Spectral correlations understanding oscillatory contributions

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Spectral correlations:understanding oscillatory contributions

B.Mehlig 1and M.Wilkinson 2

1

Max-Planck-Institut f¨u r Physik komplexer Systeme,N¨o thnitzer Str.38,01187Dresden,Germany 2

Department of Physics and Applied Physics,University of Strathclyde,Glasgow,Scotland,UK

(February 1,2008)

We give a transparent derivation of a relation obtained using a supersymmetric non-linear sigma model by Andreev and Altshuler [Phys.Rev.Lett.72,902,(1995)],which connects smooth and oscillatory components of spectral correlation functions.We show that their result is not speci?c to random matrix theory.Also,we show that despite an apparent contradiction,the results obtained using their formula are consistent with earlier perspectives on random matrix models.In particular,the concept of resurgence is not required.

Spectral correlations of complex quantum systems,such as disordered metals and classically chaotic quan-tum systems,are known to be nearly universal.For small ranges of energy they are well approximated by the Gaus-sian invariant ensembles of random matrix theory (GXE,where X=O,U or S stands for orthogonal,unitary and symplectic invariance)[1,2].Deviations from GXE be-haviour at larger energy scales may be consistently incor-porated using semiclassical or perturbative approaches [3–5].An interesting development in this ?eld was a pa-per by Andreev and Altshuler (AA)[6],who introduced a relation which suggests a degree of non-universality in short ranged spectral correlations.Their calculations are based on the non-linear sigma model.Our paper will give a transparent physical insight into their relation.

AA [6]consider the spectral two-point correlation func-tion,de?ned as

R β(?)=?2 d (E +?/2)d (E ??/2) ?1.(1)

Here d (E )=

n δ(E ?E n )is the density of states,E n

are the eigenvalues of a Hamiltonian H

and ?(E )= d (E ) ?1is the mean level spacing which is assumed to be independent of E in the following.The index β=1,2,4distinguishes orthogonal,unitary or symplectic symme-try classes respectively.

AA divide Eq.(1)into a smooth and an oscillatory contribution,and propose that (for β=2)these are re-lated as follows

R av 2(?)???2

??2

log D (?)(2)

R osc 2(?)

?

1

N

N ?1 n =0

?E n exp(?2πi kn/N )(4)

where ?E n =E n ?n ?,N is the number of energy levels,

and k takes N successive integer values.We will take the maximal k to be int(N/2).The long wavelength modes evolve almost independently,with a long relaxation time,which scales as k ?1[12,4].The short wavelength modes remain strongly coupled.The stochastic perturbation will bring the short wavelength modes into equilibrium,giving statistics of the a k which are identical to the ran-dom matrix ensemble.1

These arguments were later supported and extended with the aid of semiclassical estimates of matrix elements [4].The argument in[12]assumes that the matrix ele-ments of the perturbation are uncorrelated.It was shown that semiclassical estimates are consistent with this hy-pothesis when considering the stochastic force driving the large k modes.The forces driving the small k modes are modi?ed by the classical dynamics of the system.The resulting picture is that long wavelength?uctuations are non-universal,but that at short wavelengths the exci-tations of the modes are precisely the same as for the appropriate GXE.In particular,there is no modi?cation of the statistics of the Fourier coe?cients a k unless|k|/N is small.This appears to be in contradiction with the AA relations,which suggest that non-universal corrections to the smooth part of the correlation function are echoed by oscillations with large wavenumbers k.

One resolution of this contradiction would be that there are previously un-suspected correlations between matrix elements due to‘resurgence’,which are not cap-tured by the semiclassical approximations in[4].We will however show that there is no need in invoke this princi-ple.We will?rst describe a simple derivation of the AA result,and comment on its applicability.We will then describe the Brownian motion model and use it to derive the correlation function Rβ(?)for the case of a system with di?usive electron motion.In this calculation we assume that the statistics of the Fourier coe?cients are unchanged for large|k|.The fact that we reproduce ex-isting results veri?es that resurgence is not an essential ingredient in their derivation.Finally we will comment on the correlation function for classically chaotic systems. Our starting point is the following general expression for the correlation function Rβ(?):

Rβ(?)=?1+?

∞

n=?∞pβ(n,?)(5)

where pβ(n,?)d?denotes the probability of?nding that the di?erence between E0and E n is in the interval [?,?+d?].We will show below that in many cases,the pβ(n,?)are well approximated by Gaussians for large n. We will estimate the varianceσ2β(n)for di?erent systems. First,however,we show how the relations(2)and(3) can be derived from(5).We write

Rβ(?)=R avβ(?)+R oscβ(?).(6) Here R avβ(?)is de?ned as

R avβ(?)= ∞?∞d?′w(???′)Rβ(?′)(7)

where w(?)is a suitable window function(which could be a Gaussian centred around zero with variance much larger thanσ2β(n)and normalized to??1).R oscβ(?)is the remaining oscillatory https://www.docsj.com/doc/1510667540.html,ing(5)we have,upon expanding the slowly varying window function in (7)

R avβ(?)=?1+?

∞

n=?∞ ∞?∞d?′w(???′)pβ(n,?′)(8)??1+?

∞

n=?∞ w(??n?)+1

2?2

?2

C diag mn =C diag

m?n

(and C o?mn=C o?m?n).Using second order

perturbation theory leads to a Langevin equation

δE n= m=n|δH mn|2

E m?E n,(14)

δE mδE n =2δτβ?1C diag m?n.(15)

Semiclassical estimates[14]indicate that the

C

o?

n

de-

crease for large values of n,i.e.the repulsive interaction

is screened at long range.This e?ect was considered in

[15];for our purposes it is not signi?cant,and we will set

C o?n=1throughout.We now use(4)to obtain the same

equations of motion in terms of the Fourier variables a k.

Usingδ?E n=δE n,we have

δa kδa?l =2δτβ?1I kδkl(16)

where I k=N?1 n C diag n exp(?2πi kn/N).The trans-

formation of the drift term is less straightforward.In

general the expectation value ofδa k is a complicated

function of all of the a k,but in the limit|k|/N→0

the equations decouple and obey[4,11]

δa k =?2π2k

2π2βk

.(18)

We cannot determine |a k|2 for larger values of|k|from

this approach.We now make our key assumption,that

I k～N?1when|k|/N is not small.This corresponds

to assuming that there are no short-ranged correlations

between the diagonal matrix elements,i.e.C diag

n?m～δnm.

Semiclassical arguments which support,but do not prove,

this assumption are given in[4].(The existence of short

ranged correlations would represent a type of‘resur-

gence’,in the sense discussed earlier).Under this as-

sumption,the equations of motion of the a k are identical

to those for the Brownian motion model describing Gaus-

sian invariant ensembles.We therefore conclude that

when|k|/N is not small,the mode intensities |a k|2 are

identical to those of the Gaussian invariant ensembles.

For large n,the level spacings E n?E0are seen to be a

sum of many independent random variables and are thus

Gaussian distributed

pβ(n,?)=[2πσ2β(n)]?1/2e?(??n?)2/2σ2β(n)(19)

with mean n?and with variance

σ2β(n)=4 k |a k|2 sin2 πnk

βπ2 t H/20d t2ˉh)(21)

where Jβ(t)=I(t)for t?1,and Jβ(t)takes a univer-

sal(but to us,unknown)form when t is not small.For

the Gaussian invariant ensembles(where I(t)=1),the

variances clearly grow logarithmically with n.They are,

asymptotically for large n,

σ2β(n)?

2?2

2,C2=0and C4=log(4/π).Using

this expression in(2)and(3)gives the correct leading

order contributions to R avβ(?)and R oscβ(?)in the limit

?/?→∞(c.f.[2]).

Next we consider how the function I(t)must be modi-

?ed at small t to take account of classical dynamics.For

small values of|k|,the amplitudeδa k can be estimated

semiclassically[4]

δa k～1?k N(23)

where?U(t)=exp(?i Ht)is the evolution operator.We

consider?rst di?usive systems(electrons in disordered

metals),then systems with a chaotic classical limit.

Di?usive systems.In this case we may consider the

perturbationδ H to be uncorrelated random changes of

the site energies V n in an Anderson tight-binding model

[15]

δH= nδV n P n, δV nδV n′ =δn,n′(24)

where P n is the projection for locating an electron on lat-

tice site https://www.docsj.com/doc/1510667540.html,ing the semiclassical approximation(23),

I(t)is seen to to be proportional to the probability of

returning to the original site after time t.Normalising so

that I(t)approaches unity for large t,we have

I(t)=

∞

ν=0e?Dk2νt(25)

where the sum is over the eigenmodes of the Helmholtz

equation(?2+k2ν)ψν(r)=0with Neumann boundary

conditions.In a quasi-one dimensional system,kν=

πν/L.In this case we obtain from Eq.(21)

3

σ2β(n)=2?2

2

∞

ν=1log 1+n2

βπ2 Ci 2πnt0t H ?γ+Cβ

.(27)

This model has the feature thatσ2β(n)is?nite in the limit n→∞,corresponding to the behaviour of Dyson’s?3 statistic for systems with a smooth classical limit[3].We note that this implies[using(2)and(3)]that the oscil-latory part of the correlation function does not decay to zero.The form factor is seen to have a delta function at the Heisenberg time t H,with a magnitude proportional to(t0/t H)4/β.This feature has not been remarked upon in earlier papers which have discussed the form factor for chaotic systems[7–9].

A more precise estimate ofσ2β(n)for speci?c systems can be obtained using periodic orbit theory,following the approach used in[3].The conclusions are unchanged:σ2β(n)remains?nite as n→∞,and there must exist oscillations in the correlation function which do not de-cay.One di?erence is that for large n the pβ(n,?)are dominated by the shortest orbits.Because only a?nite number of components are signi?cant,the central limit theorem cannot be used to assert that these distributions are Gaussian.Moreover small deviations from a Gaus-sian distribution can have a large e?ect on the Fourier transform of the distribution,which determines the mag-nitude of the oscillations in(10).We infer that the AA relations may need to be modi?ed when applied to the spectra of systems with a smooth classical limit. Conclusions.We have shown that the AA relations have a simple interpretation,independent of random ma-trix theory and supersymmetric techniques.We have shown how spectral correlations of di?usive systems are obtained using from Dyson’s Brownian motion model, indicating that the AA relation is consistent with this approach.

Acknowledgements.It is a pleasure to acknowledge il-luminating discussions with O.Agam.BM was partially supported by the SFB393.MW was supported by the Max Planck Institute for the Physics of Complex Sys-tems,Dresden,and the EPSRC,grant GR/L02302.