Two-eigenfunction correlation in a multifractal metal and insulator

Two-eigenfunction correlation in a multifractal metal and insulator

E.Cuevas Departamento de Física, Universidad de Murcia, E30071 Murcia, Spain.    V.E.Kravtsov The Abdus Salam International Centre for Theoretical Physics, P.O.B. 586, 34100 Trieste, Italy,
Landau Institute for Theoretical Physics, 2 Kosygina st., 117940 Moscow, Russia.

We consider the correlation of two single-particle probability densities at coinciding points as a function of the energy separation for disordered tight-binding lattice models (the Anderson models) and certain random matrix ensembles. We focus on the parameter range close but not exactly at the Anderson localization transition. We show that even away from the critical point the eigenfunction statistics exhibit the remnant of multifractality characteristic of the critical states. This leads to an enhancement of eigenfunction correlations and a corresponding enhancement of matrix elements of the local electron interaction at small energy separations. This enhancement is accompanied by a depression of correlations at large energy separations, both phenomena being a consequence of the stratification of space into densely packed but mutually avoiding resonance clusters. We also demonstrate that the correlation function of localized states in a -dimensional insulator is logarithmically enhanced at small energy separations provided that . A simple and general physical picture of all these phenomena is presented.

Finally by a combination of numerical results on the Anderson model and analytical and numerical results for the relevant random matrix theories we identified the Gaussian random matrix ensembles that describe the multifractal features both in the metal and in the insulator phases.

localization, mesoscopic fluctuations
72.15.Rn, 72.70.+m, 72.20.Ht, 73.23.-b

I Introduction

Eigenfunction and spectral statistics in quantum systems with quenched disorder were a subject of intense study Mirlin2000 () in the context of mesoscopic fluctuations of conductance and density of states (DoS), in particular in quantum dots ABG (). For this application the most relevant is the regime of weak deviation KM (); MF () from the Wigner-Dyson statistics given by the conventional random matrix theory (RMT) Mehta (). Disordered multi-channel quantum wires is the most important example of systems where single-particle eigenstates are all localized. Here the statistics of eigenstates require a non-perturbative treatment using the formalism of nonlinear sigma-model Efet-book () or banded random matrices Mirl-Fyod91 (). A special class are systems with the critical, multifractal (MF) eigenstate statistics KMut (); CKL (); Mirlin2000 (). Two-dimensional disordered metals fall in this class EF () provided that effects of localization are suppressed by magnetic field. Otherwise, one can speak only on weak multifractality which turns to localization before being fully developed. The true physical realizations of the critical, MF eigenstate statistics are systems at the critical point of the Anderson localization transition Wegner1980 (); Ch1990 () and the integer Quantum Hall systems at the center of Landau band ChDan (). Importantly, the class of systems with MF eigenstate statistics also allows for a random matrix representation KMut (), in particular using the power-law banded random matrices (PLBRM) MFSeil ().

Another field of intense research is the interplay between disorder and electron interaction with the seminal results on quantum correction to the tunnel DoS and conductivity AAL () of disordered two-dimensional metals and the correction to superconducting transition temperature due to a simultaneous effect of disorder and the Coulomb interaction Finkel (). In all those works disorder and interaction are taken into account essentially perturbatively along the lines given in GLK (). Recently there was an attempt FIKY () to consider the problem of superconductivity near the Anderson transition in which disorder has been treated non-perturbatively by postulating the MF statistics of one-particle states that enter the matrix element of a phenomenological electron attraction:


In particular, the simplest quantity of interest is the disorder average matrix element at a given energy separation between one-particle energies and . For real eigenfunctions (orthogonal symmetry class) it is proportional to the correlation function , where


and is the spectral correlation function which is close to 1 for much greater then the mean level spacing. The correlation function is the main subject of the present paper.

The correlation function defined by Eq.(2) is a measure of overlap of two different eigenfunctions. For truly extended normalized states (e.g. in a quantum dot) and thus , where is a system volume. Remarkably, is also valid for classical examples of localized states, e.g. in a quantum disordered wire. In this case two states are typically not overlapping but with a small probability of (where is the localization radius) they are localized in the same place and then the integral in Eq.(2) is of the order of the inverse localization volume .

There are cases, however, when an eigenfunction does not occupy all the available volume or all the localization volume and the typical amplitude is not just the inverse volume (for extended states) or the inverse localization volume (for localized states). In this case a non-trivial behavior of the correlation function is expected. Such situation is realized near the critical point of the Anderson localization transition. In the vicinity of this point in the region of extended states (multifractal metal) or in the region of localized states (multifractal insulator) the system retains the characteristic features of the critical multifractal statistics of eigenstates which makes it qualitatively different from both a normal metal or a normal Anderson insulator.

In this paper we will identify and quantify such characteristic features in the correlation function and give their interpretation in terms of the typical behavior of single-particle states. To attain this goal we will combine new analytical results for the PLBRM with numerics on the PLBRM and the Anderson model. We specially focus on the dependence of on the energy difference in the cross-over region in the vicinity but not exactly at the Anderson transition point which has not been studied so far.

The paper is organized as follows. In section II we give a brief introduction into the subject of multifractality of critical eigenstates focusing on the main effect of multifractality which is the critical enhancement of eigenfunction correlations. In section III we give a cartoon of the off-critical states in a multifractal metal and a multifractal insulator and introduce the random matrix theories which may describe them. In Sec.IV we present the results of an analytical theory of eigenfunction correlations for a class of almost diagonal Gaussian random matrices which all the RMT’s suggested to describe strong multifractality fall into. In Sec.V we consider the two-eigenfunction correlation function exactly at the critical point of the localization transition in the 3D Anderson model and for the critical random matrix ensemble in the limit of strong multifractality. We show that the dynamical scaling relationship suggested by Chalker is not violated even in the limit when the fractal dimensions are very small. In Section VI we describe the new phenomenon of eigenfunction mutual avoiding and present a qualitative picture that simultaneously explains the enhancement of eigenfunction correlations at small energy separations and the eigenfunction mutual avoiding at large energy separations. In section VII we consider the properties of eigenfunction correlations in a multifractal insulator. In particular, we describe the new phenomenon of logarithmic enhancement of eigenfunction correlations at small energy separations in the 2D and 3D Anderson insulators and show the absence of such enhancement in the quasi-1D case. We also suggest a Truncated Critical RM ensemble that describes all the principal features of eigenfunction correlations in the 3D multifractal insulator. Section VIII is devoted to the random matrix description of the multifractal metal. We show that the sub-critical PLBRM suggested in Ref.MFSeil () gives a reasonable agreement with the 3D Anderson model. By analytical treatment of this RM model we found the region of parameters where the eigenfunction correlations become effectively short-range in the energy space which may poit out on the existence of a new metal phase above some critical dimensionlity in the multi-dimensional Anderson model. In the Conclusion we list all the principal results of this paper.

Ii Multifractality of critical eigenfunctions

The ”standard” model (the Anderson model) for the Anderson localization transition in dimensions is the tight-binding model with the hopping constant and random on-site energies characterized by the distribution function which is frequently chosen constant in the interval and zero otherwise. There is a vast literature (see e.g. Oht () and references therein) on numerical investigation of the Anderson localization transition in this model on a 3D lattice. Recently also higher dimensions become accessible to modern computers MEMir (). While the earlier studies of this model were focused on the critical behavior of the localization/correlation length near the critical disorder , the recent works were mostly related with the statistics of critical eigenfunctions. The multifractality of critical eigenfunctions predicted in Wegner1980 () almost immediately after emergence of scaling theory of localization has been confirmed and quantified in detail.

The results obtained for the Anderson model exactly at the critical point seem to be very well described MirEv () by the critical PLBRM model MFSeil (); KMut (). This model is defined as an ensemble of random Hermitean matrices which entries fluctuate independently around zero with the variance:


where for the Dyson orthogonal, unitary, and symlectic symmetry classes Mehta () and is the parameter that controls the multifractality exponents. This model has been studied and its comparison with the Anderson model in dimensions has been done predominantly for the statistical moments of a single eigenstate at a given energy :


The best known example is the inverse participation ratio (IPR) given by the second moment . The multifractal statistics of a single eigenstate is characterized by the moment that scales with the system volume or the total number of sites as:


where is the fractal dimension corresponding to -th moment. The existence of the scaling law Eq.(5) and the dependence of the exponent on are the principle features of eigenfunction multifractality. The fractal exponents depend also on the symmetry class and the space dimensionality . For the critical PLBRM Eq.(3) the dependence on is modeled by the dependence of on the parameter .

The critical scaling Eq.(5) with respect to the system size has its dynamical counterpart when instead of one single eigenfunction one considers the correlation function Eq.(2) of two eigenfunctions at an energy separation between them. This scaling has been suggested by Chalker Ch1990 (); ChDan () many years ago:




is the mean level spacing and is the upper cut-off of multifractality. Numerics on the integer Quantum Hall systems and in the critical point of the 3D Anderson model was consistent Huck (); SchPot () with this scaling.

An important feature of Eq.(6) is that the exponent in the -dependence is smaller than 1. Even in the limit of infinitely small correlation dimension the correlation function decays slowly as . This implies that the sparse critical states separated by large energy distance are still well overlaping FMover (), in contrast to strongly localized states which typically do not overlap even for nearest neighbors in the energy space. The reason for such a behavior and the physical meaning of the energy scale will be discussed in Section VI.

As the correlation function is equal to both for the truly extended and the ideal localized states, Eq.(6) implies the critical enhancement of eigenfunction correlations for . This enhancement is crucially important for electron interaction near the Anderson localization transition, in particular for the superconducting transition temperature FIKY (). To illustrate this point we present in Fig.1 the result of numerical diagonalization of the critical PLBRM, the classical Wigner-Dyson RM with extended eigenstates, and the ensemble of conventional banded random matrices Mirl-Fyod91 () with exponentially decreasing entries which describes strongly localized eigenstates in quasi-1D disordered systems. The critical enhancement of eigenfunction correlations is evident from this plot.

Figure 1: Critical enhancement of eigenfunction correlation. Results of exact diagonalization of the critical PLBRM at , the banded random matrices with B=5, and Wigner-Dyson RM are shown in red, blue and green, respectively, and are represented by squares (N=200), circles (N=1000) and stars (N=2000).

The physical origin of the enhancement is two-fold: (i) a critical eigenfunction ”occupies” only part of the available space which by normalization enhances its amplitude, and (ii) the supports (the manifold of where is essentially non-zero) of different critical eigenfunctions are strongly overlapping. It is important that both conditions are fulfilled simultaneously. For instance the condition (i) is fulfilled for localized states even better than for the critical ones but the lack of the condition (ii) levels off the gain in the correlation function . On the contrary, in a metal the condition (ii) is trivially fulfilled, but the eigenfunction amplitude is small.

Iii Off-critical states and their random matrix representations

Gaussian random matrix models proved to be an efficient and universal theoretical tool for describing complex systems. The success was partially due to the available analytical solutions Mehta (); Mirl-Fyod91 () and partially due to efficient algorithms of numerical diagonalization of matrices. Therefore it is highly desirable to have random matrix models that describe not only the critical MF eigenstates but also localized and extended eigenstates in the vicinity of the Anderson transition. The criterion to select such models is a qualitative and (when possible) a quantitative agreement with the results on the 3D Anderson model.

As will be demonstrated below, the correlation function in the 3D Anderson model contains the critical power-law behavior Eq.(6) well beyond the Anderson transition point. As a matter of fact the correlation function is indistinguishable from the critical one until the dynamic length ( is the mean DoS) exceeds the localization/correlation length . For , or smaller than the level spacing in the localized volume , the correlation function loses its critical features and shows typical features of a metal or an isulator.

This allows us to suggest the following cartoon of typical eigenfunctions in the vicinity of the localization transition shown in Fig.2. Namely, a typical localized state in a ”multifractal insulator” can be viewed as a ”piece of multifractal” of the size of the localization radius (Fig.2b.), in contrast to a conventional localized state where all the localization volume is more of less homogeneously ”filled” (Fig.2a.). In the same way, typical extended states on the metallic side of the localization transition (”multifractal metal”) should look like a mosaic made of such ”pieces of multifractal” (Fig.2c.).

Based on the persistence of the critical behavior beyond the critical region it is natural to assume that the random matrix model for the extended states near the critical point and the localized states on the other side of the transition should bear features of the critical RMT.

Figure 2: 2D cartoon of a) conventional localized state; b) localized state in a multifractal insulator; c) extended state in a multifractal metal. The darker regions correspond to higher eigenfunction amplitude. The localization/correlation radius is shown in each case.

Let us start by constructing a random matrix model for the multifractal insulator. Given that the quasi-1D insulator is well described by the banded random matrices Mirl-Fyod91 () with exponentially decaying variance we suggest the following hybrid RM model as a model for the multifractal insulator:


As compared with the critical PLBRM model Eq.(3), the model Eq.(8) contains an additional parameter which sets in a finite localization radius . It also contains an exponent which depends on the space dimensionality of the disordered lattice model we would like to model by the RMT. In Sec.VIII we give both analytical and numerical arguments in favor of the choice

Another candidate has been suggested in Ref.MFSeil ():


In this case the localization radius is controlled by the variable exponent of the power-law. For a multifractal insulator .

The possible RM models for multifractal metal are also constructed as deformations of the critical PLBRM. The model Eq.(9) for is believed MFSeil () to describe the multifractal metal. One can also think that the Gaussian RMT


which is a hybrid of the critical PLBRM and the WD RMT, is also suitable for this purpose. Below we will study all those RM models in detail and compare the corresponding results for the correlation function with the results obtained by numerical diagonalization of the -dimensional Anderson model.

Iv Almost diagonal Gaussian RMT: analytical results for

The characteristic properties of multifractal statistics of critical and off-critical states are best seen when the multifractality is strong. This is the case where the parameter in Eqs.(3, 8, 9, 10) is small. On the other hand, this is exactly the limit where the typical off-diagonal elements of are small compared to diagonal ones. Such matrices (referred to as almost diagonal random matrices (ADRM)) may possess a non-trivial statistics of eigenfunctions which justifies their special study KYev (); YOs (). The idea of analytical treatment of ADRM, first suggested in Ref.Levitov () and used in MirEv () to compute the correlation dimension for the critical PLBRM model Eq.(3), is similar to the virial expansion in dilute gases. However, instead of taking into account two-, three- and multiple-particle collisions, one considers progressively increasing number of interacting resonance sites coupled by a small off-diagonal matrix element . Recently the virial coefficients for the Gaussian ADRM with an arbitrary (but small) variance were expressed through the supersymmetric field theory YOs () and the correlation function has been explicitly calculated in the two-state approximation for the unitary symmetry class . The result is the following:






In Eqs.(12,13) we denote


and .

The result given by Eqs.(12,13) is valid in the limit when:


For the RMT defined by Eq.(8) and Eq.(9) with which are suggested to describe the multifractal insulator, the sum over in Eq.(15)converges. Then the validity of Eqs.(12,13) is independent of the matrix size in the limit and is controlled only by a small parameter . On the contrary, for the models of the multifractal metal described by Eq.(10) and Eq.(9) with the sum in Eq.(15) diverges at large . Then Eqs.(12,13) are only valid for where we define the correlation radius as follows:


We will show below that a good qualitative description of the metal phase in the limit can still be obtained from the above theory if one substitutes for in Eqs.(12,13).

Eqs.(12)-(16) will be used throughout the paper to analize different random matrix ensembles suggested as possible models for critical eigenstates and the off-critical states in a multifractal metal and insulator.

V Two eigenfunction correlations at criticality

It is not a priori clear that the critical power-law behavior Eq.(6) and the dynamical scaling relationship Eq.(7) hold true for all systems where Eq.(5) is valid. In particular it is interesting to study the correlation function Eq.(2) in the limit of strong multifractality when . Below we will derive an analytical formula for the critical PLBRM in the limit which corresponds MEMir () to and confirm the scaling law Eq.(7) by numerical diagonalization of PLBRM with very small .

One can easily see from Eqs.(12)-(14) in which we plug in Eq.(3) that in the interval the correlation function given by these equations has an asymptotic power law behavior Eq.(6) with :


Applying Eq.(15) to the critical PLBRM Eq.(3) gives the criterion of validity in which case one cannot distinguish between and , or, correspondingly, between and . Thus the analytical formulae Eqs.(12,3) is consistent with the scaling relationship Eq.(7), given that as .

A comparison of the analytical results and the results of numerical diagonalization of the critical PLBRM with and is shown in Fig.3. The coincidence is very good for large energy separations. The deviation at small energy separations is due to the difference in the values of . The exponent for the analytical curve and for the numerical curve which is very close to the prediction of Eq.(7) , where is found from the numerical data for and Eq.(5).

The scaling relationship Eq.(7) is further checked in Fig.4 where the numerical data for and is plotted as a function of . The fulfillment of this relationship down to as small as 0.005 and an agreement with the theoretical prediction of Ref.MirEv () is spectacular.

Figure 3: Two-eigenfunction correlation function for the critical PLBRM with , and (square),1000(circle) and 2000 (star). The analytical curve at given by Eqs.(11-13) is shown by a solid line.
Figure 4: The scaling relationship between and for the critical PLBRM. The solid line is the prediction based on Ref.MirEv () .
Figure 5: Two-eigenfunction correlation function for the 3D Anderson model (orthogonal symmetry class) with a triangular distribution of random on-site energies (solid symbols) and the critical PLBRM Eq.(3) with and (open symbols). The energy difference is measured in units of mean level spacing. The insert shows the mean density of states; the mobility edge corresponds to . The energies were taken from the window for the 3D Anderson model and for the critical PLBRM. The slope of the critical power-law Eq.(6) is 0.52 in both cases which corresponds to .

Thus from the combination of analytical and numerical results we conclude that the Chalker’s scaling Eqs.(6),(7) is valid for arbitrary small and thus for arbitrary strong multifractality.

Finally we demonstrate how well the critical PLBRM Eq.(3) describes the two-eigenfunction correlations in the 3D Anderson model at the mobility edge. To this end we modify the distribution of the on-site energies in the Anderson model from the standard rectangular box distribution to the triangular distribution where the mobility edge corresponds to . The correlation function with near the mobility edge is shown in Fig.5. It coincides almost exactly with the corresponding curve resulting from numerical diagonalization of the critical PLBRM ensemble with only one fitting parameter .

Vi Eigefunction mutual avoiding and stratification of coordinate space

Results of both numerical and analytical calculation presented in Fig.3 reveal another unexpected feature of eigenfunction correlation which appears to be common to all ADRM. Surprisingly it is also present for the 3D Anderson model both in the metal and in the insulator phase (see Fig.6). This is the negative eigenfuncton correlations for . Indeed, one can see from Fig.3 and Fig.6 that for large enough the correlation function goes below the uncorrelated limit which corresponds to . We denote by the value of where this limit is reached. For the correlation function decreases down to zero.

Figure 6: Eigenfunction avoiding for the PLBRM with b=0.42(stars), 3D Anderson insulator (red triangles) and metal (blue triangles). The dotted line corresponds to the limit of uncorrelated eigenfunctions; the solid line corresponds to the power law . Points below the dotted line correspond to eigenfunction avoiding.

Such a behavior implies that two eigenfunctions separated by an energy difference try to avoid each other. That is, if a site is occupied in one of the states it should be predominantly empty in the other.

To explain such a behavior the following cartoon is useful. Let us define a support of an eigenfunction as the manifold of sites where is essentially non-zero. To construct such a support starting from a given site with the on-site energy we find all the sites in resonance with the site , i.e. such sites which on-site energies obey the relationship . Then the procedure should be repeated for all sites and so on. It is important that so obtained manifold does not always include all the sites of the system. If this is the case, the whole coordinate space is stratified into a set of mutually non-intersecting supports (Fig.7).

Figure 7: A cartoon of stratification of the coordinate space: different non-intersecting supports shown by different colours. Each support corresponds to a shell of states occupying this support and thus strongly overlapping; states belonging to different shells do not overlap. The stratification of space explains both strong correlations of states at energy separation smaller that the single-shell bandwidth and mutual avoiding of eigenstates for .

Once the support is defined, one can build a shell of states on this support by making a linear combination of on-site states, pretty much in the same way as in building the conduction band states out of the on-site states in the tight-binding model. Then by construction the eigenfunctions belonging to the same shell are well overlapping but those belonging to different shells do not overlap.

From this cartoon it is clear that the physical meaning of the scale is the width of the energy band corresponding to a single shell. Indeed, if the energy separation greatly exceeds the typical single shell bandwidth, the two eigenfunctions must belong to different shells and thus do not significantly overlap in space. On the contrary, if is smaller than the single shell bandwidth, the two states typically belong to the same shell and thus overlap strongly no matter how sparse the shell support.

The new energy scale , which is the upper energy cut-off of the multifractal correlations, corresponds to a new length scale


which has a meaning of the minimum length scale of the fractal texture. In the -dimensional Anderson model the energy scale can be estimated as

where is the total bandwidth. Estimating the DOS as we find , where is the lattice constant.

Clearly the picture with a stratified coordinate space is possible for PLBRM Eq.(3) with small enough when the single shell bandwidth is small compared to the total bandwidth . Amazingly, the 3D Anderson model which low-frequency critical features are well described by the critical PLBRM with , also follows the predictions of the critical PLBRM for high frequencies . This is a consequence of a relatively large value of the critical disorder which results in considerably smaller than the conduction bandwidth . In particular its coordinate space must be stratified to explain the observed (see Fig.6) mutual avoiding of eigenstates.

Vii The multifractal insulator

Figure 8: Eigenfunction correlation in the 1D Anderson insulator with rectangular distribution of on-site energies and periodic boundary conditions. The disorder strength is (circles), (stars). The inverse participation ratio is equal to 0.23 and 0.46, respectively.

As has been demonstrated in Sec.V critical eigenfunctions with are strongly correlated in space. Here we consider the case of multifractal insulator where the localization radius is large compared to relevant microscopic lengths (the lattice constant or elastic scattering length) but is much smaller than the system size . We will identify a suitable random matrix model to describe this case and compare the properties of eigenfunction correlation in this model with those of the 3D Anderson model.

vii.1 The ideal insulator limit

We start by considering a limit of strong disorder when the localization length and the multifractal nature of eigenstates does not show up. A common wisdom is that in the strongly localized regime the positions of the localization centers are completely uncorrelated. As it is shown in the Introduction, this leads to


which we will refer to as the ideal insulator limit. Fig.1 shows how this limit is reached in the ensemble of banded random matrices.

Figure 9: Eigenfunction correlation in the 3D Anderson insulator with rectangular distribution of on-site energies and periodic boundary conditions. The disorder strength is (green), (blue), (red), (purple). The system size is for filled symbols and for open symbols. The inverse participation ratio for the four insulating systems is , , , which corresponds to , , , according to . The change of the slope occurs at . The slope for larger energy separations progressively increases with increasing remaining smaller than 1. The insert shows the result for , for the periodic (upper blue curve) and the hard wall (lower red curve) boundary conditions.

Note that in this limit is much smaller than the self-overlap of given by the inverse participation ratio Eq.(4). Only for very small energy separations (typically ) which we will not be considering here, the IPR limit can be approached.

Now let us see how does the correlation function look like for the strong Anderson insulator. The corresponding plot for 1D Anderson model is shown in Fig.8. It coincides almost exactly with the ideal insulator limit Eq.(19).

The plot for the 3D Anderson model is shown in Fig.9. On can see that is significantly enhanced at small energy separations and does not resemble at all the correlation function in 1D Anderson insulator.

vii.2 Repulsion of centers of localization for

In order to understand why the ideal insulator limit is not reached in the 3D case despite the ratio we compute numerically the probability distribution (PDF)


of the distance between the points and in real space (centers of localization) where has an absolute maximum, provided that the energy separation between the states is .

The results are shown in Fig.10. It is seen that the function is far from being independent of (which would imply the lack of correlations between centers of localization). In fact, there is a repulsion of centers of localization at distances which shows up in the decreasing probability density to find two centers of localization close to each other. Note that is almost 10 times larger than the localization radius estimated from the inverse participation ratio .

Figure 10: The probability density Eq.(20) of having two centers of localization at a distance in real space and at a distance in the energy space computed for the 3D Anderson model in the strong localization regime (, , ). The repulsive core exceeds the ”hard ball” limit by a factor of 5-6.

An explanation to this fact of repulsion between centers of localization and is based on the resonance interaction between states and if the energy distance between them is smaller than the typical overlap integral .

The size of the repulsion core can be estimated from the equation:


The characteristic energy scale is the mean level separation for states localized in the same volume . Thus the repulsion of centers of localization is a direct consequence of repulsion of energy levels for states confined in the same volume . The energy scale depends on the strength of disorder and is of the order of the Fermi energy for strongly localized states. At the size of the repulsion core may considerably exceed the localization radius.

The qualitative picture of repulsion of centers of localization can be quantitatively confirmed using the analytical theory Eqs.(11-14) for the almost diagonal Gaussian RMT. To this end we look at the contribution of to the sum in Eq.(12). Replacing the summation over by integration we obtain the contribution to :


This equation can be easily interpreted using an elementary perturbation theory. Indeed, for strongly localized states the eigenfunction correlation function can be represented as follows:

The amplitude at the tail of the wavefunction with the maximum at a point can be computed from the elementary perturbation theory in which the wavefunction of the zero-th approximation corresponding to the energy is :


The fluctuating on-site energy is the main part of the eigenvalue for a sufficiently strongly localized state. Thus we come to a conclusion that the amplitude of the wavefunction at a center of localization of the wavefunction is inversely proportional to and thus is strongly enhanced when . At the first glance this is in a contradiction with the common wisdom that which is apparently -independent. The point is that the quantity has many accidental spikes due to resonances between on-site energies. The measure of such resonance points is small and for some (but not all) purposes one can neglect them and to approximate . The best known example when the two-spike eigenfunction makes the main contribution is the low-frequency conductivity in the localized phase Mott (). As we will see below, here we deal with a very similar phenomenon.

Now the correlation function can be computed just by averaging over disorder and the distance :

where is the PDF defined by Eq.(20).

Comparing Eq.(VII.2) for with Eq.(22) we see that:


where is found from the condition similar to Eq.(21).

In the opposite limit , or , we have a resonance enhancement and . Then the comparison of Eq.(VII.2) with Eq.(12) yields:


vii.3 Logarithmic enhancement of correlations of localized eigenfunctions and the Truncated Critical RMT

Figure 11: A comparison of eigenfunction correlation functions for 3D Anderson model with , and the truncated critical RMT Eq.(7) with , , and . The IPR takes values of and , respectively. The scale of is different in those two cases by approximately a factor of 11.

Now let us consider Eq.(VII.2) for assuming that all states are exponentially localized and thus . We also assume for simplicity that , where . Then one immediately obtains from Eq.(VII.2) that due to the phase volume factor the correlation of exponentially localized eigenfunctions depends crucially on the dimensionality of space. Namely, for the ideal insulator limit Eq.(19) is reached for sufficiently small (see Fig.1), while for and the correlation function acquires a logarithmic in enhancement factor:


The physics behind this result is similar to the one which leads to the selebrated Mott’s law Mott (); Ber () for the ac conductivity at . The difference is that the contribution to conductivity from the resonance states with the distance between the points of maximal amplitude is proportional to the square of the dipole moment , so that the phase volume factor gets multiplied by resulting in emergence of the logarithmic factor instead of in our case.

Below we obtain this result for the truncated critical RMT defined by Eq.(8). The phase volume factor can be formally taken into account in the random matrix formalism Eq.(12) if one assumes the following relationship between the -dimensional vector and the difference of matrix indices :


where is the total solid angle in the -dimensional space.

In particular Eq.(29) suggests that for exponential localization the correct truncating factor in Eq.(8) has the form:


This sets the exponent in Eq.(8) equal to:


Then Eq.(12) can be used which is convenient to rewrite in the following form:


where is found from the equation



For the truncated critical RMT Eq.(8) with one finds:



The integral in Eq.(32) is well convergent and thus mainly contributed by . This makes it possible to obtain a simple analytical expression for :



The first line of Eq.(35)is consistent with Eq.(28) in which and . The power-law behavior in the second line of Eq.(35) is a remnant of the critical behavior Eq.(6). It exists only for considerably large where , i.e. only in the multifractal insulator. For a very strong insulator with , the localization radius is smaller than the minimal length scale of the fractal texture. This is the region of an ordinary insulator where the entire localization volume is more or less homogeneously filled.

Thus the eigenfunction correlation function for the truncated critical RMT Eq.(8)interpolates between the behavior given by Eq.(28) at (or ) and the critical behavior Eq.(6) which is valid for . At both asymptotic forms are apparently matching with each other.

The physical picture that leads to such a behavior is the following. There are two distinct regions and where physics of eigenfunction correlations is entirely different. In the case the characteristic length has a meaning of the period of beating in the overlap of two fractal eigenfunctions inside the localization volume. The regions where two fractal supports match well with each other alternate with the regions with a strong mismatch between them, very much like in the case of two grids with slightly different periods. The regions of strong overlap make the main contribution

to the eigenfunction correlation function which is of the order of the IPR of a multifractal metal (see next Section for more details) with the system size equal to the localization radius and the correlation length equal to the size of the well overlapping regions. To obtain the correlation function one has to multiply by the probability for the entire localization volumes to overlap. This probability is , as for there is no correlations in the positions of the localization volumes. Thus we obtain the critical power-law Eq.(6): .

For physics of eigenfunction correlations changes drastically. Now localization volumes are statistically repelling each other and the overlap is only due to the tails. In this region the length scale loses its physical meaning which is taken over by the length scale given by Eq.(21).

The overall shape of with the logarithmic enhancement factor obtained within the Truncated Critical RMT describes the numerical results on the 3D Anderson model very well (see Fig.11). The absence of this factor at explains the qualitative difference between the case (see Fig.8) and (see Fig.9). This difference is essentially due to a competition between two effects (i) repulsion of centers of localization and (ii) resonance enhancement of overlap by tails. The first effect tends to decrease the probability of the overlap of localization volumes. The second effect increases the eigenfunction overlap by means of tails. In the 1D case these two effects compensate each other and the result is the same as one would obtain for completely uncorrelated positions of localization volumes and the typical exponentially decreasing tails. In higher dimensions the enhancement of overlap in the tail region prevails because of the increased volume of those regions.

Concluding this subsection we claim that the truncated critical RMT provides an excellent description of the 3D Anderson insulator both in the strong localization region (see Fig.9) and in the region of multifractal insulator where the localization radius is large and the corresponding scale is small compared with the upper cutoff of multifractal correlations. Because of the limited size of the 3D lattice this latter region is out of reach for numerical simulations on the 3D Anderson model, and the random matrix theory is the only mathematical model which properly describes physics of the multifractal insulator.

vii.4 Super-critical PLBRM

Note that there is another RMT Eq.(9) suggested in Ref. MFSeil () as a candidate to describe eigenfunction correlations in the multifractal insulator. Below we show that this super-critical PLBRM is principally flawed, as it corresponds to a power-law localization which is not the case in the 3D Anderson model.

This can be best demonstrated by Eq.(24) in which with . Accordingly, the typical scale for the repulsion of centers of localization is:


In Fig.12 we plot the results of numerical calculation of the PDF for the super-critical PLBRM Eq.(9). The characteristic scale where reaches its maximum, is well seen in this plot.

Figure 12: The correlation function of centers of localization for the super-critical PLBRM Eq.(9) with , , for (black), (blue), (green). The insert shows the -dependence of where reaches its maximum. The finite slope of for is a finite size effect which was neglected in Eq.(26).

The analytical treatment based on Eq.(32) yields for this model:


where , and

As well as the entire approach based on Eq.(12), the above results are valid when is the smallest relevant parameter. In the problem of PLBRM with close to 1, there is a competition between the small parameters and , so that the validity of Eqs.(36),(37) requires also .

We see that in the infinite system the power-law Eq.(37) in is not restricted at small , and no energy scale similar to emerges. This can be explained only if we assume that the localization length for is of order one. Then for all energy separations the repulsion core , and no qualitative change in the correlation function occurs until hits the system size . For smaller the correlation function is almost a constant. This quantitative analysis is illustrated by Fig.13.

Figure 13: Two-eigenfunction correlation for the super-critical PLBRM Eq.(9) with calculated analytically using Eqs.(11-14). The power law is valid for all energy separations corresponding to . The onset of the plateau moves to in the limit . The ideal insulator limit is reached by decreasing the slope with increasing .
Figure 14: Two-eigenfunction correlation for the super-critical PLBRM Eq.(9). Numerical results for (this plot corresponds to , , ) show that the exponent of the power-law changes at with the larger value corresponding to smaller energy separations.

The region of that could describe the multifractal insulator with large corresponds to . In this case an energy scale similar to should appear. It can be found from the condition


At the slope on the log-log plot of should change from the critical value at to a different (but constant) -dependent value at . This change of the slope is clearly seen in the numerical simulations on the super-critical PLBRM presented in Fig.14. It appears that in all cases studied the slope at is larger than that at . This is in a clear contradiction with the results (see Fig.9) obtained in the 3D Anderson insulator.

An important conclusion we can draw from the above analysis of the super-critical PLBRM is that the correlation function for is the power-law in this model. This can be traced back to the power-law character of localization in the super-critical PLBRM which is not the case in the disordered lattice models (such as the 3D Anderson model) with short-range hopping integrals. This is the reason why the super-critical PLBRM is not suitable to describe the insulating phase of the 3D Anderson model.

Viii Search for random-matrix model for a multifractal metal

viii.1 Anti-truncated critical RMT

Surprisingly, the natural counterpart to the truncated critical RMT Eq.(8) which is defined by Eq.(10) (”anti-truncated” critical RMT) does not describe extended states in the multifractal metal. The reason is that this model possesses two low-frequency system-size independent energy scales instead of the single scale which is associated with the size of a multifractal cell in Fig.2c. In order to see this we analyze the analytical formulae Eqs.(11-14) with the variance defined by Eq.(10).

To this end we expand the summand of Eq.(12) in to arrive at the formula similar to Eq.(22) but with the upper limit of integration equal to the correlation radius :


where according to Eq.(16).

Substituting Eq.(10) for we arrive at:


Eq.(40) is valid at when the upper limit of integration is larger than the lower limit. This sets the energy scale .

Another scale gives the cross-over scale that separates the critical behavior and the behavior that takes place for indermediate frequencies . While the scale (similar to the scale in a 1D insulator) determines the onset of the low-frequency plateau, the second relevant scale that appears in the model Eq.(10) seems to have no physical meaning. Indeed, the existence of this scale leads to a characteristic form of the correlator which log-log plot has a significant slope increase just before it drops to zero at the plateau (see Figs.15, 16).

Figure 15: Numerics on the anti-truncated critical RMT of the orthogonal symmetry class. The predicted analytically non-monotonous behavior of log-log slope controlled by the energy scale is well seen both for small and large system sizes. The solid line is a fit according to Eq.(40).

We did not find the behavior of such type in the 3D Anderson metal (see Fig.17). The plot in Fig.17 clearly shows a saturation A-m () at :


However, there is no evidence of a maximum in the slope just above the onset of the plateau.

Figure 16: Eigenfunction correlation for the anti-truncated critical RMT Eq.(8) (red curve) and for the sub-critical PLBRM Eq.(9) (blue curve) of unitary symmetry class computed analytically from Eqs.(11-14). For the anti-truncated critical RMT the non-monotonous behavior of log-log slope is similar to the one in Fig.15.
Figure 17: Eigenfunction correlation in the 3D Anderson model: extended states. The disorder strength is (green), (red), (blue). The system size is (diamonds) (circles) and (squares). The correlation length is estimated from the inverse participation ratio as follows (, ) and is equal to: , , at , , , respectively. In the insert we compare results for , for the periodic (red filled squares) and the hard-wall (red open squares) boundary conditions. For hard-wall boundary conditions the correlation function looks ”more critical”, as the critical point in this case is closer to .

viii.2 Sub-critical PLRBM

Now we consider the sub-critical PLBRM ensemble defined by Eq.(9) with . In this case analytical arguments similar to Eqs.(39,40) predict only one relevant energy scale such that for the correlation function is constant and for (but ) it is a pure power law . Thus the sub-critical PLBRM is free from the drawback related with the unphysical second energy scale. For comparison we plotted the analytical results for the anti-truncated critical RMT and for the sub-critical PLBRM in Fig.16. It is seen that the overall shape of the blue curve for sub-critical PLBRM is much closer to the results of 3D Anderson model of Fig.17.

Note however, that the power-law emerging in the analytical results for the sub-critical PLBRM has an exponent which is larger than the critical exponent. Computer simulations (see Fig.18a) on the sub-critical PLBRM with very small confirm this analytical result as extrapolation and show that the slope increases with increasing the system size N. However, the slope of the corresponding curves for the 3D Anderson model of Fig.17 is almost independent of the system size and is equal or smaller than the critical slope.

Figure 18: Eigenfunction correlations in the sub-critical PLBRM of the orthogonal symmetry class. Left panel (a): the small- limit , at (squares) and (diamonds). The critical correlations for , and are shown by stars. The solid line corresponds to the power-law . The slope of the sub-critical curve is larger than 1 (correlations are short-range in the energy space). This case is relevant for the metallic phase of the Anderson model in very high dimensions. Right panel (b): the case , at (squares) and (diamonds) is relevant for the Anderson model in (”multifractal” metal). The slope is less than the critical (which in turn is less than 1) and is almost size-independent. Correlations are long-range in the energy space.

The reason for the discrepancy is that the analytical result for the slope corresponds to the limit . At a finite the slope decreases with increasing and at a sufficiently large may become smaller than the critical one. It is reasonable to assume that at small and this happens at . The relevance of the parameter is also seen from the expression for the correlation length


which was found (up to a constant of order one) analytically from Eq.(16). This expression is apparently meaningless for where may become negative.

Numerical simulations on the sub-critical PLBRM with (e.g. for and relevant for the 3D Anderson model) show (see Fig.18b) that the log-log slope of is somewhat smaller than the critical one and is almost independent of the matrix size . Thus the sub-critical PLBRM shows exactly the same character of eigenfunction correlations as in the 3D Anderson metal (see Fig.17).

Two parameters of the sub-critical PLBRM allow to simulate the effect of the finite correlation length (choice of ) and the dimensionality of space (choice of ). Note in this connection that for the disorder strength significantly smaller than the critical value , not only but also is -dependent. The point is that in the 3D Anderson model the variance of the on-site energies fluctuations is proportional to , while the off-diagonal hopping integral is equal to 1. This implies that the ratio of a typical off-diagonal to a typical diagonal elements controlled in Eq.(9) by the parameter should scale like . As the log-log slope of decreases with increasing , moving away from the Anderson transition into the metallic phase has an effect of decreasing the slope. On the insulator side of the transition the situation is opposite and one should expect an increase of the slope (for ) with increasing . Fig.9 shows that it is apparently the case.

Another relevant note is that for the Anderson model in higher dimensions the correlation dimension decreases. This can be modeled by a decreasing parameter . Then the analogy with the sub-critical PLBRM suggests that for sufficiently high dimensions the behavior in the -dimensional Anderson model should become similar to the one in Fig.18a. Namely, the exponent in Eq.(7) may become larger than 1. This changes qualitatively the eigenfunction correlations, as they become effectively short-range in the energy space. In particular, the return probability CKL () which is proportional to the Fourier transform of behaves in the time interval as for and is a constant for .

We believe that this qualitative change in the eigenfunction statistics (if confirmed for a -dimensional Anderson model with ) should lead to dramatic physical consequences marking a transition to a new metallic state.

Ix Conclusion

In conclusion we list the main results obtained above. The most important of them is the persistence – beyond the point of localization transition– of the critical power-law in the dependence of the eigenfunction correlation function on the energy separation and the related enhancement of at , where is the mean level spacing in the localization/correlation volume and is the upper energy cut-off of multifractality. This enhancement leads to an enhancement of matrix elements of local electron interaction which may result in, e.g. an enhancement of the superconducting transition temperature in the vicinity of the Anderson localization transition FIKY (). Another important observation is that the enhancement of correlations at is always accompanied by the depression at , both phenomena being the consequences of the stratification of the coordinate space into mutually avoiding supports of the fractal structure with well overlapping eigenfunctions living on each of them. An independent – but also important– phenomenon is the logarithmic enhancement of in the 2D and 3D Anderson insulator at (and the absence of such enhancement in the quasi-1D disordered wire). It is a result of a competition of two simultaneous phenomena: the repulsion of centers of localization and the resonance enhancement of the eigenfunction overlap by tails. Both phenomena are studied quantitatively within the Truncated Critical Random Matrix model which is suggested as a universal tool to describe the localized eigenfunctions with a multifractal texture. We also show that the sub-critical Power-Law Banded Random Matrix Ensemble suggested in Ref.MFSeil () describes the multifractal metal reasonably well. From the analytical solution for this RMT we conclude that a critical dimensionality may exist above which the -dimensional Anderson model has an unusual metal phase characterized by an effectively short-range correlation function with .

Acknowledgement.– The authors are grateful to B.L.Altshuler, M.V.Feigel’man, A.Silva and V.I.Yudson for stimulating discussions and especially to O.Yevtushenko for a collaboration at an earlier stage of this work and for a help in preparing figures. E.C. thanks the FEDER and the Spanish DGI for financial support through Project No. FIS2004-03117.


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