We investigated numerically the distribution of participation numbers in the
3d Anderson tight-binding model at the localization-delocalization threshold.
These numbers in {\em one} disordered system experience strong level-to-level
fluctuations in a wide energy range. The fluctuations grow substantially with
increasing size of the system. We argue that the fluctuations of the
correlation dimension, $D_2$ of the wave functions are the main reason for
this. The distribution of these correlation dimensions at the transition is
calculated. In the thermodynamic limit ($L\to \infty$) it does not depend on
the system size $L$. An interesting feature of this limiting distribution is
that it vanishes exactly at $D_{\rm 2max}=1.83$, the highest possible value of
the correlation dimension at the Anderson threshold in this model

A. Cohen

B. L. Altshuler

B. Shapiro

C. Castellani

C. M. Soukoulis

D. A. Parshin

D. Braun

F. Wegner

F. Yonezawa

H. Aoki

H. Grussbach

H. R. Schober

J. T. Chalker

M. Janssen

M. Schreiber

P. A. Lee

P. W. Anderson

S. N. Evangelou

S. Yoshino

T. Brandes

T. Ohtsuki

T. Terao

V. E. Kravtsov

V. I. Fal'ko

V. N. Prigodin

Y. V. Fyodorov

English

arXiv

Parshin, D. A.

Schober, H. R.

Juelich Shared Electronic Resources

VOLUME 83, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 29 NOVEMBER 19994590Distribution of Fractal Dimensions at the Anderson TransitionD. A. Parshin1,2 and H. R. Schober31Theoretical Physics Institute, University of Minnesota, 116 Church Street, S.E., Minneapolis, Minnesota 554552State Technical University, 195251 St. Petersburg, Russia3Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425, Germany(Received 6 July 1999)We investigated numerically the distribution of participation numbers in the 3D Anderson tight-binding model at the localization-delocalization threshold. In a single system these numbers fluctuatestrongly from level to level. The fluctuations grow with system size. We argue that this mainly resultsfrom the fluctuations of the correlation dimension D2 of the wave functions, whose distribution at thetransition is calculated. In the thermodynamic limit (L ! `) it does not depend on the system size L.This limiting distribution vanishes exactly at D2max  1.8, the highest possible value of the correlationdimension at the Anderson threshold in this model.PACS numbers: 63.50.+x, 61.43.Fs, 61.43.Hv, 73.20.JcThe localization-delocalization Anderson transition hasposed for a long time a fascinating problem. In systemswith short range interaction, purely diagonal disorder,and unbroken time-reversal symmetry, without spin-orbitinteraction, it occurs for dimensions d . 2 [1]. Inthe thermodynamic limit (i.e., infinite system size) thetransition point separates the systems where all wavefunctions are localized from the systems where somepart of them is extended. Exactly at the transitionone finds in the center of the band extended wavefunctions. Because of the proximity of the unavoidablelocalization at one side of the transition they have a self-similar fractal (actually multifractal) structure. This is adirect consequence of quantum critical fluctuations at thetransition point.This multifractality of the wave functions at the local-ization threshold is one of the most important features dis-covered [2,3] since the pioneering work of Anderson [4].This fruitful idea has became widely recognized (see, e.g.,Refs. [5–7]) and has considerably helped our understand-ing of different phenomena in mesoscopic systems relatedto electron localization. For instance, the well known log-normal distribution of the conductance in disordered met-als [8] can be taken as a fingerprint of multifractal wavefunctions which survive in the weakly disordered state(so-called prelocalized states [7]).Usually the multifractal (as well as fractal) structure ofa wave function manifests itself in the size dependence ofthe participation number (PN)N µZjcrj4 dr∂21~ LD2 , (1)where L is the system size and D2 , d is the correlationdimension of the wave function cr. For a localizedstate D2  0, and N does not depend on L. Onthe other hand D2  d for a delocalized wave functionwhich extends uniformly over the sample. The inequality0 , D2 , d means that a multifractal wave function isdelocalized but, in the thermodynamic limit, neverthelessoccupies only an infinitesimal fraction of the sample.0031-90079983(22)4590(4)$15.00Because of strong level-to-level fluctuations [9] it isvery hard to verify the size dependence of N for aparticular state in a computer experiment. At the tran-sition these fluctuations increase with increasing systemsize. Therefore, one has to study the size dependence ofthe averages of N or, preferably, the distribution func-tion. The size dependence of the fluctuations of N isthen converted to the size dependence of this distribu-tion function. Choosing suitable size dependent variablesone can collapse these distributions (for different systemsizes) to one universal curve. However, to do this sys-tematically one needs to understand the origin of thesefluctuations. In our opinion the main source of the abovementioned giant fluctuations of PN at the transition arethe fluctuations of the fractal dimension D2 of the wavefunctions.In a disordered system, exactly at the transition [10],critical wave functions with very different degrees of de-localization (participation numbers) should coexist inde-pendent of their energy. This is a direct consequence ofan infinitesimal neighborhood of the localized behaviorat one side of the transition and delocalized behavior atanother side. Since the critical energy region, where thecorrelation length Lc ¿ L, shrinks more slowly with Lthan the number of states increases there will, in the ther-modynamic limit, be an infinite number of critical states[11]. Accordingly each delocalized wave function shouldhave a different D2. If the distribution function P D2becomes size independent in the thermodynamic limit ourhypothesis will be correct.In the present paper we present numerical resultsfor this distribution (more exactly for the distributionof logarithms of the PN) and show that it is indeeduniversal, i.e., it does not depend on the system size inthe thermodynamic limit. This is somewhat reminiscentof the previous idea of Shapiro [12,13] about the existenceof universal distributions at the Anderson transition. Thisproblem has recently been addressed analytically in acouple of papers, but far from the transition point.© 1999 The American Physical SocietyVOLUME 83, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 29 NOVEMBER 1999In Ref. [14] the variance of the inverse participationratio (IPR) was calculated for disordered metallic samplesin leading order of the small parameter 1g2, where g ¿1 is the dimensionless conductance of the system. It wassuggested that the relative value of the IPR fluctuationsshould be of order unity at the transition point and thattheir distribution should be universal. A similar questionwas raised recently in Ref. [15] where the distribution ofthe IPR was calculated for a large but finite conductanceg of small metallic grains.To investigate this problem at the transition we numeri-cally solve the standard Anderson tight-binding modelwith diagonal disorder on a 3D simple cubic lattice withthe HamiltonianH Xi´iji ij 1Xifijtijji  jj . (2)To model a disorder we distribute the site energies´i uniformly in the interval 2W2 , ´i , W2. Forthe off-diagonal elements we took tij  1 for nearestneighbors and otherwise zero. The Anderson transitionis in this model at a critical value, Wc  16.5 (see, e.g.,Ref. [16], and references therein).Diagonalizing the matrix Hij for a cubic sample withN  L3 sites and open boundary conditions we find a setof N orthonormal eigenvectors es j and correspondingeigenvalues, Ej . The participation number for a state j isdefined as usual byNj √NXs1e4s  j!21. (3)Figure 1 shows the participation numbers Nj versusenergy Ej for a 3D system with N  8000 sites. Atthe transition there is a strong level-to-level fluctuationof the PN. In the whole energy range states are foundwhich are very close in energy but whose Nj differby up to 2 orders of magnitude. This means that theenergy of a particular state is not indicative of the spatialbehavior of the wave function. In a next step we,therefore, calculate the distribution function of the PN inan energy band around zero energy (where the Andersontransition takes place). We take the width of this bandto equal 10; i.e., we include all states with jEjj , 5.The distribution remains practically the same if we take amuch narrower strip, jEjj , 1, but the computation timeincreases significantly.The normalized distribution functions of the participa-tion numbers for different system sizes are shown in Fig. 2.As expected the distributions are strongly size dependent.Increasing the system size the position of the maximumshifts to higher values, approximately as ~N0.3, and theamplitude of the maximum decreases as ~N20.46.Let us suppose now that the fluctuation of the PN atthe transition is due to fluctuations of D2 of the wavefunctions. According to Eq. (1), without loss of gener-ality, we can take this relation in the form N ~ ND23.Then if FN N  is the normalized distribution function ofPN for an ensemble of systems with N sites and differentFIG. 1. Participation numbers versus energy for a 3D systemof 20 3 20 3 20  8000 sites at the Anderson transition.disorder, we can extract the distribution function of thecorrelation dimensions, D2, in this ensemble,PN D2  13FN ND23ND23 lnN . (4)Figure 3 shows that this distribution of correlation dimen-sions is much less sensitive to the system size than theone of the PN. Moreover, with increasing N it obviouslyapproaches some size independent function, P`D2; i.e.,in the thermodynamic limit a universal distribution func-tion of correlation dimensions does exist at the Andersontransition. For a given model of disorder it should be thesame for different realizations. In other words we believethat the distribution function P`D2 is a self-averagedquantity [17] and can be obtained by analyzing one suffi-ciently large system.Figure 3 shows an additional interesting feature ofthis distribution: with increasing argument the functionPN D2 rapidly decays to zero. The drop gets steeperwith increasing size N . This is seen more clearlyin a logarithmic plot of PN D2. For systems withFIG. 2. Distribution functions of participation numbers at theAnderson transition for different system sizes. In the bracketsthe numbers of realizations used in the averaging are shown.4591VOLUME 83, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 29 NOVEMBER 1999FIG. 3. Distribution functions of the correlation dimensionPN D2 at the Anderson transition for different system sizes.N $ 512, PN D2 drops 4 orders of magnitude in thesmall interval 1.5 , D2 , 2. We think, therefore, that inthe thermodynamic limit there is a point, D2max, where thefunction P`D2 approaches exactly zero. In such a casethis would be the highest possible value of the correlationdimension D2 at the Anderson transition. In the followingwe present additional evidence in support of this idea.For that purpose we calculate the size dependence ofthe average PN and of the PN of the most extended statein the system, i.e., for a given realization of disorderthe state with maximal participation number, Nmax. Theensemble averages of both quantities (denoted by aapnand ampn, respectively) against system size N are plottedin Fig. 4. First, both quantities scale with N as apower law, aND23. Second, the value D2  1.26 for theaverage participation number (aapn) is very close to theposition of the maximum in the distribution function ofcorrelation dimensions shown in Fig. 3. It is surprisingthat averaging over all states does not affect the powerlaw behavior given by Eq. (1) for a single state. For theensemble average of the maximum participation number,Nmax, we find D2  1.8. This is the correlationdimension of the most extended state in the system at thetransition. To answer the question whether it really is themaximal correlation dimension, D2max, we should studythe fluctuations of this quantity. A maximal correlationdimension D2max does exist if the fluctuation of thisquantity goes to zero with N ! `.Figure 5 shows the relative fluctuations of the maxi-mum participation number versus system size. First, thefluctuations are rather small and decrease slowly with in-creasing N . According to Eq. (1) we can relate them tothe fluctuations of the correlation dimension dD2max,dmpn  dNmaxNmax  dD2max3 lnN . (5)We conclude from the above that, in the thermodynamiclimit (N ! `), the fluctuation of the correlation dimen-sion for the most extended state dD2max ! 0 at least notslower than 1 lnN . Therefore D2max  1.8 is indeed4592FIG. 4. Averaged maximum participation number (ampn) andaveraged average participation number (aapn) at the transitionversus system size. Lines are the best least squares fit withN   aND23, where N  is “ampn” or “aapn,” respectively.the highest possible value of the wave function corre-lation dimension at the Anderson transition. The distri-bution function P`D2 should approach exactly zero atthis value.There are numerous previous calculations of thecorrelation dimension D2 at the 3D Anderson transi-tion. From the scaling with N of the density-densitycorrelation of the wave functions, the correlation di-mension was estimated to be D2  1.7 6 0.3 (forW  16) [18]. Scaling with size of the participationnumbers has been investigated in Ref. [19]. Averagingboth, over different wave functions in a small energyinterval 0.25 around zero and over disorder (with aGaussian distribution of site energies), D2  1.6 6 0.1was found [20]. In Ref. [21] from the spectral com-pressibility, x , and using x  1 2 D2d2, derivedin Ref. [22], D2  1.4 6 0.2 was derived. FromFIG. 5. Relative root mean square deviation of the maximumparticipation number against system size. The solid fitting linecorresponds to a power law behavior, ~1N0.14.VOLUME 83, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 29 NOVEMBER 1999the decay of the temporal autocorrelation function,Ct ~ t2D23, a value D2  1.5 6 0.2 was extracted[23]. Using box-counting procedures D2  1.7 6 0.2[24], D2  1.52 6 0.11 [25], and D2  1.68 [26] wascalculated. Again using box counting and averaging theresults over wave functions in a small energy interval,DE  0.01, around zero and over five different realiza-tions of disorder D2  1.46 was found [16]. Again usingbox-counting techniques in Ref. [27], D2  1.33 6 0.02was obtained.This shows clearly the quite large uncertainty, in theliterature, in the existing values of D2 at the transition.The reason is obviously that one deals with a distributionof this quantity. Among different characteristics of thisdistribution one can consider, for example, the position ofthe maximum, at about 1.3, the average value, D2  1.26,and the correlation dimension of the most extended stateat the transition, D2max  1.8.Similarly the distribution function of the informationdimension D1 as well as of the other generalized dimen-sions Dq at the transition can be obtained. The results ofthis analysis will be published elsewhere. In the thermo-dynamic limit all these distribution functions are expectedto be universal and self-averaged quantities. Using theLegendre transformation the multifractal spectrum fjafor each wave function with energy Ej can be calculated(compare Ref. [28]). In other words critical wave func-tions at the transition should be characterized by a wholedistribution of fa’s which is already a functional. Inparticular, the positions of the maxima, a0’s of these func-tions should be distributed.In conclusion, we investigated numerically the distribu-tion of the correlation fractal dimension D2 for the criticalwave functions at the 3D Anderson transition. This dis-tribution appears to be universal; i.e., it no longer dependson the system size in the thermodynamic limit. Extrapo-lating these results to other fractal dimensions we con-clude that each critical wave function should possess itsown (infinite) set of generalized fractal dimensions, Dd ,at the transition. And one should rather speak about thedistribution functional of these functions.The authors are very grateful to B. I. Shklovskii forfruitful discussions. One of us (D. A. P.) gratefullyacknowledges the financial support and hospitality ofthe University of Minnesota where part of this workwas done.[1] P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57,287 (1985).[2] F. Wegner, Z. Phys. B 36, 209 (1980).[3] H. Aoki, J. Phys. C 16, L205 (1983); Phys. Rev. B 33,7310 (1986).[4] P. W. Anderson, Phys. Rev. B 109, 1492 (1958).[5] C. Castellani and L. Peliti, J. Phys. A 19, L429(1986).[6] M. Janssen, Int. J. Mod. Phys. B 8, 943 (1994).[7] V. I. Fal’ko and K. B. Efetov, Phys. Rev. B 52, 17 413(1995).[8] B. L. Altshuler, V. E. Kravtsov, and I. V. Lerner, Phys.Lett. A 134, 488 (1989).[9] S. Yoshino and M. Okazaki, J. Phys. Soc. Jpn. 43, 415(1977); F. Yonezawa, J. Non-Cryst. Solids 35&36, 29(1980).[10] In fact, it is sufficient that the correlation length Lc wasmuch bigger than the sample size L.[11] V. E. Kravtsov, in Correlated Fermions and Transport inMesoscopic Systems, edited by T. Martin, G. Montam-baux, and J. Tran Tanh Van (Editions Frontieres, Gif-sur-Yvette, 1996).[12] B. Shapiro, Phys. Rev. B 34, 4394 (1986); Philos. Mag. B56, 1031 (1987); Phys. Rev. Lett. 65, 1510 (1990).[13] A. Cohen and B. Shapiro, Int. J. Mod. Phys. 6, 1243(1992).[14] Y. V. Fyodorov and A. D. Mirlin, Phys. Rev. B 51, 13 403(1995).[15] V. N. Prigodin and B. L. Altshuler, Phys. Rev. Lett. 80,1944 (1998).[16] H. Grussbach and M. Schreiber, Phys. Rev. B 51, 663(1995).[17] To calculate the distribution functions PN D2 forthe rather small sizes shown in Fig. 1 we had, ofcourse, to average over many different realizations ofdisorder.[18] C. M. Soukoulis and E. N. Economou, Phys. Rev. Lett. 52,565 (1984).[19] M. Schreiber, Phys. Rev. B 31, 6146 (1985).[20] M. Schreiber, Physica (Amsterdam) 167A, 188(1990).[21] D. Braun, G. Montambaux, and M. Pascaud, Phys. Rev.Lett. 81, 1062 (1998).[22] J. T. Chalker, V. E. Kravtsov, and I. V. Lerner, JETP Lett.64, 386 (1996).[23] T. Ohtsuki and T. Kawarabayashi, J. Phys. Soc. Jpn. 66,314 (1997).[24] T. Brandes, B. Huckestein, and L. Schwetzer, Ann. Phys.(Leipzig) 5, 633 (1996).[25] T. Terao, Phys. Rev. B 56, 975 (1997).[26] M. Schreiber and H. Grussbach, Phys. Rev. Lett. 67, 607(1991).[27] S. N. Evangelou, Physica (Amsterdam) 167A, 199(1990).[28] H. Grussbach and M. Schreiber, Phys. Rev. B 48, 6650(1993).4593

Distributions of fractal dimensions at the Anderson transition

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Distribution of Fractal Dimensions at the Anderson Transition

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Distribution of fractal dimensions at the Anderson transition

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