Nonuniversality in level dynamics

PHYSICAL REVIEW E MARCH 1997VOLUME 55, NUMBER 3

Nonuniversality in level dynamics

Paweł Kunstman, Karol Z˙ yczkowski, and Jakub ZakrzewskiInstytut Fizyki Mariana Smoluchowskiego, Uniwersytet Jagiellon´ski, ulica Reymonta 4, 30-059 Krako´w, Poland

~Received 7 October 1996!

Statistical properties of parametric motion in ensembles of Hermitian banded random matrices are studied.We analyze the distribution of level velocities and level curvatures as well as their correlation functions in thecrossover regime between three universality classes. It is shown that the statistical properties of level dynamicsare in generalnonuniversaland strongly depend on the way in which the parametric dynamics is introduced.@S1063-651X~97!04803-4#

PACS number~s!: 05.45.1b, 95.10.Fh, 47.52.1j

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I. INTRODUCTION

A link between random matrix theory~RMT! @1# and thestatistical properties of spectra of quantum systems is westablished. Depending on the symmetry of a classically cotic quantum system, its spectral fluctuations are descrby the Gaussian orthogonal~GOE!, the Gaussian unitary~GUE!, or the Gaussian symplectic ensemble~GSE! @2,3#.

Quite often the physical systems depend on some exteparameter, say,l; therefore, it is interesting to study thlevel dynamics, i.e., the motion of eigenvaluesEi(l) as afunction ofl. Among the first parametric properties studiwere the investigations of the avoided crossings gaps@4–6#,the parametric number variance@7#, or the curvature of thelevels ~i.e., the second derivatives of their energies withspect to the parameter! @8–13#. It has been claimed that thstatistical properties of level dynamics are universal for dordered or strongly chaotic systems@8,14# provided thechange ofl does not modify global symmetries propertieTo reveal the universality one has to both unfold the enelevels @3# and appropriately rescale the parameterl@8,11,14#. Other statistical measures of parametric dynamsuch as the level slope~velocity! distribution ~Gaussianshaped for random systems@8,11,14#!, the velocity-velocitycorrelation function@14–17# in the bound spectrum, or parametric conductance fluctuations@18# and fluctuations in theWigner time delay@19# for scattering systems have also bediscussed.

A word of caution is, however, necessary at this poiEven for the nearest-neighbor spacing distribution, widconsidered to be universal, exceptions from the RMT predtion may be quite significant for real physical systems@20#.Much more pronounced and common are the deviations fthe RMT predictions for the parametric motion of levels.particular, as shown by Takami and Hasegawa@10#, the cur-vature distribution shows nonuniversal behavior for smcurvatures even for the mixing system~a Bunimovich sta-dium!. Similarly, non-Gaussian slope distributions as wellstrongly non-Cauchy-like curvature distributions were oserved for a magnetized hydrogen atom@11#. The origin ofthese deviations has been linked to partial wave-functioncalization on unstable periodic orbits. Thus the nongenfeatures of parametric statistics may provide most interesinformation about the physics of a given physical system

Most of these studies considered pure symmetry ca

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i.e., systems pertaining to a given, e.g., GOE universaclass. This is often not the case in a realistic situation.particular, in the presence of the magnetic field or tAharonov-Bohm flux, the time-reversal invariance~TRI!symmetry becomes broken; such a situation correspondscrossover between the GOE and the GUE for a randomtem. In this context the velocity-velocity correlation functiohas been studied intensively@21,22,17# as well as the veloc-ity distribution@23# or the curvature distribution@24,25#. Theauthors considered mostly the situation when the increasthe external parameterl ~e.g., the magnetic field! destroysthe time-reversal invariance, although, importantly, it hbeen noticed@23# that the parametric velocity distributiomay strongly depend on the nature of the perturbation. Rtively less frequent were studies of the parametric dynamin the transition region between completely delocalized alocalized spectra~see, however, e.g., the treatment of tvelocity distribution for the broken-TRI-symmetry case@26,27#!.

In order to model the spectra of quantum systems icrossover regime~a weak localization or a partially brokesymmetry! one may utilize random matrix ensembles thinterpolate between the canonical ensembles. For examreal symmetric band random matrices are capable of moing the transition between the localized and the delocaliregime. Statistical properties of their eigenvalues and eigvectors depend on a single scaling parameterx5b2/N @28#,whereN denotes the matrix size andb the bandwidth. Al-lowing the matrices to be Hermitian and changing the retive weight of the imaginary componenta, one can modelthe effect of the time-reversal symmetry breaking andtransition from an orthogonal to a unitary universality claThe corresponding scaling parametery is proportional toNa @29# for small perturbations. An ensemble of Hermitiaband random matrices~HBRMs! can therefore be completelcharacterized by two scaling parameters (x,y) @30#. ThePoissonian, strongly localized spectrum is obtained inlimit x!1, while in the opposite delocalized limitx@1 themodel reduces to the GOE fory50 and the GUE fory@1.

This work is intended as a systematic study of the pametric dynamics and the corresponding statistical measin the transition region and in the localized regime. Our wodiffers from most of the analyses mentioned above inway the parametric dependence is introduced. We ass

2446 © 1997 The American Physical Society

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55 2447NONUNIVERSALITY IN LEVEL DYNAMICS

that the changes ofl leave the global properties of the sytem unaffected. In the random matrix approach thisequivalent to the assumption that the statistical propertiethe ensemble of matrices do not depend on the value ofparameterl determining the parametric dynamics. Fphysical system applications this is equivalent to sayingthe symmetry properties of the system considered, in ation to the character of the underlying classical dynam~say, the fraction of the phase-space volume that is chao!,are invariant with respect tol or, from a practical point ofview, the the dynamics changes only weakly withl, in theinterval of l values considered in each case. The levelnamics is, in a sense, ‘‘perpendicular’’ to crossovers betwcanonical ensembles, as schematically shown in Fig. 1. Sa physical situation may correspond to a variation ofdisorder parameter in a mesoscopic system, for whichother parameters are kept constant. Special attention is dto the localized case, characterized by small values oxwhere some analytic predictions obtained using supersmetric calculus exist@26,27#.

The interest in such a study is twofold. First, it is inteesting to see how the parametric properties of the sysfollow the transition between different pure universalclasses. Second, the results obtained in the HBRM momay serve in the future as a reference for a comparisonstatistics obtained in real physical systems. It is then ofmost importance to know what one may expect frompurely random model. This may enable one to isolatenongeneric, i.e., characteristic for a given system, proper

The paper is organized as follows. In the next sectiondescribe the model and the parametric dynamics. The dibution of level velocitiesP(v) is analyzed in Sec. III. Levecurvatures are discussed in Sec. IV. Section V is devotethe velocity-velocity correlation functionCv(l). Section VIconsiders higher-order statistical measures. Finally, wecuss the consequences of the results obtained for varstatistical measures in Sec. VII.

II. PARAMETRIC DYNAMICS FOR HERMITIAN BANDRANDOM MATRICES

Hermitian band random matrices are defined by

Hi j5~j i jR1 i j i j

I !Q~b2u i2 j u!, i , j51, . . . ,N, ~2.1!

FIG. 1. Scheme of the space of random matrices~and dynamicalsystems!. Three circles represent universality classes~Poissonian,orthogonal, and unitary! and dashed lines represent crossoverstween them. Solid arrows stand for perpendicular transitions, alyzed in this paper. A broken arrow exemplifies a parallel transitinot treated here.

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R andj i jI are distrib-

uted according to Gaussian distributions with the zero-mroot-mean-square values equal tos i j

R ands i jI , respectively.

The parametera measures the relative size of the imaginapart of the off-diagonal matrix elementsa5(s i j

I /s i jR)2, i

Þ j ~the notation has been simplified with respect to R@30#!. A normalization condition Tr(H2)5N11 keeps allthe eigenvalues in a constrained energy range. It also allus to express the variances of the real and imaginary parmatrix elements in terms of matrix sizeN, integer bandwidthb, and real parametera,

~s i jR!25

~N11!

2N1~a11!~2N2b!~b21!~11d i j !, ~2.2!

~s i jI !25

a~N11!

2N1~a11!~2N2b!~b21!~12d i j !. ~2.3!

For a diagonal random matrix (b51) the density of ei-genvalues is Gaussian and the level spacings are distribaccording to the Poisson distribution, independently ofparametera. In the opposite limiting case of the full matri(b5N), variations of the parametera correspond to the process of the time-reversal symmetry breaking in a dynamsystem and control the transition between orthogo(a50) and unitary (a51) ensembles.

Statistical properties of the spectrum and eigenvectorreal symmetric band matrices depend only on a scalingrameterx5b2/N. This scaling law, observed initially by numerical computation of the localization length@28#, was re-ported to describe also the distribution of eigenvalues@31#and eigenvectors@32# and was subsequently explained theretically @33#.

The same scaling law holds also for Hermitian matric@33,34#. Moreover, effects of the time-reversal symmetbreaking are controlled by another scaling paramey52Na/(12a) @30#, stemming from the universal propeties of the orthogonal-unitary transition founded by Pandand Mehta@29#. The structure of eigenfunctions of HBRMand the distribution of the inverse participation ratio haalso been studied recently@35#.

Let us now consider the parametric random matrix

H~l!5H1cosl1H2sinl. ~2.4!

Both matricesH1 andH2 are taken from the same ensembof HBRMs. Hence the spectral properties ofH are stationaryand do not depend onl. Moreover, during the transition, thmotion of eigenvalues is restricted to a bounded energyterval for arbitraryl. This model of parametric dynamicwas already used for the investigation of level curvatu@11# and velocity correlation functions@16,17#. The dynam-ics of eigenvalues as a function ofl may be treated as thdynamics of interacting particles~eigenvalues! with l play-ing the role of fictitious time@2#. This allows us to interpretthe slope of the levels as the velocity of the particles atheir curvature as the corresponding accelerations.

Parametric dynamics defined above can be studiedmerically in a straightforward way. For several valuesensemble parameters (N,b,a) we have generated random

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2448 55KUNSTMAN, ZYCZKOWSKI, AND ZAKRZEWSKI

matrices according to Eqs.~2.1!–~2.4!. Diagonalizations ofresulting matrices for several values ofl have allowed usthen to find level velocities and curvatures by a finidifference method. Special attention has been paid to obreliable values of velocities and curvatures, especiallyvery small and very large values, by varying the size ofstep inl @36#. Before computing the derivatives of the egenvalues with respect tol, the standard unfolding technique was applied@3# to set the mean level spacingD tounity. We have considered matrices of sizeN varying be-tween 50 and 500, velocities and curvatures have been cputed at about 200 different values ofl, and the typicalnumber of independent realizations of the dynamics@Eq.~2.4!# in each case studied has varied with matrix sizeensure at least 200 000 data points in each statistics. In owords, we have simultaneously performed the averagover the energy~data from different energy levels of a givematrix H) and the averaging over the disorder parame~several realizations of the dynamics for the same valueN,b, anda).

To check the reliability of the numerical procedure whave setb5N and we have reproduced known results cocerning the distribution of velocities and curvatures as was the velocity correlation function for the GOE (a50) andthe GUE (a51). Moreover, we have verified that both scaing parametersx andy correctly describe the parametric dnamics. The statistical properties of all quantities studhave been found to be independent of the matrix dimensN ~for sufficiently largeN) provided the parametersx andy have been kept constant.

In the following sections we describe results obtaineddifferent statistics, commencing with the distribution of firderivatives, i.e., velocities. To avoid any misunderstandilet us repeat again that all the data presented are obtaineperpendicular transitions@both H1 andH2 in Eq. ~2.4! be-long to thesamerandom matrix ensemble# as exemplified bydouble-sided arrows in Fig. 1. Thus, for all values ofl thescaling parametersx andy have the same values. We shanot consider here the case when the parameter change mfies the global symmetry properties, a situation exemplifiby broken line arrow in Fig. 1.

III. DISTRIBUTION OF LEVEL VELOCITIES

For level dynamics within the GOE or the GUE the dtribution of level velocitiesP(v) is Gaussian@8,11,14#. Thisfact is easy to explain using first-order perturbation theoFor l50 the derivativedEi /dl is equal to the diagonaelement of matrixH2 expanded in the eigenbasis ofH1.Since both matrices are drawn independently from the sensemble, the matrix elements are Gaussian random numleading to the Gaussian velocity distribution.

On the other hand, in the strongly localized limit an anlytical formula forP(v) given by Fyodorov@26# for systemswith a broken TRI symmetry strongly differs from a Gausian. A non-Gaussian character of the velocity distributionthe GOE to GUE transition, corresponding to the TRI symetry breaking, has been discussed in@23–25#.

We have analyzed the transition between localizeddelocalized spectra both for random systems with a broTRI symmetry~i.e., the ensembles interpolating between

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Poisson ensemble and the GUE! and for ensembles interpolating between the Poisson ensemble and the GOE.former allows us to test the analytical prediction of Fyodor@26#.

The theoretical prediction, as presented in@26#, has nofree parameters; both the shape of the distribution andscale~determined by the velocity variance! are determinedby the theory. Surprisingly, the direct comparison of thdistribution with the numerical data obtained has been higunsatisfactory. The agreement is recovered~see Fig. 2! whenboth the theoretical distribution and the numerically obtaindata are rescaled with respect to the velocity variasv5A^v2& ~note that the mean velocity vanishes by the costruction of the ensemble!. Thus the apparent disagreemeoriginally observed is due to the difference between the thretical and numerically obtained values of the velocity vaance~the ratio of the numerical value to the theoretical pdiction being about 13!. We do not have a clear explanatioof this disagreement. It may be due to the fact that the bawidth in our HBRM ensemble is sharply defined@compareEq. ~2.1!#, while Fyodorov@26# assumed a smooth decreaof the random matrix elements variance with increasing dtance from the diagonalu i2 j u.

The theoretical prediction@26#, represented by a smootline in Fig. 2, takes the form

P~w!5p

6

pwcoth~pw/A6!2A6

sinh2~pw/A6!, ~3.1!

where the rescaled velocityw5v/sv . Similar qualitativeagreement is obtained for different values of the scalingrameter up tox of the order of unity, corresponding to thtransition to a delocalized case. Then the numerical datato show a Gaussian~typical for the GUE! large velocity tailinstead of the exponential tail corresponding to fully locized situation, as exemplified by the lack of large velocitin the numerical data presented in Fig. 2 as a thin line hisgram.

FIG. 2. Velocity distribution in a semilogarithmic scale for thHBRM model interpolating between the Poisson ensemble andGUE (a51, a fully broken TRI symmetry!. The thick ~thin! linehistogram corresponds to numerical data obtained in the localcasex50.126 ~the delocalized casex51.408) from diagonaliza-tions of 10 000 matrices of rankN571. A dashed thick line represents the theoretical prediction, Eq.~3.1!, while a thin line repre-sents the Gaussian distribution.

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Although the theoretical prediction is obtained for tcase of a fully broken TRI symmetry, our numerical daindicate that it works extremely well also for preserved Tsymmetry~real symmetric matrices! provided that again thevelocity variance is appropriately adjusted. The exempldata are presented in Fig. 3 for two cases correspondinstrong localization and a transition to the delocalized regimHere the numerically obtained variance is twice as largethe theoretical value calculated in the same way as forbroken-TRI-symmetry ensemble. It seems, therefore, thatsame distribution, Eq.~3.1! describes the velocity distribution for both the TRI-symmetry case and the no-TRsymmetry situation. The difference between the twosembles ~the former interpolating between the Poissensemble and the GOE, the latter between the Poissonsemble and the GUE! appears in the numerical value of thvelocity variance only. It is clear the variance is a uniqparameter that determines the appropriate velocity scsimilarly as for the pure GOE and GUE@8#.

IV. DISTRIBUTION OF LEVEL CURVATURES

Let us consider now the distribution of curvaturK5d2E/dl2. As shown by Gaspard and co-workers@8#, thetail of the distribution decays algebraically asK222b. Thisuniversality has been verified for different systems@9–11#.At the same time, the small curvature behavior has bfound to be nongeneric even for strongly chaotic syste@10,11# and reflecting the system-dependent wave-functlocalization properties~scarring by periodic orbits!.

On the other hand, the scaled curvature

k5KD

bpsv2 ~4.1!

for pure random ensembles obeys the generalized Cadistribution @11–13#

P~k!5Nb

1

~11k2!~b12!/2 ~4.2!

~whereb51,2,4 for the GOE, GUE, and GSE, respectiveandNb denotes the normalization constant!. Here we dem-

FIG. 3. Same as Fig. 2, but for real symmetric matricesa50:thick line histogram, a localized casex50.056; thin line histogram,a delocalized casex55.63.

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onstrate that if one allows the parameterb to acquire realvaluesbP(0,2#, the same distribution may be used ingeneral case of the intermediate ensemble interpolatingtween Poisson ensemble, GOE, and GUE pure cases~pro-vided we consider the perpendicular transition!. The normal-ization constant is then equal to

Nb51

Ap

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2 D ~4.3!

and the rescaling Eq.~4.1! holds almost everywhere.To test this conjecture we have generated several p

metric dynamics perpendicular to crossovers betweenPoisson ensemble and the GUE, the Poisson and betweeGOE, and the GOE and the GUE using, as before, themulation of Sec. II, Eqs.~2.1! and ~2.4!. The numericallyobtained histograms of curvatures in the double-logarithmscale have been used to fit the algebraic decay of the tathe distribution by the formulaP(K);K2m. Then b hasbeen found asb5m22 @compare Eq.~4.2!#. The same valueof b has been used together with the numerically obtainvelocity variance to rescale the curvatures according to~4.1!. The exemplary results of such a procedure togetwith the conjecture~4.2! are presented in Figs. 4 and 5double-linear and double-logarithmic scales, respectivObserve the excellent agreement between the numericasults and the proposed distribution.

While Eq. ~4.2! seems to describe well, at least appromately, the numerical data for curvatures everywhere intween pure cases of the GOE, the GUE, and the Poislimit, the scaling ~4.1! works best for the delocalized oweakly localized spectra. For the Poisson ensemble to Gcrossover, close to the Poisson limit, the scaling obtaiusing Eq.~4.1! is incorrect. The agreement with the generized Cauchy distribution~4.2! is obtained only if the numeri-cal data are rescaled additionally by a numerical factor of

FIG. 4. Exemplary curvature distribution for the ensemble intpolating between the localized Poisson case (b50) and the GOE(b51) obtained numerically for matrices of rankN571 and band-width b55 corresponding tox50.352 ~histogram!. The thickdashed line represents the fitted distribution, Eq.~4.2!, withb50.59. Thin solid and dotted lines represent the limiting distribtions for the Poisson ensemble and the GOE, respectively.

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order of unity~typically 1.5–2, depending onb). Putting itdifferently, the parameterb in the denominator of Eq.~4.1!should be replaced by another function ofb that goes tobwhen the transition to the delocalized TRI-symmetry-brokcase~i.e., the GUE! is fully accomplished. This indicates thathe proposed distribution~4.2! is most probably the approximate one only. Still, we find it quite remarkable that thsimple analytic expression, with the proper rescaling, woso well for the interpolating ensembles.

Let us mention that the power of the algebraic tail behior may be analytically related to the level repulsion paraeterb by a simple consideration of 232 random matrices@8# yielding m5b12. A comparison ofb values obtainedfrom the tail of the distribution withb8 values obtained fromthe independent fit of the Izrailev distributionPb8(s) @38# ispresented in Fig. 6 in the whole interval of the intermediab values. The agreement is quite good~and of a similarquality to that obtained for the Fourier transform of tvelocity-velocity correlation function! considering that boththe spacing distributionPb8(s) and the proposed curvaturdistributionP(k) are most probably good approximationsthe true distributions only.

It is worth noting that the distribution~4.2! works well forthe ensemble interpolating between the GOE and the G

FIG. 5. Same as Fig. 4, but in a double-logarithmic scale

FIG. 6. Parameterb obtained from the decay of the tail of thcurvature distribution against the level repulsion parameterb8 ob-tained from the independent fit of the Izrailev distribution@38#.Each dot represents one ensemble interpolating between the Poensemble, the GOE, or the GUE.

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for the perpendicular action of the parameterl. On the otherhand, ifl is responsible for the TRI symmetry breaking,has been shown that the tails of the curvature distributionexponential@24,25# and not algebraic, as observed in thwork. Parametric statistics are therefore sensitive to the wthe parameter acts. Another example of this sensitivityavailable from the earlier studies of periodic band randmatrices@37# and three-dimensional Anderson model@39#,where the curvature distribution close to a log-norm disbution has been observed in the localized case, whileanother distribution has been proposed in transition reg@40#.

V. VELOCITY-VELOCITY CORRELATION FUNCTION

In a series of papers, Simons and Altshuler@14# havediscussed the universality of parametric statistical properfor disordered samples as well as for Gaussian ensemblereveal the universality both the eigenvalues~to unit meanspacing! and the parameterl ~asX5svl) have to be res-caled@8,11,14#. We have observed the power of such a recaling already in the previous sections.

Consider next the velocity-velocity correlation function,frequent subject of recent investigations@14–18,21,24#,

Cv~l!:51

2pD2 K E0

2p

v i~l8!v i~l81l!dl8L , ~5.1!

where^& denotes ensemble averaging andD stands for themean level spacing. By definitionCv(0)5sv

2 , thus the ap-propriately rescaled correlation functions take the foCv(X)5Cv(X)/sv

2 . Moreover, several models of timereversal symmetry breaking due to the Aharonov-Bohm fllead to a correlation function practically indistinguishabfrom the one characteristic for the GUE.

It was shown@14,21,15# that for all three universalityclasses the rescaled correlation function

Cv~X!;AbX22, X→`, ~5.2!

with the proportionality coefficientAb dependent on the ensemble ~we denote byb the level repulsion parameteb51,2,4 for the GOE, the GUE, and the GSE, respective!.

Explicit expressions have been obtained@14# for closelyrelated@but distinct fromCv(l)# autocorrelation functions afixed energy. A global approximation forCv(X) has beenproposed@16#. For the case of a classically chaotic systesubject to an Aharonov-Bohm flux Berry and Keating@22#obtained a semiclassical approximation forCv(l) having theform of an everywhere analytic function ofl. Yet it wasdemonstrated@17# that Cv(l) is not analytic and suffers alogarithmic singularity atl50.

Analytic properties of correlation functions are convniently studied using the periodicity inl. In the Fourier do-main

Cv~l!5 (n50

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cncos~nl!. ~5.3!

The mean-squared velocity, determining the scale, is giby the sum of all coefficientssv

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~5.1! it is easy to see thatcn5n2ûanu2&/D2.The asymptotic behavior ofCv(l);2l22 corresponds

to a linear increase of Fourier coefficientscn for smalln. Onthe other hand, for largen it was shown @17# thatûanu2&;n242b and, consequently,

cn;n222b. ~5.4!

This result was obtained by extending the parameterl intothe complex plane and analyzing the distribution of branpoints and anticrossings@4,6#.

Thus, despite the nonanalytic character of the correlafunction, bothCv(l) and its Fourier transform have simpasymptotics given by Eqs.~5.2! and ~5.4!, respectively. It isinteresting to see whether a similar behavior may be fofor the interpolating ensembles. To this end we have studthe asymptotics ofCv for all three possible transitions, i.eensembles interpolating between the Poisson ensemblethe GOE, the Poisson ensemble and the GUE, as well asGOE and the GUE. In all cases the parameterl acted per-pendicular to a given transition~compare Fig. 1!.

We have observed the sameX22 large-X behavior Eq.~5.2! independently of the ensemble studied. As an examFig. 7 shows the rescaled velocity-velocity correlation funtions Cv(X) corresponding to five different cases along tPoisson-GUE crossover.

Algebraic decay of the corresponding Fourier transforis visualized in Fig. 8. Observe the continuous change ofslope, growing from24 for the GUE until22 for the Pois-son limit. This corresponds to the continuous change ofrepulsion parameterb between 2 and 0 in the level spacindistribution. Independently one may fit the nearest-neighspacing distribution obtained numerically to the Izrailev dtribution @38# Pb8(s), which provides an excellent approxmation for the nearest-neighbor spacing distribution forinterpolating ensembles.Pb8(s);sb8 for small s. We haveverified thatb values obtained by fitting the straight linethe tail of log(cn) equalb8 values obtained from the fits othe spacing distribution within 5%. Therefore, we conclu

FIG. 7. Rescaled velocity correlation functionCv(X) obtainedfor five transitions perpendicular to the GUE-Poisson crossoN571,c52.0: b571 (n) ~GUE!, b510 ~1!, b57 ~h!, b55 ~s!,andb54 ~L!. Ensemble averaging is performed over 100 matricthe lines are drawn to guide the eye.

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that the validity of Eqs.~5.2! and ~5.4! extends to the inter-mediate ensembles and fractional values of the repulsionrameterb.

We stress again that this result is restricted to the perpdicular transitions only. If parameterl is responsible for thetransition between the ensembles~e.g., the magnetic flux inthe Aharonov-Bohm effect! the velocity-velocity correlationfunction obeys thecn;n24 algebraic decay, independentof the degree of localization@17#. Thus, similarly to the ve-locity distribution itself@23#, also the velocity-velocity cor-relation functionCv(X) is sensitive to the nature of pertubation generating the parametric dynamics.

In a full analogy with the velocity-velocity correlationfunction ~5.1! we define the curvature correlation function

Ck~l!:51

2pD2 K E0

2p

Ki~l8!Ki~l81l!dl8L . ~5.5!

However, this function does not provide us with any neinformation. This fact is easy to understand studyingFourier expansionCk(l)5(n50

` kneinl. As for velocity cor-

relation function one uses mean Fourier coefficients of invidual energy levels and obtains relationkn5n4ûanu2&/D2.A comparison with the velocity correlation function Fouricoefficients yields immediately

Ck~l!5]2

]l2Cv~l!, ~5.6!

which easily yields the properties ofCk(l) from knownproperties~e.g., the asymptotic behavior! of Cv(l). Equation~5.6! holds for an arbitrary matrix ensemble. For compleness we present the numerically obtainedCk(l), rescaledwith respect toCk(0), for the GOE and the GUE inFig. 9.Notice a much faster decay of the correlation between cvatures as compared to the velocity correlation function. Aymptotically, using Cv;X22 and Eq. ~5.6! we get Ck}X24 for the large rescaled parameterX, in full agreementwith Fig. 9. The curvature correlation function, in view oEq. ~5.6!, may be used, together with the velocity correlatifunction, for numerical tests of the accuracy of the curvatevaluation ~which may be quite tricky using the finite

r

;

FIG. 8. Fourier transform of the velocity correlation functiondisplayed in Fig. 7 in the log-log scale. Lines represent slopes cacteristic for the GUE (24), the GOE (23), and the Poisson ensemble (22). The data forb510 are not plotted to improve thelegibility of the figure.

r

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oereeleonird

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difference method since small and large curvatures mayquire different steps in the parameter!.

VI. HIGHER DERIVATIVES OF ENERGYWITH RESPECT TO THE PARAMETER

Algebraic decay of the Fourier transform of the veloccorrelation function, on the one hand, provides informatabout the singularity of some higher derivative ofCv(l) atl50 and, on the other hand, indicates the possibility odivergence of a distribution variance of some higher deritives of the energy levels with respect to the parameter@17#.In particular, for the orthogonal ensemble (b51), the vari-ance of the curvature distributionK2& does not exist and inorder to characterize the mean curvature one uses the mabsolute valueuKu& instead@37#. Moreover, the second moment of the distribution of the third derivatives of enerlevels L:5d3E/dl3 was predicted@17# to diverge forb51 andb52.

To test this prediction we have studied the distributionthese third derivatives. The most difficult part here is to dtermine what to call them, using the level motion pictuwhere the curvature of the level is identified with the acceration of the fictitious particle, the third derivative of thenergy will correspond to the derivative of the acceleratiIn the spirit of this mechanical analogy we refer to the thderivative as a change in acceleration.

We have restricted the numerical study of the distributof the third derivatives to canonical orthogonal and unitaensembles~the GOE and GUE!. The obtained numerical results are displayed in Fig. 10. As expected, the distributiof the third derivatives are characterized by the algebtails; the numerically obtained power-law decay yielP(L);L (b13/2), which confirms the divergence of the varances both for the GOE and the GUE.

It is interesting that the distribution of the third derivtives may be quite nicely approximated by a very simansatz

P~L !5N b8

~11BbLA~b!!~b13!/2A~b! , ~6.1!

FIG. 9. Curvature-curvature correlation functionCk(X) ob-tained forN550 (L), 60 ~s!, 70 ~h!, 80 ~n!, andN590 ~K/'!for the GOE~open symbols! and the GUE~full symbols!. Universalrescaled velocity correlations are represented for comparisonthick dashed~GOE! and solid~GUE! lines.

e-

n

a-

an

f-

-

.

ny

sic

whereN b8 is a normalization constant andBb andA(b) arefree parameters. In Eq.~6.1! the third derivatives are conveniently rescaled taking the unfolded spectrum~i.e., with themean level spacing equal to unity! with derivatives calcu-lated with respect to the rescaled parameterX5A^v2&l.

We have fitted the distribution of this form to the numecal data for both the GOE and the GUE. The resultsrepresented in Fig. 10 as smooth curves and quite succfully represent the numerical data. The obtained values ofparameters are equal for the GOE toA151.67 andB159.08, while for the GUE we obtainA252.50 andB250.84. The obtained values ofAb are close to simplefractionsA155/3 andA255/2, the corresponding curves aindistinguishable from best fits within the accuracy of odata.

VII. CONCLUDING REMARKS

We have analyzed various aspects of parametric dynamin the space of Hermitian random matrices. Such a momay be applied to study transitions between Poissonian,thogonal, and unitary universality classes. We have analynumerically the situations when the parameter change dnot modify the global properties of the ensemble studied,case labeled as a ‘‘perpendicular’’ transition to contraswith the ‘‘parallel’’ case when the parameter is responsifor the breakup of the symmetry or other change of the prerties of the ensemble studied. We have, however, compour results with predictions of other works where often sua parallel parameter action was considered.

We have paid particular attention to the study of the trasition between the Poissonian ensemble characterizedstrongly localized wave functions and the delocalized Gauian ensembles~the GOE or GUE!. In particular, the numeri-cal tests of the analytic predictions for the distributionlevel velocities in the case of broken time-reversal invarian@26# confirmed the predicted shape. We have observed aagreement between the theory@26# and the numerical datahowever, as far as the prediction for the velocity variance

by

FIG. 10. Distribution of third derivatives of the eigenenergwith respect to the parameterP(L) in the double-logarithmic scaleHistograms correspond to numerically obtained data for the G~thick line! and the GUE~thin line!, both obtained for random matrices of rankN571. Dotted and dashed lines represent the besof the proposed distribution, Eq.~6.1!, for the GOE and the GUErespectively.

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considered. We have discussed the possible origin ofdifference. We have shown that the distribution of the safunctional form works for the delocalization transition alfor the real symmetric random matrices. This calls for textension of the theory to such a case.

We have studied in detail also the distribution of levcurvatures. Here no analytic prediction is available. We hfound that the numerically obtained distribution of curvtures is well approximated by the generalized Cauchy disbution ~4.2! shown earlier to be exact@11–13# for the ca-nonical ensemble, the GOE, and the GUE. The only requmodification is to take the fractional value of the level repsion parameterb in accordance with the spacing distributioWe have found also that the same rescaling~4.1! holds ev-erywhere except for the localized, no-TRI-symmetry esemble. Then the agreement with Eq.~4.2! requires addi-tional multiplication of all curvatures~rescaling! by a factorof the order of unity and dependent onb.

This form of the curvature distribution implies its algbraic tails of the formP(K);K222b. Similarly, we havefound that the same level repulsion parameterb governs thetails of the Fourier-transformed velocity correlation funtions. Explicitly, the corresponding Fourier coefficients sisfy to a good precisioncn;n222b.

A comparison with other works, where mostly the paraltransition have been studied@23–25,35,37,39# in variousmodels, indicates strong differences with the parallel trantions. This difference has been observed for the velocity crelation function in the case of a partially broken TRI symmetry in a fully delocalized case@23#. Here we have shownthat the sensitivity of level dynamics to the way in which tparameter acts extends also to other parametric statismeasures as well as to other ensembles interpolating betw‘‘pure’’ cases of the Poisson ensemble, the GOE, andGUE. This has an important consequence: it shows thatuniversality of parametric dynamics is more limited that aticipated before@14#.

Finally consider the consequence of the presented refor studies of realistic systems. Consider the semiclasslimit when the system is ‘‘large,’’ with a high density ostates and many highly excited levels. In the generic sittion a small change of the parameter cannot induce sigcant changes in system symmetries and global propersmall changes of a parameter may be thus considereperpendicular cases. This indicates that the perpendic

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transitions studied in this paper are typically generic.important exception is the change of the magnetic fieldssystems with no additional symmetries where the fieldduces the breakup of the time-reversal symmetry acttherefore, in a parallel way.

Thus the fact that the curvature distribution for the trasition between the Poisson ensemble and the GOE, reavia the banded matrix model studied here, is described bygeneralized Cauchy law Eq.~4.2! has important consequences. It has been suggested~see@30# and references citedtherein! that banded matrices may be used to simulate statical properties of partially chaotic systems interpolating btween the integrable case~with Poisson level spacing statistics! and the fully chaotic case~the GOE!. This has beenpartially based on the similarity of the level spacing distbution observed in both cases, well approximated byIzrailev ansatz@38# or for TRI symmetry systems by thBrody distribution. However, even for chaotic systems,shown before, small curvature behavior may be abunddue to isolated avoided crossings and scarring of wave futions @11#. For the mixed-phase-space systems the avoicrossings are typically quite narrow and isolated, betwethem the levels can be adiabatically followed as a paramis varied. Generically, small changes of a parameter arecompanied by small changes of eigenenergies that maytreated by a Taylor-series expansion with a leading linterm. It shows that such systems will exhibit a great abdance of small curvatures, accompanied possibly by a sinlarity of the distribution atK50 ~see also@11# for an addi-tional discussion and numerical examples!. Banded randommatrices show a curvature distribution different from whatexpected for a quantum system with a mixed phase spThus this ensemble is not adequate for simulating the statical properties of partially integrable or weakly chaotic sytems at least in cases when parametric dynamics is ccerned. On the other hand, the HBRM ensemble seems tvery useful for obtaining predictions for random systems texhibit a transition to localization, well into the localizatioregime.

ACKNOWLEDGMENTS

It is a pleasure to thank Italo Guarneri and Yan Fyodorfor fruitful discussions. We acknowledge financial suppfrom Polish Committee of Scientific Research under GrNo. 2P03B 03810.

.

K.

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Nonuniversality in level dynamics

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