Physica E 9 (2001) 548–553www.elsevier.nl/locate/physe

Eigenvalue correlations for bandedmatrices

Pragya Shukla ∗

Department of Physics, Indian Institute of Technology, Kharagpur, India

Abstract

We study the evolution of the distribution of eigenvalues of a N × N matrix ensemble subject to a change of the distributionparameters of its matrix elements. Our results indicate that the evolution of the probability density is governed by a Fokker–Planck equation similar to the one governing the time evolution of the particle distribution in Wigner–Dyson gas, with afunction of distribution parameters now playing the role of time. This equivalence alongwith the already known particledensity correlations of Wigner–Dyson gas can therefore help us to obtain the eigenvalue correlations for various physicallysigni�cant cases modeled by random banded and sparse Gaussian ensembles. ? 2001 Elsevier Science B.V. All rightsreserved.

PACS: 05.45+b; 03.65 sq; 05.40+j

Keywords: Random matrix theory; Localization

It is nowwell established that the statistics of energylevels of complex quantum systems e.g. chaotic sys-tems, disordered systems can be well-modeled by theensembles of large random matrices [1]. The natureof underlying quantum dynamics of these systems di-vides the applicable random matrix ensembles (RME)into two major classes. The ensembles of the full ran-dom matrices (RFME) [2] have been very successfulin modeling the statistical properties of systems withdelocalized quantum dynamics. On the other hand,presence of the localized quantum dynamics requiresa consideration of the Band and Sparse random matrixensembles [1]. The former (RBME) can generally be

∗ Tel.: 91-3222-83850 (O�ce), 91-3222-83851 (Home); fax:91-3222-55303 (also 55239).E-mail address: shukla@phy.iitkgp.ernet.in (P. Shukla).

described as a N × N matrix with non-zero elementse�ectively within a band of the width b¿ 1 aroundthe main diagonal while the latter (RSME) consists ofnon-zero elements not con�ned only to the band. Ouraim in this paper is to suggest a general method forthe statistical studies of RBME and RSME.Recent research on complex systems has indicated

the presence of various RBM-type structures in manyphysical contexts such as nuclei, atoms [3], solidstate [4], quantum chaos [5]. For example, RBME,with zero mean value of all the matrix elements andthe variance 〈|Hij|2〉˙ a(|i − j|), can serve as goodmodel for the spectral statistics of the tight bindingHamiltonian of a quantum particle in a 1-D sys-tem with long range random hoppings [6] and alsofor quasi 1-D disordered wires [7]. Here the func-tion a(r) decays with r either as a power law or

1386-9477/01/$ - see front matter ? 2001 Elsevier Science B.V. All rights reserved.PII: S 1386 -9477(00)00261 -7

P. Shukla / Physica E 9 (2001) 548–553 549

exponentially (or faster) at r/1. Another example isthat of WRBM (W stands for Wigner) [8,9] wherethe mean value of diagonal elements increases lin-early along the main diagonal (〈Hnn〉= �n). Thesematrices have been shown to be a good model, forexample, for the tight-binding Hamiltonian of a quan-tum particle in a 1-D disordered system subject to aconstant electric �eld as well as for heavy atoms andnuclei [4]. The ensemble of banded matrices withdiagonal elements uctuating much stronger thanthe o�-diagonal ones (〈|Hii|2〉=〈|Hij|2〉˙ b/1), alsoknown as RBM with preferential basis, are quite use-ful for physical understanding of the two interactingparticles propagating in a quenched random potential[10]. The RBME with correlated diagonals can modelwell the level-statistics in a single Landau level ofQuantum Hall systems with disorder [11].Various studies of the eigenfunctions of complex

systems have revealed the existence of various typesof localizations, intermediate between fully localizedeigenfunctions (occurs only when system size L→∞) and fully extended eigenstates (implying that thelocalization length¿ the system size). For example,the eigenfunctions, in a weakly disordered potentialexhibiting the Anderson metal–insulator transition[12], are essentially structureless and extended andthe statistics here can be well modeled by FRME[13]. In the insulator regime, a complete localizationof eigenfunctions leads to a zero correlation amongthem and a Poisson statistics for energy levels. How-ever the eigenfunctions in the critical region nearAnderson transition reveal a special feature “mul-tifractality” in their structure [13,14]. In disorderedsystems, the localization → delocalization (L→ D)transition can be brought about by various ways. Forexample, a Quantum Hall system without disorder hasall the states degenerate within a single Landau level.The introduction of disorder leads to a broadening ofthe levels (also termed as diagonal disorder) as wellas random hopping between them (o�-diaonal disor-der) and a competition between the two leads to aL→ D transition. On the other hand, in the Andersonmodel, the L→ D transition is caused by the com-petition between diagonal disorder and non-randomhopping (bandwidth) [12].The nature of a localized dynamics leaves its im-

prints on the distribution properties of the matrix ele-ments of the generator of motion e.g. Hamiltonian. It

is therefore desirable to study RME with various dis-tributions and recently a lot of e�ort has been appliedin this direction. However the techniques suggested sofar [6,4,15] depend strongly on the type of distribu-tion chosen and mostly give approximate results. Ouraim in this paper is to suggest a method which leadsto results in a generic form valid for most of the Gaus-sian RM models. This we achieve by mapping theproblem to the study of a particular class of transitionensembles, the one arising during the Poisson→ stan-dard Gaussian transitions under a perturbation takenfrom the standard Gaussian ensembles (SGE). Thepre-existing information regarding the correlations inthe latter case therefore can help us to obtain the samefor RBME and RSME.We proceed as follows. Our interest is in the evo-

lution of the eigenvalue distribution of H , taken froman ensemble of hermitian matrices (real symmetric),due to variation of the variances of its matrix ele-ments. We choose the distribution �(H) of matrixH tobe a Gaussian, �(H; y; b) = C exp(−∑k6l �kl(Hkl −bkl)2) with C as the normalization constant and y asthe set of the coe�cients ykl = �klgkl with gkl = 1 +�kl. Let P(�; y; b |H0) be the probability of �ndingeigenvalues �i ofH between �i and �i + d�i at a giveny and b (with H0 as an initial condition),

P(�; y; b |H0) =∫

N∏i=1�(�i − �i)�(H; y; b) dH: (1)

As the �-dependence of P in Eq. (1) enters onlythrough �(H) and

9�9�kl

= [(2�kl)−1 − (Hkl − bkl)2]�

= (2�kl)−1[�+ (Hkl − bkl) 9�9Hkl

]

with9�9Hkl

=− 9�9bkl

a derivative of P with respect to �kl can be written asfollows:

9P9�kl

=P2�kl

+12�kl

∫N∏i=1�(�i − �i)Hkl 9�9Hkl dH

+12�kl

∫N∏i=1�(�i − �i)bkl 9�9bkl dH: (2)

550 P. Shukla / Physica E 9 (2001) 548–553

The second integral in Eq. (2) is equal tobkl(9P=9bkl). The integration by parts of the �rstintegral, followed by a use of the equality

9∏Ni=1 �(�i − �i)9Hkl

=−N∑n=1

9∏Ni=1 �(�i − �i)9�n

9�n9Hkl

;

then gives 2�kl(9P(�; y)=9�kl) = Ikl + bkl(9P=9bkl),where

Ikl =N∑n=1

99�n

∫N∏i=1�(�i − �i) 9�n9Hkl

Hkl� dH: (3)

Our aim is to �nd a function Y of the coe�cients�kl’s and bkl’s such that the evolution of P(�; Y ) interms of Y satis�es a F–P equation similar to thatof Dyson’s Brownian motion model (Wigner–Dysongas) [2]. For this purpose, we consider the sum2∑

k6l( − gkl�kl)�kl(9P=9�kl) which can be rewrit-ten as follows:

∑k6l( − ykl)

[2ykl

9P9ykl

− bkl 9P9bkl

]=

∑k6lIkl −

∑k6lyklIkl: (4)

Here is chosen to be the minimum value taken byany of the ykl’s (for reasons to be explained latter).As �n =

∑i; j OinHijOjn = 2

∑i6j OinHijOjn=gij

(eigenvalue equation, O being the orthogonal, eigen-vector matrix), one can obtain the rate of change of�n with respect to Hkl: 9�n=9Hkl = (2=gkl)OknOln.Using this relation, the �rst term on the right-handside (Eq. (4)) can further be simpli�ed,

∑k6l Ikl =∑

n 9=9�n(�nP) (equality A). The second term cansimilarly be rewritten as follows:

∑k6lyklIkl =−∑

k6l

N∑n=1

9Fkl9�n

+∑k6lJkl (5)

where

Fkl =∫

N∏i=1�(�i − �i)OknOln 9�9Hkl dH (6)

=∫ 99Hkl

[N∏i=1�(�i − �i)OknOln

]� dH: (7)

Here Jkl = yklbkl∑N

n=1 9=9�n∫ ∏N

i=1 �(�i − �i)× (9�n=9Hkl)� dH and an integration by partscan be used to show that Jkl =−yklbkl(9P=9bkl).

Eq. (5) can further be reduced by di�erenti-ating the terms inside the brackets, giving ustwo integrals along with the term

∑k6l Jkl. The

use of orthogonality relation of matrix O inthe �rst integral so obtained and the relation∑

k6l 9OnkOnl=9Hkl =∑

m 6=n 1=(�n − �m) in the sec-ond [16] gives

∑k6l yklIkl =−∑n 9=9�n[9=9�n +∑

m 6=n �=(�m − �n)]P −∑k6l yklbkl(9P=9bkl) (equal-ity B). Using both the equalities A and B in Eq. (4)we obtain the desired F–P equation

9P9Y =

∑n

99�n

(�nP)

+∑n

99�n

[99�n

+∑m 6=n

��m − �n

]P; (8)

where � = 1 and the left-hand side of above equation,summing over all ykl and bkl, has been rewritten as9P=9Y with Y given by the condition that 9=9Y =2∑

k6l ykl( − ykl)9=9ykl −∑

k6l bkl(9=9bkl) (con-dition C). By using the unitarity of eigenvectors andfollowing the same steps, it can be proved for com-plex Hermitian case too (now � = 2) (see [17, fordetails]).The de�nition of Y depends on the relative val-

ues of the coe�cients �kl’s and bkl’s and can beobtained as follows. We de�ne M = N (N + 1)variables (Y; Y1; : : : ; YM−1) as the functions of allykl’s and bkl’s such that the condition C is satis-�ed. This is indeed possible by using the orthog-onal (Jacobi) coordinate transformation betweenvariables {Y; Yi}i=1; :::;M−1 and {Xi}i=1; :::;M given byY = 1=M

∑Mj=1 Xj and Yi =

∑Mj=1 aijXj for i = 1→

M − 1 with ∑Mj=1 aij = 0, Xj =

12 lnyj=|yj − |+ cj

for j6N (N + 1)=2 and Xj =−(1= ) ln bkl + cj forj¿N (N + 1)=2 with cj as arbitrary constants of in-tegration. The form of Y , ful�lling condition C, cantherefore be given as

Y =1

N (N + 1)∑k6l

[12ln

ykl|ykl − | −

1 ln bkl

]+ C

with C =∑jcj:

The steady state of Eq. (6), P(�;∞) ≡ P∞ =∏i¡j |�i − �j|�e− =2

∑k�2k , is achieved for Y − Y0 →

∞ which corresponds to almost all ykl → (for �nitebkl values). This indicates that, in the steady-statelimit, system tends to belong to the standard

P. Shukla / Physica E 9 (2001) 548–553 551

Gaussian ensembles. Eq. (6) (later referred asvariance–variation or VV case) is formally the sameas the F–P equation governing the Brownian mo-tion of particles in Wigner–Dyon gas [2] with thetransition parameter being Y in the former and timein the latter. This is also similar to the F–P equa-tion for the eigenvalue distribution of a hermitianmatrix H = H0 +

√YV undergoing random pertur-

bation V of strength Y 1=2 and taken from a SGE

(〈V 2ii 〉= 2〈V 2ij〉= −1 with arbitrary initial conditionH0 (later referred as perturbation variation or PVcase [18,19], also known as Rosenzweig Porter orRP model); here Y acts as the transition parameter.P(�; Y ) can therefore be obtained by the same pro-cedure as used in PV case which is brie y given as

follows. The transformation = P=√P∞ reduces

Eq. (6) to the form 9=9Y = H where the “Hamil-tonian” H turns out to be a Calogero–Moser (CM)Hamiltonian, H =

∑i 92=9�2i − 14

∑i¡j �(� − 2)=

(�i − �j)2 − 14

∑i �2i , and has well-de�ned eigen-

states and eigenvalues [20–24]. The “state” orP(�; Y |H0) can therefore be expressed as a sumover its eigenvalues and eigenfunctions which onintegration over all the initial conditions H0 leads tothe joint probability distribution P(�; Y ) and therebyeigenvalue correlations. Unfortunately, due to thetechnical problems, the latter could be evaluatedonly for the transitions with �nal steady state (limit�→ ∞) as GUE [23,24]. However a set of hier-archic relations among the correlators for all tran-sitions in PV case, and therefore for VV case, canbe obtained by a direct integration of the F–P equa-tion for P (Eq. (6)) [23,24]. For example, in largeN -limit, the evolution of the �rst-order correlatorR1(�; Y ) =

∫ ∏Ni=2 d�iP(�; �2; : : : ; �n) is governed by

the Dyson–Pastur equation [23,24] which results inits semi-circular form (see Eq. (5) of Ref. [25]):

R1(�) = 2N=�a2[a2 − �2]1=2, a2 = 4N −1(1 + Y ).For localization studies, it is appropriate to choose

the initial ensemble H0 as that of diagonal matrices(P(H0)˙ e−

∑i[H0]

2ii) which corresponds to Poisson

distribution for the eigenvalues and fully localizedeigenstates. The equilibrium distribution (Y − Y0 →∞) is given by SGE and corresponds to completedelocalization of eigenstates. The localization →delocalization transition can therefore be mappedonto a Poisson → SG transition, of the PV type,

with Y as a transition parameter and the interme-diate states of localization (represented by variousRMEs depending on the values of ykl’s and bkl’s)mapped to the “Brownian” ensembles (intermediateensembles in the PV case) corresponding to variousY values. The eigenvalue correlations for variouslocalization cases will therefore be similar to thosefor the Brownian ensembles appearing in the Pois-son → SGE transition of PV type. We already knowthat the transition for PV case is abrupt for large di-mensions and �nite Y and a rescaling of Y by meanspacing D(D = 1=R1) is required to make it smooth,the new transition parameter being �= Y= D2. Asimilar rescaling should also be applied to Y in VVcase for large systems. For example, for an ensem-ble with the exponential decay of variances awayfrom the diagonal i.e �kl = e|k−l|=b; k6l; 1.b.N ,the parameter Y =−1=(2N (N + 1))∑N

k6l ln|1− y−1kl |+ C. Here if H0 is chosen an ensemble ofdiagonal matrices with P(H0)˙ e−Tr H

20 , this gives

Y0 =−(1=2N (N + 1))∑Nk6l ln|1− y−1kl [H0]|+ C

(with ykl[H0]→ ∞ for k 6= l and ykk [H0]→ 2).As =min{Ykl}= 2, we get Y − Y0 =−1=2N (N +1)∑N

r=1(N − r)ln|1− 2e−r=b| ≈ b=N and R1(�) =[2N (1 + Y )− �2]1=2=4�(1 + Y ) ≈ √

N [28]. Thetransition parameter � for large systems therefore,is �= (Y − Y0)=D2 ≈ b which recon�rms that, inlarge systems, the transition is governed only bythe localization length b2. (Note that Y = b=N inthis case governs the transition of levels with leveldensity R1 ≈

√N and therefore di�ers from the

transition parameter b2=N obtained by SUSY tech-nique [26] where R1 ≈ N=

√b. However a proper

unfolding of the transition parameter by the leveldensity in each case gives the same transition pa-rameter, namely b, for unfolded levels. Another caseof importance is the ensemble with power-law de-cay of variances Hij = Gija(|i − j|) with G a typicalmember of SGE (〈G2ii〉= 2〈G2ij〉= v2) and a(r) = 1and (b=r)� for r6b and r ¿b (b/1), respectively(known as PRBM model with P stands for power)[6]. This correspond to yij = 1=(v2a2(|i − j|)) andtherefore =min{yij}= 1=v2. Again as for the ex-ponential case, the choice of initial and the �nal en-semble remains the same. Now as yij ≡ yr = (r=b)2�(with r = |i − j|), we get �= D−2(Y − Y0) =−(1=4N )∑N

r=b+1(N − r)ln(1− (b=r)2�). Thus, for

552 P. Shukla / Physica E 9 (2001) 548–553

large N -values and �¡ 1=2, �(˙ N 1−2�) is su�-ciently large and the eigenvalue statistics approchesSG limit. At � = 1

2 , the statistics is governed bythe parameter b2=N instead of N only. For � = 1,the non-zero, �nite �-value (�˙ b even whenN → ∞) leads to an eigenvalue statistics intermedi-ate between that of SGE or Poisson. For �¿ 3

2 withN → ∞; �→ 0 therefore, the eigenvalue statisticsapproches Poisson limit, � being very small. Allthese results are in agreement with those obtained inRef. [6] by SUSY technique.

Similarly the transition parameter for L→ D tran-sition in a d-dimensional Anderson model can also beobtained. The presence of just the diagonal disorderand non-random hopping between neighboring sitesin this system (of size L) can be modeled by writingthe N × N (N = Ld) Hamiltonian matrix in site ba-sis and by choosing �kl → ∞ for all k 6= l; bkl =−bfor {k; l} representing any of the neighboring sites ind-dimensions, �kk = � and bkk = 0 for all k. Now ifinitially the system is in insulating regime (P(H0)˙e−�0

∑iH

2ii ), the transition can be viewed in terms of the

parameter

�=Y − Y0D2

=14

[ln�|�0 − �M |�0|�− �M | −

KN�M

ln b]:

As obvious from the above, the transition is gov-erned by relative values of the disorder and thehopping. Here �→ 0 leads to fully localized regimewhich corresponds to following condition on � andb, ln (�=�0) + (�− �0)=�M = (K=N�M ) ln b, which is,therefore, the condition for the critical region ormobility edge (see Ref. [17] for details).

Fortunately the two-point correlation R2(r;�) forPoisson → GUE transition in PV case has alreadybeen obtained [23,24] and, as discussed above, is alsovalid for the VV case (now �= (Y − Y0)=D2).R2(r;�)− R2(r;∞)

=4�

∫ ∞

0dx∫ 1

−1dz cos(2�rx)f(z; x); (9)

where

f(z; x) = exp[− 8�2�x(1 + x + 2z√x)]

×(√

(1− z2)(1 + 2z√x)1 + x + 2z

√x

)

and R2(r;∞) = 1− sin2(�r)=�2r2 (the GUE limit).As can be easily checked, above equation has the cor-rect limiting behavior, that is, R2 = 0 for �→ 0 (thePoisson case Y = Y0) and R2 = R2(r;∞) for �→ ∞.As obvious from Eq. (7), R2 depends on Y and there-fore on the nature of localization which, for �nite�-values (in limit N → ∞), results in various typesof the level-statistics intermediate between Poissonand GUE. This result is in agreement with the earliernon-linear sigma model (NLSM) studies on RBMs,for example, the eigenvalue statistics of PRBM casefor � = 1 is already known to be an intermediate one[6]. However there seems to be some contradiction re-garding the nature of eigenvectors. The NLSM studyof PRBMs has indicated the presence of multifrac-tality among the eigenvectors. In disordered systemstoo, the eigenfunctions near mobility edge are foundto be multifractal (here again the level statistics is in-termediate between Poisson and GUE) [27]. As ourtechnique is equally well applicable to both these sys-tems, therefore, for the corresponding �, the Poission→ GUE transition in PV case (equivalently the RPmodel) should also support multifractaity. But somestudies [28,14] have claimed the absence of multifrac-tailty in RP model; their argument is based on thenumber variance�2(r)-study of the eigenvalues whichis believed to be an indicator of mutifractal nature ofthe eigenfunctions. However, as indicated by a recentstudy [15], the results of Ref. [28] are not valid ingeneral. Furthermore Section IV.B of Ref. [15] along-with the numerical analysis presented there also leavesthe impression that the �2-behavior for RP model isnot yet fully understood and therefore the questionof multifractality is not yet settled. Our results indi-cate that either the multifractality is a common featureof all the Gaussian models (including RP model andAnderson transition), making its appearance for some�nite �-value (in limit N → ∞) or it does not ex-ist in any of them. It may also be possible that the�2-behavior of eigenvalue statistics is not always anecessary indicator (although su�cient in some cases)of multifractal nature of eigenfunctions.In this paper, we have analytically studied the re-

sponse of energy levels of complex quantum systemsto the changing distribution of the matrix elementsof their hamiltonians. Our results indicate that thenth-order correlations at a given set of distribution pa-rameters are the same as the corresponding eigenvalue

P. Shukla / Physica E 9 (2001) 548–553 553

correlations of hamiltonianH = H0 +√YV with Y as

a function of various distribution parameters,H0 takenfrom a Poisson ensemble and V taken from a SG en-semble. This analogy also extends to the second-orderparametric correlators however the Markovian natureof F–P equation restricts from making similar conclu-sions about higher orders. This technique can also beextended to non-Gaussian distributions [17]. The in-termediate ensembles arising in the VV transition cor-respond to various types of RMEs, and as our mappingsuggests, the results for most of them can be obtainedby studying just one transition in full detail, namely,an appropriate initial ensemble→ SGE caused due toa SG perturbation; the knowledge of their correlationscan help us in statistical studies of many importantphysical properties of complex systems. Further, asdiscussed in Ref. [14], some of the RBMEs can alsobe connected to various other types of ensembles; aknowledge of the properties of former will thereforehelp in statistical analysis of the latter. It should alsobe possible to apply this technique to the ensemblesof Unitary matrices (periodic RBMs) too.Our study still leaves many questions unanswered.

At present, it is not clear whether our method can beextended to non-Hermitian distributions as well. Weare also unable to say whether a similar mappingcan be done for the eigenvector statistics too. In thisconnection, however, it should be mentioned thatanalogy of equations governing the eigenvalue distri-bution leads to a similarity of equations for the dis-tribution of matrix elements [2] and, therefore, itshould somehow manifest itself in eigenvector distri-butions too.

I am greatful to B.S. Shastry, B.L. Altshuler,V. Kravtsov and N. Kumar for various useful sugges-tions during the course of this study.

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