the K fixed limit the functional integral’s spins do not exhibit long distance fluctuations, i.e. suchfluctuations are massive and therefore decay exponentially, which signals conduction. In contrastthe N K fixed limit preserves the massless character of certain spin fluctuations, allowing them tofluctuate over long distance scales and cause Anderson localization.
I. THE SITE-DIAGONAL-DISORDER MODEL
In the first sections of this article we study Wegner’ssite-diagonal-disorder model with N orbitals on each site,which at N = 1 reduces to the Anderson model. Theon-site Hamiltonian H i for site i is an N × N randommatrix which describes interactions between the N or-bitals at site i . It is the natural N × N generalizationof the Anderson model’s disorder potential. The meansquared value of H i ’s matrix elements is proportional to N − , so that when a single site is taken in isolation, itsspectral density (except for tail states) is in the inter-val E = [ − , +2]. This interval is independent of N ’svalue. The averaged eigenvalue density ν ( E ) of a singlesite obeys the semicircle law, ν ( E ) = N √ − E π . (1)Like the Anderson model, the site-diagonal-disordermodel combines random on-site disorder with determin-istic hopping between sites. The site-diagonal-disordermodel’s hopping connects each site with its nearest neigh-bors. The hopping matrix is proportional to the N × N unit matrix I N , i.e. H i,j = K I N , (2)where K is the hopping strength, and i, j are the positionindices of two nearest neighbors. This hopping matrix isa natural N × N generalization of the Anderson model’snon-random hopping.We will examine the site-diagonal-disorder model withtwo alternative symmetry classes. The first is the orthog-onal symmetry class with real matrices obeying H i = H Ti , and the second is the unitary symmetry class with Hermitian matrices H i = H † i . The orthogonal class de-scribes systems with time reversal symmetry, while theunitary class describes systems where time reversal sym-metry is broken, for instance by a magnetic field.Our numerical work will concentrate on the one di-mensional case where the sites are arranged in a long onedimensional chain, and the position of each site is de-noted by the index i . The nearest neighbors of the i -thsite are at i ± K between sites is kept fixed as N is var-ied, Wegner showed that in the N → ∞ limit the site-diagonal-disorder model always has finite conductivity; iis always in the conducting phase, for all energies withinthe band. Later Khorunzhy and Pastur supplied a moremathematically precise proof. This result implies thatthe localization length ξ is infinite in the K fixed, N → ∞ limit. Our numerical results in section II confirm the in-finite localization length and absence of localization inthe K fixed, N → ∞ limit.This result, the absence of a localized phase in an dis-ordered model with short-range hopping and no disordercorrelations, is a bit peculiar because it is independentof the dimensionality. In particular, Wegner’s proof ofthe absence of a localized phase applies to 1-D wires. Hisproof contrasts with rigorous analytical proofs that 1-Dwires with uncorrelated disorder and short range hoppingare localized, except in the special case of certain en-sembles with perfectly conducting channels. Includedamong such proofs is a recent one concerning the samesite-diagonal-disorder model discussed here, in 1-D wires,and proving that the model’s spectrum is pure point ifthe number of orbitals N is kept fixed. The object of this paper is to show that Wegner’s ab-sence of a localized phase is a peculiarity of the limit that a r X i v : . [ c ond - m a t . d i s - nn ] J a n Wegner took, fixing the hopping strength K while taking N to infinity. In section III we will show numerically thatin the double scaling limit where the product of N and K is kept fixed while taking N to infinity, the site-diagonal-disorder model does have a localized phase where thelocalization length is finite. In other words, the K fixed, N → ∞ limit deletes the physics responsible for Ander-son localization, while the N K fixed, N → ∞ doublescaling limit preserves that physics.In subsequent sections we will analyze Wegner’s gauge-invariant model, which differs from the site-diagonal-disorder model only in that the hopping between sitesis random. After writing a functional integral which ismathematically equivalent to the gauge-invariant model,we will use the saddle point approximation and perturba-tion theory to analyze the model’s excitations and theirenergy and length scales. Our analysis will show that inthe
N K fixed limit certain excitations are massless andthus can cause localization, while in the K fixed limitthose excitations are made massive with a mass propor-tional to N . Thus in the K fixed limit these excitationsare removed from the model, forcing it to conduct.A similar result about N -orbital models has recentlybeen presented, proving exponential decay of matrix el-ements of the resolvent in a finite volume, which is asignature of Anderson localization. Notably, the au-thors showed that exponential decay could be provedwhen
N K is smaller than a bounding function, pointingto the double scaling limit discussed here. Their resultcontrasts with ours in two respects: theirs concerns thedecay of the resolvent while ours concerns the localiza-tion length, and theirs is a fully rigorous mathematicalproof while we rely on both numerical calculations anda non-rigorous analysis of a functional integral which isrigorously equivalent to an N -orbital model. Anotherdifference is our focus directly on the model’s behaviorin the N → ∞ limit where N K is kept fixed; we per-form this limit numerically, and we use
N K as a controlparameter for perturbative analysis of the functional in-tegral.
II. NUMERICAL CALCULATION OF THELOCALIZATION LENGTH WHEN THEHOPPING STRENGTH K IS KEPT FIXED.
In Figure 1 we present the localization length ξ ,which we calculate using the transfer matrix method .The transfer matrix method for calculating localizationlengths operates by building a long chain with length L and measuring the average decay within the chain. Wekeep the energy fixed at E = 2 .
8, fairly close to the bandedge which is near E = 3 . ξ with the hopping strength K fixed atthree values K = 1 , ,
3, and for both the unitary andorthogonal ensembles. In each of these six cases we cal-culate ξ at several values of N ranging up to N = 36. TABLE I: Linear fitting parameters, K fixed, Unitary Ensem-ble. The data that was fitted had statistical errors of about0.2%. k a , Unitary a , Unitary1 -0.0453032 0.8992641.5 -0.0614256 5.587062 0.212931 16.9203 This upper limit on N was determined by the numericalcost, which grows with N for two reasons. The first rea-son is that the numerical cost of adding one site to thechain scales with N . The second is that, in the K -fixedlimit, the scattering length and localization length bothincrease with N , so longer chains are required to obtainacceptable numerical accuracy. We extended the chainsuntil the statistical error of the localization length, quan-tified as a standard deviation, was reduced to 0.2% per-cent of the localization length’s value. At large K = 2 . N = 36 this standard of numerical accuracyrequires chain lengths L longer than 10 sites.Figure 1 presents the ratio of the localization length ξ to the number of orbitals N . We normalize ξ by thefourth power K of the hopping strength K , which is keptfixed at 1, 2, or 3. To compare the unitary ensemble tothe orthogonal ensemble in the same graph, we dividethe unitary results by s = 2 and the orthogonal resultsby s = 1.Each of the six data sets converges to a constant be-tween 0 .
40 and 0 .
60 as N becomes large, which showsthat when N is large the localization length ξ grows lin-early with N . The black lines are fits of the orthogonalensemble data to the linear relation ξ ( N ) = a + a N . a is the proportionality constant between ξ and N , and a is the extrapolated value of ξ at N = 0. The nu-merical values of these fitting parameters are given inTables I and II. Figure 1 shows excellent agreement be-tween the linear fits and the data, and χ -squared analysisalso confirms that the fits are good. This gives additionalevidence that at large N the localization length ξ is pro-portional to N .Since ξ is proportional to N at large N , when N istaken to infinity the localization length diverges and thereis no localized phase. Since this numerical result was ob-tained in one-dimensional systems, and localization ef-fects are generally strongest in 1-D and weaker in higherdimensions, it is a strong indication that the localizedphase is absent in all dimensions. This is consistent withWegner’s demonstration, based on perturbation theory,that the site-diagonal-disorder model is always in the con-ducting phase in the limit of N → ∞ with K fixed. N / ( s N K ) Ensemble
K=2 s=2 Unitary K=2 s=1 Orthogonal K=1.5 s=2 Unitary K=1.5 s=1 Orthogonal K=1 s=2 Unitary K=1 s=1 Orthogonal
FIG. 1: (Color online.) The ratio of the localization length ξ to the number of orbitals N , normalized by s K , in the K -fixed limit. The ratio’s convergence to fixed values at large N shows that in the K -fixed limit the localization length ξ isproportional to N . In consequence ξ diverges and there is noinsulating phase when N is taken to infinity. Black lines arefits of ξ to linear functions of N .TABLE II: Linear fitting parameters, K fixed, OrthogonalEnsemble k a , Orthogonal a , Orthogonal1 1.109 0.4522751.5 1.06834 2.79292 0.622876 8.47447 III. NUMERICAL CALCULATION OF THELOCALIZATION LENGTH IN THE DOUBLESCALING LIMIT WHERE NK IS KEPT FIXED.
Now we turn to the more interesting case where theproduct
N K is kept fixed while taking the N → ∞ limit.Figure 2 shows the localization length ξ with N K fixed at
N K = 1 , , N K is fixed to these values the scatteringlength always remains small and does not significantlyaffect the computational cost of calculating the localiza-tion length. Therefore we fix the energy at the bandcenter E = 0 and calculate ξ at large values of N up to N = 192.Figure 2 plots the numerical data on the localizationlength as colored symbols connected by line segments.In the N K fixed limit the localization length ξ does notdiverge with N , and instead converges to a finite andnon-zero value at N → ∞ . Therefore Figure 2 plots ξ vs. N − / , so that the left border of the graph correspondsto N → ∞ . This graph gives visual proof that ξ doesnot diverge with N → ∞ . If it did diverge, then ξ ’s slopewould increase as it approached the left border of thegraph. Moreover ξ itself would become either very largeor very small near the left border and would escape theupper or lower boundary of the graph. This is not thecase: ξ ’s slope becomes smaller near the left border, and N Ensemble
NK=3 Unitary NK=3 Orthogonal NK=2 Unitary NK=2 Orthogonal NK=1 Unitary NK=1 Orthogonal
FIG. 2: (Color online.) Localization length ξ as functionof N , in the double scaling limit where NK is held fixed.The black lines are fits to quadratic or cubic polynomials in1 / √ N . The points where the fits intercept the y-axis give the N → ∞ limit of the localization length.TABLE III: Fitting Coefficients, Nk fixed, Unitary Ensemble.The data that was fitted had statistical errors of about 0.2%. Nk b b b b all curves approach values in the interval between zeroand one.Figure 2 plots also black lines which are fits of thedata to polynomial functions of N − / . Both χ -squaredanalysis and visual examination show that these polyno-mials are excellent descriptions of the localization lengthat small 1 / √ N . The fitting formula is ξ = b + b ( N k ) N − / + b ( N k ) N − + b ( N k ) N − / . (3)The fitting parameters are listed in Tables III and IV.When N K is small the cubic term b in the fitting poly-nomial is not needed to obtain a good fit, so in this casewe leave it at zero.Because these fits are successful, the localizationlength’s N → ∞ limit can be read off from the b values in the left columns of Tables III and IV. For N K = 1 , ,
3, and for both unitary and orthogonal en-sembles, ξ ’s N → ∞ limit lies between 0 .
25 and 0 . TABLE IV: Fitting Coefficients, Nk fixed, Orthogonal En-semble Nk b b b b IV. DISCUSSION OF THE NUMERICALRESULTS
These results are a numerical proof that the one-dimensional site-diagonal-disorder model is localized inthe double scaling limit where
N K is kept fixed and N istaken to infinity. This localized property is expected ofone-dimensional wires with short-range uncorrelated dis-order and short-range hopping. Wires in this class havebeen shown both numerically and analytically to be lo-calized, except in the special case of certain ensembleswith perfectly conducting channels. These numerical results demonstrating a localizedphase in the site-diagonal-disorder model are in contrastwith Wegner’s proof that the same model is always con-ducting. Although there is a contrast, there is no contra-diction: Wegner’s proof concerns the N → ∞ , K fixedlimit, which removes the physics that is responsible forlocalization.These results also suggest that in three dimensions thesite-diagonal-disorder model may exhibit both localizedand conducting phases, and an Anderson transition be-tween them. If so then its behavior would be substan-tially the same as the three-dimensional Anderson model. V. THE GAUGE INVARIANT MODEL
In order to go beyond one dimension, and to obtainnon-numerical insight, we will now analyze both the K fixed limit and the N K fixed limit using analyti-cal analysis of a functional integral. In order to easeour mathematical analysis, we will switch from Wegner’ssite-diagonal-disorder model to Wegner’s gauge invariantmodel, both of which were introduced in the same paper. The only difference between the two models is that inthe gauge invariant model the hopping between sites ismediated by random matrices, while in the site-diagonal-disorder model hopping is mediated by deterministic ma-trices proportional to the identity. We will use however amore flexible notation for the hopping, which is generalto any geometry, connectivity, or dimension.Like the site-diagonal-disorder model and the Ander-son model, the gauge invariant model is a tight-bindingHamiltonian; i.e. the electrons live on a lattice or graphgeometry with V sites and N electron orbitals at eachsite. There are N basis elements at each site, so thetotal basis size is N × V . We will use the lower case let-ters n, v respectively to denote the orbital index and theposition.We develop the unitary ensemble version of the gaugeinvariant model, where the Hamiltonian is hermitian.This work can easily be generalized to the orthogonalensemble and probably also to the symplectic ensem-ble. In the orthogonal ensemble variant the spin variablesoccuring in the functional integral are 4 × × H , including both its on-site and hop-ping parts, is random and fully described by the secondmoment ˜ H n n v v ˜ H n n v v = ˜ (cid:15) N − (1 − k ) v v δ n n δ n n δ v v δ v v (4)The system geometry, including the number of dimen-sions and all other structural details, is encoded in thepositive indefinite operator k , which is the only nonlocaloperator in the model. k controls kinetics and hopping.We require that the 1 − k operator be positive definitein order to assure convergence of a Hubbard-Stratonovichtransformation used to derive the functional integral rep-resentation of this model. We require that k be a Lapla-cian, meaning that k | (cid:126) (cid:105) = 0, where | (cid:126) (cid:105) is the spatiallyuniform vector. The requirement that k be a Laplacianis physically the minimal requirement for it to be able tocapture the physics of hopping and some kind of graph orlattice geometry, and is mathematically required to cap-ture the global continuous symmetry which is key to thephysics of conduction and localization. It can generallybe understood that k is small compared to 1, though wedo not require this except when taking the N K fixed, N → ∞ limit. ˜ (cid:15) is the energy scale of the Hamiltonian.The site-diagonal-disorder model and the gauge invari-ant model are very much alike. The main physical differ-ence between the two models is in the scattering length,which is zero (on-site) in the gauge invariant model, butcan have any value in the site-diagonal-disorder model.In other words, the gauge invariant model is an effectivefield theory: it describes the physics of diffusive conduc-tion and of localization at scales longer than the scatter-ing length, but omits ballistic physics at scales shorterthan the scattering length. There are two notable mathe-matical simplifications in the gauge invariant model. Thefirst is that its energy band is the semicircular one of ran-dom matrix theory, while the site-diagonal-disorder en-ergy band can have any shape at all, according to the ki-netic operator (hopping matrix) which one chooses. Thesecond is that the gauge-invariant model’s long distancephysics, i.e. movement of its cooperon and diffuson, isdirectly controlled by the kinetic operator k , which is aninput parameter. In the site-diagonal-disorder model thekinetics of the cooperon and diffuson is regulated by morecomplicated mathematics. VI. FUNCTIONAL INTEGRAL FORMULATIONOF THE GAUGE INVARIANT MODEL
The conversion of disordered models to functional in-tegrals is a well developed topic that originated withSchafer and Wegner in 1980.
The main point ofthese functional integrals is that their spin variables obeya continuous symmetry and therefore are capable of atransition from a spontaneously-broken-symmetry phaseto a symmetric phase. In the symmetry-broken phasethe spins are correlated over long distances, which cor-responds to electronic conduction over long distances.In the symmetric phase spins have a finite correlationlength, so when two spins are separated by a distancelonger than that length their correlator is exponentiallysmall. This symmetric phase corresponds to the Ander-son localization phase, where conduction over distanceslonger than the localization length is exponentially small.The correlation length of the spins is the same as thelocalization length. Thus functional integrals bring tolight the spontaneous symmetry breaking that lies at thecore of the phase transition from Anderson localizationto long-distance conduction.A general feature of these functional integrals is thatthe conversion from disordered Hamiltonians to func-tional integrals can be accomplished if desired with math-ematical rigor, i.e. there is a strict equivalence betweenthe original ensemble of Hamiltonians and the seeminglymuch different functional integral. Another general fea-ture is that the spin variables in the functional integralsare always matrix variables, and that the matrices mustbe decomposed into eigenvalues vs. rotations. The rota-tions possess the global continuous symmetry at the heartof the Anderson phase transition, while the eigenval-ues always remain massive, reflecting the short-distancephysics associated with the scattering length.Although these features are universal, for each disor-dered model there are several alternative functional in-tegrals. The oldest one is based on the replica tech-nique, which is more flexible but relies on the replicatrick where the number of replicas is taken to zero.
The other alternative functional integrals generally in-volve graded matrices, i.e. matrices in which half thevariables are anticommuting (Grassmann) variables andthe other half are the more commonly known commutingvariables. The first such integral was developed by Efe-tov, who used the Hubbard-Stratonovich transformationto derive the functional integral, adapting that trans-formation to graded matrices.
More recently super-bosonization was introduced, which avoids the Hubbard-Stratonovich transformation in favor of a mathematicalstep of pushing one integration domain onto a more re-stricted domain of graded matrices. Here we will use a hybrid approach pioneered byFyodorov.
This technique judiciously combines botha Hubbard-Stratonovich transformation and a push. Itsprincipal advantage is that half of the variables - the an-ticommuting variables - can be integrated exactly. Theresult of this integration is that the resulting functionalintegral contains two different matrix spins, plus a de-terminant which is a direct result of the integration.While this form is a bit more complicated than alter-native functional integrals, it does have the advantagethat the remaining variables in the model are all com-muting, which simplifies their mathematical treatment.It should be noted that a very similar procedure re-sulting in a functional integral with a determinant inthe place of Grassmann variables had already been per- formed twenty years earlier by Ziegler and developed byother authors.
However Ziegler confined his work toa functional integral suitable for calculating the densityof states, while Fyodorov distinguished himself by pro-ducing a Grassman-free functional integral suitable forcalculating the advanced-retarded two point correlator R AR .Probably any of the competing functional integrals canbe analyzed in terms of energy and length scales to giveinsight into the absence of a localized phase in the N fixed, N → ∞ limit and the presence of that phase inthe N k fixed limit. We adopt the hybrid approach herefor reasons of personal experience and taste, and to show-case certain qualities which the hybrid approach offers:clear mathematics without need for modern results ongraded matrices, and a very systematic if rather compli-cated arithmetic and perturbation theory. Moreover, be-cause variables can be grouped together and manipulatedas commuting matrices, calculation of the leading-ordervalue of the advanced-retarded density-density correla-tor R AR proceeds with very quickly, with a minimum ofalgebra, and without any restrictions on the level spac-ing parameter ω = E − E ; we supply a sketch of thiscalculation in the appendix. This correlator is a well-known result that has been calculated using a variety ofapproximations and techniques. Fyodorov developed the hybrid approach for zero-dimensional (single-site) models.
To perform thepush step in deriving the functional integral, Fyodorovused an integral that was first done by David, Duplantier,and Guitter, and that was later connected to the Sin-gular Value Decomposition. Disertori generalized Fyo-dorov’s hybrid approach to the gauge invariant model inextended systems. Using Fyodorov’s hybrid approach,Wegner’s gauge invariant model transforms in a com-pletely mathematically rigorous way into the followingfunctional integral: ¯ Z = γ (cid:90) Q b ≥ dQ f dQ b e L × det( Q fv δ v v δ j j − Q bv L (1 − k ) v v δ i i ) L = ( N − (cid:88) v T r (ln Q fv ) + ( N − (cid:88) v T r (ln Q bv L )+ ıN ˜ (cid:15) − (cid:88) v T r ( Q bv L ( ˆ E − J bv )) + ıN ˜ (cid:15) − (cid:88) v T r ( Q fv ˆ E ) − N (cid:88) v v (1 − k ) − v v T r ( Q fv Q fv ) − N (cid:88) v v (1 − k ) v v T r ( Q bv LQ bv L ) γ = N NV +2 V (( N − N − − V × − V π − V (det(1 − k )) − e NV (cid:15) T r ( ˆ E ˆ E ) (5)The degrees of freedom Q f and Q b are 2 × Q b ≥ Q b is constrainedto have only positive or zero eigenvalues; the integrationmeasure contains a theta function θ ( Q b ). The L matrixis equal to the Pauli σ z matrix; it is diagonal with +1and − E operator contains the ener-gies at which the densities are measured; it is diagonalwith eigenvalues E and E . The source matrices J bv arealso diagonal matrices, with two entries on the diagonal.The matrix Q fv δ v v δ j j − Q bv L (1 − k ) v v δ i i in thedeterminant lives in a 2 × × V space - the V × V matrix k mediates transitions between positions v , v , the 2 × Q f mediates transitions between i , i , and the2 × Q b mediates transitions between j , j . The ensemble-averaged value of the density ρ ( v , E )at site v and energy E can be calculated by taking thefirst derivative of ¯ Z with respect to J bv ,j =1 and then set-ting J b = 0. Equivalently, one can just insert a factor of − ıN ˜ (cid:15) − ( Q b ) v ,j =1 into the body of the functional inte-gral and then set J b = 0.Taking this a step further, the Advanced-Retarded cor-relator R AR , which gives the correlation of the density ρ ( v , E ) at site v with the density ρ ( v , E ) at site v ,can be calculated by using the functional integral to ob-tain the average value of − N ˜ (cid:15) − ( Q b ) v ,j =1 ( Q b ) v ,j =2 .The Advanced-Retarded correlator R AR reveals whether Q b ’s values are correlated across long distances, which isthe case in the conducting phase. Alternatively R AR canshow that Q b ’s correlations die off exponentially with dis-tance, which is the case in the localized phase. Anotherway of saying this is that if Q b is everywhere pinned tothe same value then we have conduction, while if instead Q b fluctuates from point to point then we have Andersonlocalization.The development and final form of this functional inte-gral are strongly constrained by the fact that the disorderin the original gauge invariant model is static; it does notdepend on time. As a consequence, the spin variables inthe functional integral have important relationships witheach other; the functional integral possesses importantsymmetries. In alternative functional integrals for thesame gauge invariant model there is only one matrix, ei-ther living in a space of replicas, or in a graded spacecontaining both commuting and anticommuting (Grass-man) variables. In the hybrid model adopted here the Q b and Q f matrices are analogous to the commuting vari-ables in the supersymmetric formulation, and the deter-minant figuring so prominently in the functional integralis the result of performing an exact integration of theanticommuting variables.One spectacular consequence of the functional inte-gral’s symmetry is that if one does not take any deriva-tives with respect to the sources, i.e. if one does notaverage over Q b , then the integral’s value is identical toone. This result is not immediately apparent in Equation5 and a perturbative analysis does not give a clear indi-cation of this result. In order to give the correct result(one), at each order of perturbation theory large numbersof diagrams must sum to zero. Part of the reason whyperturbation theory does not easily reveal this result is that the Q f and Q b matrices are given different roles; forinstance the T r ( Q fv Q fv ) term is multiplied by (1 − k ) − while the T r ( Q bv LQ bv L ) term is multiplied by (1 − k ).However the symmetry between Q f and Q b can be par-tially restored by performing nonlocal transformations of Q f = (1 − k ) − / Q f and Q b = (1 − k ) / Q b , arriving at analternative formulation of the same functional integral:¯ Z ∝ (cid:90) dQ f dQ b e L × det( Q fv (1 − k ) − / v v δ j j − Q bv L (1 − k ) / v v δ i i ) × Π v θ ( (cid:88) v (1 − k ) − / v v Q bv L ) L = ( N − (cid:88) v T r (ln( (cid:88) v (1 − k ) / v v Q fv ))+ ( N − (cid:88) v T r (ln( (cid:88) v (1 − k ) − / v v Q bv L ))+ ıN ˜ (cid:15) − (cid:88) v T r ( Q bv L ˆ E ) + ıN ˜ (cid:15) − (cid:88) v T r ( Q fv ˆ E ) − (cid:88) v N T r ( Q fv Q fv ) − (cid:88) v N T r ( Q bv LQ bv L ) − ıN ˜ (cid:15) − (cid:88) v v (1 − k ) − / v v T r ( Q bv LJ bv ) (6)After this transformation the principal formal differencebetween Q f and Q b is that Q b ’s rotations live on a hyper-bolic manifold that is not compact, while Q f ’s rotationslive on a half-sphere manifold that is compact. The twomanifolds can be parameterized in a way that is identi-cal except for a difference in signs, i.e. a plus sign in onemanifold’s parameterization vs. a minus sign in the othermanifold’s parameterization. Similarly there is a differ-ence in signs in the kinetic terms now embedded inside ofthe logarithms and determinant. At the level of the sad-dle point approximation there is an additional difference: Q b ’s two eigenvalues at the saddle point are of the samesign, while Q f ’s two eigenvalues are of opposite signs. Inconsequence, at the level of perturbation theory aroundthe saddle point the only differences between Q b and Q f are certain minus signs. VII. TWO ENERGY AND LENGTH SCALES
In this section we will use the hybrid functional integralin Eq. 6 to show that the gauge invariant model has twonatural energy scales. The first is the scattering energyscale ˜ (cid:15)N . The second is the much smaller energy scale˜ (cid:15)N k which controls conduction and localization. Corre-sponding to these two energy scales are two length scales:the scattering length which is equal to 1 in the gauge in-variant model because scattering occurs on site, and thelocalization length which can be much larger or even di-verge. The beauty of the functional integral approach todisordered models is that these two scales can be sepa-rated from each other by breaking the matrix variablesinto two sectors: their eigenvalues and their rotations.We will first analyze the eigenvalues of Q f , Q b , whichmediate scattering physics. Their saddle-point valuesare of order O (1), except near the band edges wherethe saddle-point approximation breaks down. In con-trast, the eigenvalues’ fluctuations are small, of order O (1 / √ N ), and do not show significant inter-site correla-tion. The net effect is that the eigenvalues are basicallythe same at every site in the system.We will next analyze the rotations of Q f , Q b , whichmediate conduction and localization physics. In a con-ducting system these rotations maintain the same value(except for small fluctuations) throughout the entire sys-tem, and Q f , Q b retain the same value everywhere. In aninsulating system the fluctuations in the rotations growsteadily with distance and reach O (1) at the localizationlength scale. In the gauge invariant model the kineticterm which penalizes fluctuations in the rotations is pro-portional to N k . Therefore in the N → ∞ limit with K (the magnitude of the kinetic operator k ) fixed the kineticterm diverges. The divergence forces the fluctuations tozero, forcing the system into the conducting phase. Incontrast, when N K is kept fixed the kinetic term doesnot diverge. In this case, because the kinetic operator k isa Laplacian with a zero eigenvalue when acting on spa-tially constant rotations, it has small eigenvalues whenacting on long-wavelength rotations. It therefore can failto eliminate rotations over long length scales, resultingin the localized phase. This is the key reason why the N fixed limit eliminates the localized phase while the N K fixed limit retains it.Lastly we will analyze the determinant. This containstwo sectors: one sector with eigenvalues proportional to N , and another with eigenvalues proportional to N k .Because k has a zero eigenvalue, the latter sector includestwo very small eigenvalues signalling the same physics asthe globally uniform rotations of Q f , Q b . The two sectorsare exact analogues of the eigenvalue and rotation sectorsof the Q f , Q b matrices. A. About the style of the present analysis of thefunctional integral
The style in this present work of our analysis of thefunctional integral is not rigorous. A fully systematicand maximally rigorous analysis would start with care-ful separation of globally uniform rotations from otherrotations where some sites rotate differently from others,and would then perform a nonperturbative integration ofthe globally uniform rotations. Maximal rigor would alsorequire a detailed solution of the saddle point equationincluding all terms that contribute to it, a systematic perturbative expansion of the determinant, and a sys-tematic perturbative expansion of the corrections to thesaddle point, including fluctuations of both the eigenval-ues and the rotations. These requirements are all withinreach, except for two difficulties. The first difficulty isthat it is mathematically questionable to make a pertur-bative expansion of fluctuations in the rotations becausethose fluctuations are not around a single value of the ro-tation matrix but instead around all points on a manifoldof spatially uniform rotations. This difficulty manifestsitself in divergences of the Jacobian which is producedwhen separating the globally uniform rotations from theother rotations.A second difficulty is that the gauge invariant modelhas competing saddle points, and the choice of saddlepoint can in principle be different at each site on thelattice. Unless the globally uniform saddle points arestrongly favored over their alternatives where some sitespick one saddle point and other sites pick another, per-turbative analysis of the functional integral is impossiblebecause of the exponential proliferation of saddle points,each of which violates translational invariance. This is-sue was considered in two rigorous proofs about certainfunctional integrals used to study the density of states indisordered models . Such functional integrals used tostudy the density of states are structurally simpler thanthe integrals used to study the Advanced-Retarded twopoint correlator R AR , but unfortunately the density ofstates does not contain information about the localizedand conducting phases, while R AR does. In these twoproofs it was shown that a single global saddle point dom-inates, and also rigorous control over fluctuations aroundthe saddle point was obtained using coupled cluster ex-pansions. To the author’s knowledge, no similar efforthas been taken to control non-uniform saddle points infunctional integrals built for computing the Advanced-Retarded two point correlator R AR . Both of these twodifficulties are of a non-perturbative nature, so a narrowfocus on purely perturbative mathematics will naturallybypass them.Leaving these two difficulties aside, a maximally rigor-ous analysis of the saddle point equations and the pertur-bative corrections to the saddle points results in a mathe-matical structure with more than two dozen different ver-tices that combine to produce a wide variety of Feynmandiagrams. While there is nothing particularly difficultin principal about this mathematics, the huge variety ofvertices and diagrams is rather difficult to manage. It ispossible to classify all vertices and Feynman diagrams inpowers of N , or alternatively in powers of N K . It is eas-ily possible to show that in the K fixed limit there is a fi-nite and very small number of diagrams which survive the N → ∞ limit; in other words this limit is trivial which iswhy it is conducting. With a great deal more effort it ispossible to determine a subset of vertices which survivethe N K fixed limit, and (if the saddle point equationswere formulated and solved with sufficient care) to showthat there is no diagram which diverges in this limit. The fact that there are several vertices that remain finitein this limit and that they combine to produce an infinitenumber of Feynman diagrams is a perturbative indicationthat the
N K fixed limit produces a sigma model whichin principal is capable of producing a localized phase.In the present work we leave aside attempts at suchrigor and move less systematically. We look broadly atthe scales in play in order to show that some sectors ofthe functional integral are controlled by N while othershave kinetics that are controlled by N k . This is sufficientto provide an account of why the N fixed limit removesthe localized phase while the N K fixed limit preservesit.
B. The Eigenvalues
We begin our analysis of the functional integral by de-composing the matrix spins Q f , Q b into eigenvalues androtations. For Q f the correct change of variables is Q fv = U v x fv U † v , where the eigenvalue matrix x fv is diagonal, therotation U v is unitary, and v is the position index. Thenew integration measure is dQ f = ∆ V dM ( x f ) dU dx f where ∆ V dM ( x f ) = x f − x f is the Van der Monde de-terminant. In the previous sentence the subscript of x f distinguishes the two eigenvalues at position v . Note thatthe unitary matrix U lives on the U (2) manifold whichis compact - it can be parameterized with variables thatare all bounded.The correct change of variables for Q b requires a lit-tle more care because of the constraint that Q b must bepositive indefinite; Q b ≥
0. It is also important that Q b is always paired with L = σ z . Fyodorov showedthat Q b L factors into Q bv L = T v x bv T − v , where x b is di-agonal and constrained by x b L ≥
0, and T is a mem-ber of the pseudo-unitary hyperbolic group U (1 ,
1) de-fined by T † LT = L . The integration measure is dQ b = dx b dT ∆ V dM ( x b ). Note that T lives on the U (1 ,
1) man-ifold which is not compact and is indeed hyperbolic. Asa consequence T has parameterizations in which one ormore of its parameters is unbounded. The presence of un-bounded degrees of freedom in the model is an importantmathematical feature which can complicate analysis.The behavior of the eigenvalues can be most easily un-derstood by leaving out the rotations, so that Q f → x f and Q b → x b . The Langrangian for x f (that for x b issimilar) is L = ( N − (cid:88) v T r (ln( (cid:88) v (1 − k ) / v v x fv ))+ ıN ˜ (cid:15) − (cid:88) v T r ( x fv ˆ E ) − (cid:88) v N T r ( x fv x fv ) (7)All of these terms are proportional to N , so that as N increases toward the N → ∞ limit x f becomes progres- sively more closely pinned to its saddle point value. Ex-panding to second order in the fluctuations of x f aroundthe saddle point immediately reveals that they are of or-der O ( N − / ) and that their second moment is of order O ( N − ). Compared to these quite small on-site fluctu-ations, the correlations between fluctuations at differentsites are even smaller: they are suppressed by factors of k , which is small compared to 1.Concerning the saddle points themselves, there are twosuch saddle points that give leading contributions, whichwe will label as the + saddle point and the − saddlepoint. At both saddle points the x b values are the same: x b = (cid:112) − ( E / (cid:15) ) + ıE / (cid:15)x b = (cid:112) − ( E / (cid:15) ) + ıE / (cid:15) (8)At the + saddle point the x f values are the same as the x b ones i.e. x f = x b and x f = x b , while at the − saddlepoint they are interchanged i.e. x f = x b and x f = x b .In addition there are two suppressed saddle points wherethe sign of x f or x f is reversed. C. Global Rotations
It is necessary to make a distinction between on onehand global rotations where every site in the entire sys-tem rotates in synchrony, and on the other hand non-uniform rotations where the sites do not all rotate to-gether. This is necessary because, when E is set equalto E , i.e. when ˆ E is proportional to the identity, the in-tegrand inside the functional integral is invariant underglobal rotations. Therefore the global rotations cannotbe performed perturbatively, and must be done exactly.In contrast the non-uniform rotations may be integratedusing pertubation theory. Therefore a change of variablesmust be performed, from the original rotation variableswhich are each located at individual sites, to a set of ro-tation variables where the global rotation is distinct andseparate from all other rotation variables. Accompanyingthis change of variables, a Jacobian must be introducedto the functional integral.Fortunately, in the absence of the Jacobian, the in-tegrals over global rotations can be done exactly. Theintegral over global rotations of Q f is the famous Harish-Chandra-Itzykson-Zuber integral , and the integralover global rotations of Q f is a well-known generalizationof the same integral . The Harish-Chandra-Itzykson-Zuber integral is part of the math that produces themodel’s two saddle points, since they are related by aglobal rotation of Q f . As mentioned before, the Jaco-bian diverges at certain non-zero points on the manifoldof global rotations, and therefore must be either treatedapproximately or neglected. Once both integrals overglobal rotations have been performed, the average energy( E + E ) / x f , x b , influencing their saddle pointvalues. There remains a coupling of the energy splitting ω = E − E to the rotations and to the source J b , andthe functional integral is multiplied by a factor ω − .Strictly speaking, it is necessary to perform the in-tegration over global rotations prior to performing thesaddle point approximation of the x b eigenvalues. Ro-tations of Q b lie on a non-compact manifold, so the La-grangian’s (cid:80) v T r ( Q bv L ˆ E ) term is unbounded. For somepoints on the non-compact manifold of Q b ’s rotations thispushes x b ’s saddle points outside of the band, making itimpossible to perform the saddle point approximation.The only correct way around this problem is to performthe exact integration over global rotations prior to per-forming the saddle point approximation. Mae and Iida and Takahashi were the first to address this kind of is-sue and perform the saddle point integration earlier thanusual. However this is a mathematical nicety that doesnot affect the remainder of our discussion here. D. Non-Uniform Rotations
Next we examine all other rotations besides the globalrotation. Since the eigenvalues remain very close to theirsaddle point values and don’t show any significant longdistance behavior, we pin the eigenvalues to their saddlepoints and then analyze the rotations in isolation. We set Q fv = U v x f U † v and Q bv L = T v x b T − v , where x f , x b arethe saddle point values, and U, T are the non-uniformrotations, i.e. the global rotations have been removedas described earlier. Setting aside the source term, thefunctional integral simplifies to¯ Z ∝ (cid:90) dU dT e L × det( Q fv (1 − k ) − / v v δ j j − Q bv L (1 − k ) / v v δ i i ) L = ( N − (cid:88) v T r (ln( (cid:88) v (1 − k ) / v v Q fv ))+ ( N − (cid:88) v T r (ln( (cid:88) v (1 − k ) − / v v Q bv L ))+ ıN ˜ (cid:15) − ω (cid:88) v T r ( Q bv ) + ıN ˜ (cid:15) − ω (cid:88) v T r ( Q fv L ) (9)If there were no hopping, i.e. if the kinetic term k wereequal to zero, then this integrand would be invariant withrespect to non-uniform rotations; the Lagrangian wouldsimplify to a constant. Therefore the Lagrangian can beexpanded in powers of k : L = − N − (cid:88) v ,v k v v T r (ln( Q fv Q fv ))+ N − (cid:88) v ,v k v v T r (ln( Q bv LQ bv L ))+ ıN ˜ (cid:15) − ω (cid:88) v T r ( Q bv ) + ıN ˜ (cid:15) − ω (cid:88) v T r ( Q fv L )(10) Many alternative parameterizations of U and T ex-ist, but the following one seems to be particularly wellsuited for performing a perturbation integration of thenon-uniform rotations: T = (cid:20)(cid:112) y b ) + ( z b ) y b − ız b y b + ız b (cid:112) y b ) + ( z b ) (cid:21) ,U = (cid:20)(cid:112) − ( y f ) − ( z f ) ıy f + z f ıy f − z f (cid:112) − ( y f ) − ( z f ) (cid:21) (11) y b and z b vary from −∞ to ∞ , while y f and z f varywithin the unit circle defined by ( y f ) + ( z f ) = 1.The integration measures are dy f dz f and dy b dz b . Thesevariables are convenient for perturbation theory becausewhen the rotations U, T are equal to the identity the y, z variables are zero. We will see that fluctuations in the y, z variables are of order (
N k ) − .After expanding in powers of y, z , the quadratic partof the Lagrangian is: L ≈ ∓ πıN ω ˆ ρ ˜ (cid:15) − (cid:88) v (( y fv ) + ( z fv ) )+ 2 πıN ω ˆ ρ ˜ (cid:15) − (cid:88) v (( y bv ) + ( z bv ) ) − N π ˆ ρ (cid:88) v v k v v ( y fv y fv + z fv z fv ) − N π ˆ ρ (cid:88) v v k v v ( y bv y bv + z bv z bv )2 π ˆ ρ = (cid:112) − ( E/ (cid:15) ) (12)Here N ˆ ρ is the gauge invariant model’s density of states,and the ± sign specifies which saddle point is being used.All four terms in the quadratic part of the Lagrangianare proportional to N k , which means that fluctuationsin the y, z angular variables have a typical size whichscales with (
N k ) − / . If K is fixed during the N → ∞ limit, then fluctuations in the angular variables are forcedto zero; the angular variables have the same value at allsites on the lattice. This is why the K fixed N → ∞ limitforces the gauge invariant model into conducting regime.On the other hand, taking the N K fixed N → ∞ limit leaves the angular fluctuations with their magnitudeunchanged. Because k is a Laplacian, it acts very weaklyon long wavelength fluctuations. On a regular latticethe eigenvalues of k scale quadratically with momentum (cid:126)k , i.e. as | (cid:126)k | . Therefore in the N K fixed limit long-wavelength fluctuations can be very large. This is whythe
N K fixed limit is able to preserve the localized phase,while the N fixed limit deletes it.The integration over the angular rotations, taken toquadratic order, divides the functional integral by twospectral determinants:Π λ k (cid:54) =0 ( λ k ± ıω (2 π ˆ ρ ˜ (cid:15) ) − ) × Π λ k (cid:54) =0 ( λ k − ıω (2 π ˆ ρ ˜ (cid:15) ) − ) (13)Here λ k are the eigenvalues of the kinetic operator k ,and the spectral determinant expressed in Π λ k (cid:54) =0 includes0contributions from all of k ’s eigenvalues except the zeroeigenvalue associated with spatially uniform rotations.In addition to spectral these determinants from thequadratic part of the Lagrangian, there is an importantcontribution from the zeroth order Lagrangian, i.e. thevalue of Lagrangian when the rotations are set to theidentity and the eigenvalues are fixed at their saddle pointvalues. This Lagrangian simplifies to e ıπNV ˆ ρω ˜ (cid:15) − (1 ± (14)where ω = E − E is the energy splitting and the ± sign denotes which saddle point is being used. The La-grangian also includes factors of N and e which are how-ever cancelled out by factors from the functional inte-gral’s normalization and from integrations over Q f and Q b ’s eigenvalues and rotations. The exponential in Eq.14 encodes Wigner-Dyson oscillations, a well known man-ifestation of the repulsion between adjacent energy levelsthat occurs in disordered materials. E. The Determinant
In the previous sections we have performed a sad-dle point analysis of the Q f , Q b spins. Fluctuations inthe eigenvalues are of size N − / and are therefore sup-pressed in the both the K fixed and N K fixed limits aslong as N → ∞ . Fluctuations in the angles have size( N k ) − / and therefore are suppressed in the K fixedlimit, deleting the localized phase. In the N K fixed limitangular fluctuations are not suppressed and the localizedphase is preserved. In addition, global angular rotationshave no mass except that caused by the energy splitting ω .In this section we analyze thedet( Q fv (1 − k ) − / v v δ j j − Q bv L (1 − k ) / v v δ i i ) de-terminant which figures so prominently in the functionalintegral. This determinant has the same mass structureas the Q f , Q b spins. This can be seen by settingthe angular fluctuations to unity, in which case thedeterminant factors into four determinants:det( x fv , (1 − k ) − / v v − x bv , (1 − k ) / v v ) × det( x fv , (1 − k ) − / v v + x bv , (1 − k ) / v v ) × det( x fv , (1 − k ) − / v v − x bv , (1 − k ) / v v ) × det( x fv , (1 − k ) − / v v + x bv , (1 − k ) / v v ) (15)The only differences between the four determinants arein which of the two x f eigenvalues contribute, and whichof the two x b eigenvalues contribute.Recall that at the + saddle point the x f values are thesame as the x b ones i.e. x f = x b and x f = x b , while atthe − saddle point they are interchanged i.e. x f = x b and x f = x b . Therefore in two of the determinants x f and x b nearly cancel each other and leave a term proportional to the energy splitting ω = E − E , plus fluctuations in thesaddle point which we have seen are of order N − / , plusof course also a contribution from k . Therefore these twonear-zero determinants aredet ( k + ı ( ± − ω ((2 π ˆ ρ ˜ (cid:15) ) − ) / δx f − δx b ) (16)where δx f , δx b represent fluctuations in the saddle point,and the ± sign specifies which saddle point is in use.These two determinants contain physics which is exactlythe same as the non-uniform angular rotations of Q f , Q b .Since the kinetic operator’s action on the constant vec-tor is to multiply it by zero, each of these two determi-nants also contains a factor where the k disappears leav-ing only two factors of ( ı ( ± − ω ((2 π ˆ ρ ˜ (cid:15) ) − ) / δx f − δx b ). The two factors contain exactly the same physicsas global rotations of Q f , Q b .Lastly there are two more determinants where x f and x b add to each other instead of nearly cancelling eachother. The arguments of these determinants are roughlyˆ ρ + k . The kinetic operator k is small compared to ˆ ρ which is of order one, so these two determinants representpurely local physics and are of order one. In fact thetwo determinants represent the same physics seen in theeigenvalues of Q f , Q b . VIII. SUMMARY
In summary, we have presented evidences that two dis-ordered models, each with N orbitals per site, possess alocalized phase with finite localization length in the dou-ble scaling limit where N → ∞ and N K is kept fixed.In the site-diagonal-disorder model we directly computedthe localization length and showed that it remains finitein one dimension when
N K is kept fixed. In the gauge in-variant model, in any dimension or geometry, we showedthat the
N K fixed limit preserves dynamic spin variableswhich in the K fixed limit become very massive and areeffectively deleted. These spin variables may be capa-ble of exiting the spontaneous symmetry breaking phasewhich occurs in their absence, and thus entering the An-derson localized phase. These arguments indicate that N -orbital models are capable of exhibiting both local-ized and conducting phases, and an Anderson transition,in the double scaling N K fixed limit.
Acknowledgments
We especially thank Universit`a degli Studi di Roma”La Sapienza”, the Asia Pacific Center for TheoreticalPhysics, Sophia University, and the Zhejiang Institute ofModern Physics, where much of this work was done. Wethank T. Ohtsuki for help during the early stages of thiswork. We also acknowlege support from the Institute ofPhysics of the Chinese Academy of Sciences, NanyangTechnological University, Royal Holloway University of1London, and Rutgers University. We also thank X. Wan,G. Parisi, T. Spencer, S. Kettemann, J. Zaanen, D. Cul-cer, P. Fulde, and A. Leggett for hospitality and discus-sion, and J. Verbaarschot, J. Keating, A. Mirlin, K. Efe-tov, M. Disertori, M. Zirnbauer, M. A. Skvortsov, K.-S.Kim, B. Liu, Y. Fyodorov, M. Milletari, C. Miniatura, B.Gremaud, and J. Y. Yong’En for helpful conversations.
Appendix A: Calculation of the two point correlator
Here we discuss how the hybrid functional integral canbe used to easily calculate the two point density correla-tor. First we collect leading order results mentioned inthe text. The result of integrating the angular fluctua-tions of Q f , Q b is division by two spectral determinants,given in Eq. 13:Π λ k (cid:54) =0 ( λ k ± ıω (2 π ˆ ρ ˜ (cid:15) ) − ) × Π λ k (cid:54) =0 ( λ k − ıω (2 π ˆ ρ ˜ (cid:15) ) − ) (A1)Here λ k are the eigenvalues of the kinetic operator k ,and the spectral determinant expressed in Π λ k (cid:54) =0 includescontributions from all of k ’s eigenvalues except the zeroeigenvalue associated with spatially uniform rotations.The ± sign denotes which of the two saddle points weare computing.The Harish-Chandra-Itzykson-Zuber integral of spa-tially uniform rotations of Q f , and its analogue for Q b ’srotations, contribute a factor of ± ω − (2 π ˆ ρ ˜ (cid:15) ) .Equation 16 gives the sector of the Q f − Q b determi-nant which corresponds to angular rotations:Π λ k ( λ k + ı ( ± − ω (4 π ˆ ρ ˜ (cid:15) ) − + δx f − δx b ) (A2)The spatially uniform eigenvalue of k , λ = 0, is includedin this determinant.The value of the Lagrangian at the saddle point is e ıπNV ˆ ρω ˜ (cid:15) − (1 ± .Next we put these results together. The eigenvaluefluctuations δx f , δx b can be safely dropped where k ’seigenvalues λ k are not zero. We need retain only theglobal averages of the eigenvalue fluctuations, δx f , δx b ,which contribute to the terms where λ k = 0. We obtain:¯ Z = (cid:88) ± ± e ıπNV ˆ ρω ˜ (cid:15) − (1 ± × ( ı exp ıφ ( ± − ω (4 π ˆ ρ ˜ (cid:15) ) − + δx f − δx b ) ω (2 π ˆ ρ ˜ (cid:15) ) − × ( ı exp − ıφ ( ± − ω (4 π ˆ ρ ˜ (cid:15) ) − + δx f − δx b ) ω (2 π ˆ ρ ˜ (cid:15) ) − (A3) × Π λ k (cid:54) =0 ( λ k + ı ( ± − ω (4 π ˆ ρ ˜ (cid:15) ) − ) ( λ k ± ıω (2 π ˆ ρ ˜ (cid:15) ) − )( λ k − ıω (2 π ˆ ρ ˜ (cid:15) ) − )The phases exp ± ıφ come from the saddle point values ofthe eigenvalues x = exp ± ı φ , where φ is chosen so thatexp ± ıφ = (cid:112) − ( E/ (cid:15) ) ± ıE/ (cid:15) . The factors of Q f and Q b inside the determinant produce these phase factors. The + and − saddle points simplify to:¯ Z − = − ( ı exp ıφ ω (2 π ˆ ρ ˜ (cid:15) ) − + δx f − δx b ) ω (2 π ˆ ρ ˜ (cid:15) ) − × ( ı exp − ıφ ω (2 π ˆ ρ ˜ (cid:15) ) − + δx f − δx b ) ω (2 π ˆ ρ ˜ (cid:15) ) − ¯ Z + = e ı πNV ˆ ρω ˜ (cid:15) − × ( δx f − δx b )( δx f − δx b ) ω (2 π ˆ ρ ˜ (cid:15) ) − (A4) × Π λ k (cid:54) =0 λ k ( λ k + ıω (2 π ˆ ρ ˜ (cid:15) ) − )( λ k − ıω (2 π ˆ ρ ˜ (cid:15) ) − )This has the desired property that ¯ Z ’s average value isone, because single instances of the eigenvalue fluctua-tions average to zero, zero-ing the + saddle point andsetting the − saddle point to one. Because of the func-tional integral’s construction, it should be identical toone for all values of its parameters, as long as factors of Q b are not inserted to obtain density averages.To calculate the density at point v , we must calculatethe average of − ıN ˜ (cid:15) − Q bv, L = − ıN ˜ (cid:15) − ( x b + ˜ x b ,v − π ˆ ρ (( y bv ) + ( z bv ) )) , (A5)and take the real part. Q bv, L is the lower right entry in Q bv L , which is a 2 × v . x b = − exp − ıφ is the saddle point value of Q b ’s second eigenvalue mul-tiplied by L = σ z ’s second eigenvalue, and ˜ x b ,v is thefluctuation in the second eigenvalue at position v . Eventhough this fluctuation pairs with an instance of δx b in¯ Z + , there is no pair for the δx b , so contributions fromeigenvalue fluctuations to the density at point v are zeroat leading order, and the + saddle point contributes noth-ing. At leading order the density at point v is equalto N ˜ (cid:15) − Re (exp − ıφ ) = N ˜ (cid:15) − ˆ ρ . The angular fluctuations(( y bv ) +( z bv ) )) dress this leading order result with a termproportional to the on-site value of the Green’s function,i.e. T r ((2 π ˆ ρ ˜ (cid:15)k − ıω ) − ). This represents a diffuson re-turning to its origin and dressing the density.Going on to the two point correlator, we now have tocalculate the average of the product of − ıN ˜ (cid:15) − Q bv , L (discussed above) and − ıN ˜ (cid:15) − Q bv , L . The saddle pointvalue of this last factor’s eigenvalue is + exp ıφ . There arethree interesting terms in this product, two of which comefrom the − saddle point, and the other from the + sad-dle point. First, the saddle point values − ıN ˜ (cid:15) − x b ,v and − ıN ˜ (cid:15) − x b ,v multiply to give N ˜ (cid:15) − , which is multipliedby ¯ Z ’s average value, one. Second, the angular fluctua-tions at site v correlate with the angular fluctuations atsite v , producing a factor (( y b ) +( z bv ) ) × (( y bv ) +( z bv ) ).After performing the integral, this turns into a prod-uct of two Green’s functions connecting v to v , i.e. |(cid:104) v | (2 π ˆ ρ ˜ (cid:15)k − ıω ) − | v (cid:105)| . If one averages over position v , v to get the correlator between energy levels E , E ,2this turns into Re T r ((2 π ˆ ρ ˜ (cid:15)k − ıω ) − ). Both of theseterms involve only the − saddle point, not the + saddlepoint.The two point correlator does have a contribution fromthe + saddle point: the eigenvalue fluctuations ˜ x b ,v and˜ x b ,v both pair with the δx b and δx b in that saddle point,contributing a factor of N − . Therefore the + saddlepoint contributes to the two point correlator a term pro-portional to Re [ ω − e ı πNV ˆ ρω ˜ (cid:15) − (A6) × Π λ k (cid:54) =0 λ k ( λ k + ıω (2 π ˆ ρ ˜ (cid:15) ) − )( λ k − ıω (2 π ˆ ρ ˜ (cid:15) ) − ) ]With a little more care, and the choice of kinetic oper- ator k = ( − π ˆ ρ ) − ∇ , it can be shown that these threeterms reproduce the standard result for the two level cor-relator R , which was obtained by Andreev, Altshuler,and Shklovskii . 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N K fixed limitwhere that phase is not trivial) appears to be unreachable. We have shown calculation of the advanced-retarded cor-relator. In order to obtain R it is necessary to calcu-late also the much simpler advanced-advanced correlator,which contributes a factor of − N ˜ (cid:15) − e − ı ¯ φφ