Between Poisson and GUE statistics: Role of the Breit-Wigner width

25 Pages·0.39 MB·English

by  Klaus M. Frahm

Between Poisson and GUE statistics: Role of the Breit–Wigner width
Klaus M. Frahm1, Thomas Guhr2 and Axel Mu¨ller–Groeling2
1Laboratoire de Physique Quantique, UMR 5626 IRSAMC, Universit´e Paul Sabatier, F–31062 Toulouse, France
2Max–Planck–Institut fu¨r Kernphysik, Postfach 103980, D–69029 Heidelberg, Germany
(February 1, 2008)
Weconsiderthespectralstatistics ofthesuperposition ofarandomdiagonal matrixandaGUE
matrix. By means of two alternative superanalytic approaches, the coset method and the graded
eigenvalue method, we derive the two–level correlation function X2(r) and the number variance
Σ2(r). ThegradedeigenvalueapproachleadstoanexpressionforX2(r)whichisvalidforallvalues
of the parameter λ governing the strength of the GUE admixture on the unfolded scale. A new
twofold integration representation is found which can be easily evaluated numerically. For λ 1
≫
8 theBreit–WignerwidthΓ1 measuredinunitsofthemeanlevelspacingDismuchlargerthanunity.
9 In this limit, closed analytical expression for X2(r) and Σ2(r) can be derived by (i) evaluating
9 the double integral perturbatively or (ii) an ab initio perturbative calculation employing the coset
1 method. The instructive comparison between both approaches reveals that random fluctuations of
n Γ1manifestthemselvesinmodificationsofthespectralstatistics. Theenergyscalewhichdetermines
a the deviation of the statistical properties from GUE behavior is given by √Γ1. This is rigorously
J shown and discussed in great detail. The Breit–Wigner Γ1 width itself governs the approach to

8 the Poisson limit for r . Our analytical findings are confirmed by numerical simulations of an
→∞
2 ensembleof500 500matrices, whichdemonstratetheuniversalvalidityofourresultsafterproper
×
unfolding.

1
v PACS numbers: 05.40.+j, 05.45.+b, 72.15.Rn
8
9
2
1
I. INTRODUCTION
0
8
9 Oneofthearchetypicalproblemsinquantummechanicsconsistsofcalculating(certainpropertiesof)theeigenvalue
/ spectrum of a diagonal operator and a superimposed non–diagonal one. Little can be said in general about this
t
a problem. In our paper, we focus on the particular case where the matrix representations H and H of the above
0 1
m operators can be taken from the Poisson and the Gaussian Unitary Ensemble1 (GUE), respectively,
-
d H =H +αH , (1.1)
0 1
n
o
with α some strength parameter. The Poisson Ensemble is constructed from all those matrices whose eigenvalues
c
are independent random numbers with identical, and largely arbitrary, distribution function. Due to the rotational
:
v invarianceoftheGUE,itsufficestoconsideronlydiagonalmatricesH ofthePoissontype. Therandommatrixmodel
0
i
X (1.1) should provide an adequate description for numerous transition phenomena from regular to chaotic fluctuation
propertiesinatomic,nuclear,andcondensedmatterphysics,aswellasinquantumchaology(forareviewseeRef.2).
r
a Nuclearphysicsprovidesanimportantexample. Inheavyionreactions,fastrotatingcompoundnucleiareproduced.
The rotation is a collective motion of all nucleons which is often accompanied by single particle excitations. Thus
the total Hamiltonian can be modeled as a sum of two contributions, a regular one, H , describing the collective
0
motion,andastochasticone,H ,describingthe influence ofthe singleparticleexcitations. Thisscenariowasstudied
1
numericallyin Ref.3 for the cases thatH is drawnfroma Poissonora harmonicoscillatorensemble. Using Efetov’s
0
supersymmetry method4,5, the qualitative behavior of the two–level correlations was also discussed analytically in
Ref. 3.
As a further important example we mention the problem of two interacting particles in a random potential in-
troduced by Shepelyansky6. He predicted that two particles in a one–dimensional disordered chain can be extended
on a scale L far exceeding the one–particle localization length L . Subsequent work7–9 quickly led to a definite
2 1
confirmation and better understanding of this phenomenon. One possible approach6 to this problem is to construct
aneffectiveHamiltonianby diagonalizingthe noninteractingpartofthe two–electronproblemandexpressingthe mi-
croscopicHamiltonianin the basis oftwo–electronproductstates. The resulting representationconsistsof a diagonal
contribution containing the eigenvalues of the noninteracting problem, and an off–diagonal contribution originating
from the interaction operator. With the crucial assumption that both the diagonal and the off–diagonal matrix el-
ements can be chosen to be random variables we arrive at the above random matrix model (for system sizes L of
the order of L ). In the regime L > L the effective Hamiltonian has been studied in some detail10–12. Imry has
1 1
1
shown7 that the enhancement factor L /L is given by the “Breit–Wigner width” or “two–particle Thouless energy”
2 1
Γ measuredinunitsofthetwo–particlelevelspacingD ,L /L =Γ . Thisraised,amongotherthings,the question
2 2 2 1 2
how to identify Γ in typical spectral observables like the number variance Σ2(r) of the random matrix model (1.1)
2
considered here13. This problem, and some surprises which we encountered while studying it, have been our original
motivation in this project.
The random matrix model (1.1) has been considered by many authors14–18,20,21. Recently, one of us succeeded18
in deriving exact integral representations for the spectral k–point functions of the Hamiltonian (1.1). To this end,
the graded eigenvalue method was used18,19, which is a variant of Efetov’s supersymmetry method4,5. In an effort
to discuss certain approximations from these integral representations, two of the present authors derived20 a closed
expressionfor the spectral two–pointfunction X (r) by means of a special kind of saddle–point approximation. This
2
led to the surprising observation that the energy scale at which X (r) and the number variance Σ2(r) deviate from
2
random matrix behavior is given by √Γ and not by Γ . Here, Γ is the Breit–Wigner (or spreading) width induced
1 1 1
by the perturbation αH and corresponds to the quantity Γ in the above example. Both Γ and r are measured in
1 2 1
units of the mean level spacing D. This result was later confirmed in a perturbative calculation21 of the two–point
correlatorX (r). The saddle–pointapproximationemployedinRef. 20waslimited to certainsituations anddoesnot
2
include cases in which the energy separations r is much larger than the Breit–Wigner width Γ .
1
The purposeofthe presentpaperis threefold. First,we derive,avoidingthe above–mentionedsaddle–pointexpres-
sion, a relatively simple exact expression for X (r) for all r and all relative strengths of H and H . For Γ 1 this
2 0 1 1
leads to very compact analytical formulas for both X (r) and Σ2(r). The two scales √Γ and Γ can be id≫entified
2 1 1
and interpreted. Second, we compare both variants of the supersymmetry formalism, the graded eigenvalue method
and the coset method, by deriving our results independently for both methods. This sheds some light on the relation
between these approaches. Furthermore our comparison elucidates the role played by statistical fluctuations of the
Breit–Wigner width. Third, we confirm our result with the help of rather extensive numerical simulations of the
spectral properties of the Hamiltonian (1.1).
Our paper is organized accordingly. Following this introduction, Sec. II deals with the graded eigenvalue method
andthe improvedtreatmentofthe integralrepresentationsderivedinRef. 18. InSec.III the cosetmethod is invoked
to essentially re-derive the results of Sec. II. We have tried to keep both sections reasonably self–contained. Readers
whoarecompletelyunfamiliarwiththesupersymmetryformalismshould,however,consulttheintroductoryliterature
forthe coset4,5 andthe gradedeigenvaluemethod19,18. SectionIVisdevotedtoournumericalsimulationsandSec.V
contains a summary and the discussion.
II. GRADED EIGENVALUE METHOD
Ina firstpart,Sec. IIA, we introducesome basic terminologyandbriefly recapitulate the derivationofthe integral
representations18 of the k–level correlation functions for the convenience of the reader. In Sec. IIB the integral
representation for the two–point function is further transformed to derive a relatively simple and, most importantly,
tractable expression for the level–level correlator. Finally, Sec. IIC is devoted to a perturbative evaluation of this
expression in the limit where the Breit–Wigner width is much larger than the level spacing.
A. Integral representation for spectral correlation functions
The k–level correlationfunctions for the Hamiltonian H in Eq.(1.1) characterize the spectral statistics completely
and are defined by
k
1 1
R (x ,...,x ,α)= P (H) Imtr d[H] . (2.1)
k 1 k πk N x− H
Z pY=1 p −
The probability of finding k energy eigenvalues in infinitesimal intervals dx around x (i = 1,...,k) is given by
i i
R (x ,...,x )dx ...dx . For the case considered in this paper the probability distribution function P (H), which
k 1 k 1 k N
depends explicitly on the dimension N of the matrices H, is given by the product
P (H)=P(0)(H )P(1)(H ) (2.2)
N N 0 N 1
with
2
N
P(0)(H )= p(0)(Hnn) δ(ReHnm)δ(ImHnm) ,
N 0 0 0 0
n=1 n>m
Y Y
2N(N−1)/2
P(1)(H )= exp tr(H2) . (2.3)
N 1 πN2/2 − 1
(cid:0) (cid:1)
The function p(0)(Hnn) is smooth but otherwise arbitrary. We introduce the modified k–point correlators
0
Rˆ (x ,...,x ), which are obtained by omitting the projection onto the imaginary part in Eq. (2.1). The original
k 1 k
quantities can be reconstructed by appropriate linear combinations of the Rˆ . To perform the ensemble average we
k
write the modified correlators in terms of a supersymmetric normalized generating functional,
1 ∂k
Rˆ (x ,...,x ,α)= Z (x+J,α) (2.4)
k 1 k (2π)k ∂J1...∂Jk k (cid:12)
(cid:12)J=0
(cid:12)
(cid:12)
where the energies and source variables form diagonal2k 2k matrices accord(cid:12)ing to x=diag(x1,x1,...,xk,xk) and
×
J =diag( J ,J ,..., J ,J ), respectively. The averagedfunctional takes the form
1 1 k k
− −
Z (x+J,α)= d[H ]P(0)(H ) d[σ]Q (σ,α)detg−1[(x±+J σ) 1 1 H ] , (2.5)
k 0 N 0 k − ⊗ N − 2k⊗ 0
Z Z
where σ is a 2k 2k Hermitean supermatrix and Q (σ,α) = 2k(k−1)exp( trgσ2/α2) is a normalized graded proba-
k
× −
bility density. The subsequent steps can be summarized as follows19. The matrix x+J is shifted from the graded
determinant to the graded probability density and the supermatrix σ is diagonalized according to σ =u−1su, where
s=diag(s ,is ,...,s ,is ). The volume elementcan be rewrittenas d[σ]=B2(s)d[s]dµ(u) with B (s) the Jaco-
11 12 k1 k2 k k
bian(calledBerezinian inthis case)ofthe transformation. The non–trivialintegrationoverthe unitarydiagonalizing
supergroup with its Haar measure dµ(u) is the central step in the graded eigenvalue method and can be performed
with the supersymmetric extension of the Harish–Chandra Itzykson Zuber integral. Collecting everything we arrive
at
1
Z (x+J,α)=1 η(x+J) + G (s x J,α)Z(0)(s)B (s)d[s] ,
k − B (x+J) k − − k k
k Z
Z(0)(x+J)= d[H ]P(0)(H )detg−1[(x±+J) 1 1 H ] , (2.6)
k 0 N 0 ⊗ N − 2k⊗ 0
Z
where the kernel resulting from the group integration is Gaussian and given by
1 1
G (s x,α)= exp trg(s x)2 . (2.7)
k − (πα2)k −α2 −
(cid:18) (cid:19)
The distribution 1 η(x+J) in Eq. (2.6) ensures the normalization Z (x,α) = 1 at J = 0. We mention in passing
k
−
thatthe generatingfunctionZ (x+J,α) satisfiesanexact diffusionequationinthe curvedspaceofthe eigenvaluesof
k
Hermitean supermatrices. Here, t=α2/2 is the diffusion time and the generating function Z(0)(x+J) serves as the
k
initial condition. This diffusion is the supersymmetric analogue18,22 for the diffusion of the probability distribution
function in the space of ordinary matrices, which is equivalent to Dyson’s Brownian. Performing the source term
derivatives in Eq. (2.4) we find
( 1)k
R (x ,...,x ,α)= − G (s x,α) Z(0)(s)B (s)d[s] , (2.8)
k 1 k πk k − ℑ k k
Z
where the symbol indicates the above–mentioned proper linear combination of terms. In order to arrive at truly
ℑ
universalquantitieswehavetomeasureallenergiesinunitsofthemeanlevelspacingD. This leadstothe definitions
ξ = x /D, λ = α/D, X (ξ ,...,ξ ,λ) = lim DkR (x ,...,x ,α), and z(0) = lim Z(0)(Ds). The level
p p k 1 k N→∞ k 1 k k N→∞ k
correlation functions X defined in this way are translation invariant over the spectrum. They can be expressed as
k
( 1)k
X (ξ ,...,ξ ,λ)= − G (s ξ,λ) z(0)(s)B (s)d[s] (2.9)
k 1 k πk k − ℑ k k
Z
after the redefinition s s/D. We notice the very similar structure of the integral representations (2.8) and (2.9).
→
This is so because the above mentioned diffusion in superspace is, apart from the initial conditions, the same on all
3
scales,incontrasttothediffusioninordinaryspacewhichisequivalenttoDyson’sBrownian. Thetwo–pointfunction
X , to which we restrict our attention henceforth, depends on r = ξ ξ only and we can perform two of the four
2 1 2
−
integrations in Eq. (2.9) due to translational invariance. This leads to the integral representation18
∞ ∞
8 1 rt rt t t
X (r,λ)= exp (t2+t2) sinh 1 sin 2 1 2 z(0)(t ,t )dt dt (2.10)
2 −π3λ2 −2λ2 1 2 λ2 λ2 (t2+t2)2ℑ 2 1 2 1 2
Z Z (cid:18) (cid:19) 1 2
−∞−∞
with t =s s and t =s s . The initial condition z(0) is still arbitrary. For the case of Poisson regularity
1 11− 21 2 12− 22 2
we have
1 t2+t2
z(0)(t ,t )= Re exp iπ 1 2 1 . (2.11)
ℑ 2 1 2 2 − 2t− −
(cid:18) (cid:18) 1 (cid:19)(cid:19)
Equation (2.10)is exact but, unfortunately, difficult to evaluate as it stands. In the following subsection, we derive a
more convenient formulation of Eq. (2.10).
B. Simplification of the two–point level correlation function X2(r,λ)
As explained in the introduction, a previous effort20 to extract physical information from Eq. (2.10) led to the
discovery that the energy scale at which X (r,λ) deviates from random matrix behavior is linear in λ. For very
2
general reasons, however, the Breit–Wigner width is always quadratic in the strength of the perturbing matrix
elements,hence Γ λ2. This ledto the conclusionthat√Γ isthe importantenergyscale. The approximationused
1 1
∝
to derive this result was, however, not valid for r Γ . Here, we avoid the saddle–point approximation proposed in
1
Ref. 20 and proceed as follows. We introduce the≫new variables x =t /(πλ2) and the important abbreviations
j j
1 r
c= , κ= . (2.12)
π2λ2 πλ2
Itis instructivetonote thatπλ2 isactuallythe Breit–Wignerwidthmeasuredinunits ofthe levelspacing,πλ2 =Γ .
1
Thiscanbe deducedfromthe localdensityofstatesforthe modelstudiedhere20 andwillbecomemoreobviouswhen
we discuss the coset method in the next section. With the above notation we find
X (r,λ) X (κ,c)=X˜ (κ,c)+X˜ ( κ,c)+κ2 (2.13)
2 2 2 2
≡ −
where
∞ ∞
2ic x x
X˜ (κ,c)= dx dx 1 2 exp( (x ,x ))
2 π 1 2(x2+x2)2 −L 1 2
Z Z 1 2
−∞−∞
1 x x +i x
(x ,x )= 1 (x +i+κ)2+ 1 (x +iκ 1 )2 . (2.14)
L 1 2 2c(cid:20)x1+i 1 x+1 2 x1+i (cid:21)
To reduce the order of the pre-exponential singularity we perform an integration by parts using 2x /(x2 +x2)2 =
2 1 2
∂ (x2+x2)−1. With the additional transformation x =y x /(x +i) Eq. (2.14) takes the form
− x2 1 2 2 2 1 1
∞ ∞
i (x +i)(y +iκ)
X˜ (κ,c)= dx dy 1 2 exp( ˜(x ,y ))
2 −π 1 2 (x +i)2+y2 −L 1 2
Z Z 1 2
−∞−∞
1 x
˜(x ,y )= 1 (x +i+κ)2+(y +iκ)2 . (2.15)
1 2 1 2
L 2cx +i
1
(cid:2) (cid:3)
Now,ourstrategyisto eliminate the imaginarycontributionstothe squaresin ˜(x ,y ). To this endweperformtwo
1 2
L
shifts, namely y = x +i and u = y +iκ. This amounts to moving the integration contour from the real axis to
1 1 2 2
the lines Im(x )= 1 and Im(y )= κ, respectively, see Fig. 1. The integral (2.15) remains unchanged under these
1 2
− −
shiftsunlesstheintegrandexhibitssingularitiesintheregionbetweentheoldandthenewintegrationcontours. With
the help of Fig. 1 the readercan easily convince himself that such singularitiesdo indeed exist in our case. They give
rise to two different residuum contributions R and R . We therefore arrive at
1 2
4
∞ ∞
i y u
X˜ (κ,c)= dy du 1 2 exp( ˜(y i,u iκ)) R R
2 −π 1 2y2+(u iκ)2 −L 1− 2− − 1− 2
−Z∞−Z∞ 1 2−
0 1
κ
R = dy (y +iκ)+ dy (y +iκ)exp 2i (1 y )
1 2 2 2 2 2
c −
−Z1 Z0 (cid:16) (cid:17)
|κ| 0
κ κ+ κ κ κ κ
R = dy y κ exp (y i) | | + dy y +κ exp (y i) −| | . (2.16)
2 1 1 1 1 1 1
κ − − − c κ − − c
Z (cid:18)| | (cid:19) (cid:18) (cid:19) Z (cid:18)| | (cid:19) (cid:18) (cid:19)
0 −|κ|
We notice the appearance of sign(κ) = κ/κ which is needed to distinguish between the two cases κ < 0 and
| |
κ>0 in Eq. (2.13). The residuum contributions can be easily calculated analytically and with the final (innocuous)
transformation u =y +κ we obtain the following result for the two–point level correlation function (2.13),
1 1
1 r2 1
X (r,λ)=1+ exp 2 cos(2πr) 1 +
2 2(πr)2 − λ2 − (πλ)2
(cid:20) (cid:18) (cid:19) (cid:21)
∞ ∞
i (u κ)u 1 i
du du 1− 2 exp 1 (u2+u2) +(κ κ) (2.17)
−(π−Z∞−Z∞ 1 2(u1−κ)2+(u2−iκ)2 (cid:18)−2c(cid:20) − u1−κ(cid:21) 1 2 (cid:19) ↔− )
In the limit λ , the first two terms in Eq. (2.17) reduce to the two–point level correlationfunction for the GUE,
→∞
sinπr 2
XGUE(r)=1 . (2.18)
2 − πr
(cid:18) (cid:19)
The additional terms are corrections for finite λ. We denote the double integral in Eq. (2.17) by P(c,κ). It can be
evaluated numerically for a wide range of values of c and κ without too much effort. For our present purposes, in
particular for the comparison with the coset method in Sec. III, we prefer to derive some analytical results in the
limit c 1.
≪
C. Perturbation theory for X2(r,λ)
Inthefollowingweassumethatc 0,butκ=const.ThismeansthattheBreit–Wignerwidthismuchlargerthan
→
the level spacing, Γ 1, and that the energy difference r scales like Γ . Under these circumstances P(c,κ) can be
1 1
≫
calculated perturbatively, resulting in a power series in c. With the polar coordinates u = ρsinϕ and u = ρcosϕ
1 2
and after expanding both the exponent and the pre-exponential term in Eq. (2.17) in powers of ρ we obtain the
following expression for the double integral,
2π ∞
1 ρ ∞ ρeiϕ m iρ2 ∞ ρsinϕ n
P(c,κ)= dϕ dρρeiϕcosϕ 1 sinϕ exp
2π Z0 Z0 (cid:16) − κ (cid:17)mX=0(cid:18) 2κi (cid:19) −2cκnX=1(cid:18) κ (cid:19) !×
1κ+i
exp ρ2 . (2.19)
× −2 cκ
(cid:18) (cid:19)
The structure of the angular, i.e. the ϕ, integration indicates that only terms containing odd powers of ρ lead to
non-vanishing contributions to P(c,κ). From Appendix A it is clear that P(c,κ) comprises terms of order c1, c2,
c3, and so forth. For later reference, we will calculate the c1 and c2 contribution explicitly. To first order in c the
contribution of the double integral P(c,κ) to the two–point correlator X (r,λ) is (cf. Appendix A)
2
2
cκ cκ 1 Γ r
P(1)(c,κ)+P(1)(c, κ)= + = 1 , (2.20)
− 2(κ+i) 2(κ i) πr2+Γ2 Γ
− 1 (cid:18) 1(cid:19)
where we have used that Γ =πλ2. This resultcan be combined with the term 1/(π2λ2)=1/(πΓ ) c in Eq. (2.17)
1 1
≡
to give the full first–order contribution X(1)(r,λ) to X (r,λ),
2 2
5
1 Γ
X(1)(r,λ)= 1 (2.21)
2 πr2+Γ2
1
The second–order contribution X(2)(r,λ) is composed of four different combinations of terms from the perturbation
2
series (2.19), see Appendix A. Combined they can be written as
c2 1 2i 3
X(2)(r,λ)= + + +(κ κ)
2 4 (κ+i)2 (κ+i)3 (κ+i)4 ↔−
(cid:20) (cid:21)
1 Γ2 r2 c2 2i 3
= 1− + + +(κ κ) . (2.22)
−2π2(r2+Γ2)2 4 (κ+i)3 (κ+i)4 ↔−
1 (cid:20) (cid:21)
In the last line of the above equation we have distinguished between two contributions to X(2)(r,λ), a first piece
2
originating from the term with the quadratic denominator, and “extra” contributions arising from the terms with
higher–order denominators. The significance of this distinction will become clear in the following section, where we
essentially re-derive the present results from the point of view of the coset method.
III. COSET METHOD
The supersymmetric “coset” method4,5 has been widely employed in the last ten years or so to solve problems
of random matrix theory2. In the present context, it was already used in Ref.3 for a qualitative discussion. To
make contact with the well–established methods used in the literature over the years we will adopt normalization
conventions in this section which differ slightly from those used in Sec. II.
A. Basic definitions
We write the full Hamiltonian H in Eq. (1.1) as
H =H +H , (3.1)
0 1
thereby effectively absorbing the strength parameter α into the definition of H . The distribution p(0)(η) of the non-
1
vanishing elements Hii of H defines an important energy scale, the bandwidth B, because we have p(0)(Hii) B−1
0 0 0 ∝
forreasonsofnormalization. TheprobabilitydistributionfunctionP(1)(H )adoptedhereisaslightlymodifiedversion
N 1
of Eq. (2.3) and reads
N N2/2 N γ2
P(1)(H )=2N(N−1)/2 exp trH2 = Hij 2 = , (3.2)
N 1 2πγ2 −2γ2 1 ⇒ h| 1 | i N
(cid:18) (cid:19) (cid:18) (cid:19)
where the symbol ... denotes averaging over the H –ensemble (i.e. the GUE). The parameter γ introduced in this
1
h i
way is related to the strength parameter λ employed in Sec. II through
2 γ
λ= . (3.3)
N D
r
It is our purpose in this section to calculate the two–point correlator X (and later the number variance Σ2) for
2
the Hamiltonian (3.1) perturbatively in a certain suitable range of parameters. To define this range we note that
D = B/N is the level spacing of H and γ/√N the typical strength of the perturbing matrix elements. Hence we
0
find for the (dimensionless) induced spreading or Breit–Wigner width Γ
1
γ2 γ2
Γ D = . (3.4)
1
∝ ND B
Our calculation in this section is valid under the two conditions
γ Γ D γ2
1
= 0 , Γ 1 (but finite) . (3.5)
1
B B → ∝ BD ≫
r
6
The first of the conditions (3.5) means that the bandwidth B is infinitely larger than the Breit–Wigner width Γ D.
1
This ensures that neither B nor the level spacing D are appreciably changed by the perturbation H . The second
1
conditionensuresthatwedealwithoverlappingresonances,i.e. theoriginalstatesofH arethoroughlymixedtoform
0
theneweigenstatesofthecombinedHamiltonianH. Ourperturbativecalculationreliesonthesmallparameter1/Γ .
1
Again, as in the previous section, the parameter κ=r/Γ is held fixed so that the dimensionless energy difference r
1
scales like Γ .
1
With the Green function G =(x± H)−1 the density of states is given by
±
−
1
ρ(x)= (trG trG ) (3.6)
+ −
2π −
and hence the averageddensity–density correlationfunction takes the form
ρ (x+ω/2,x ω/2) = ρ(x+ω/2)ρ(x ω/2)
2 − − 0
1
= (cid:2)(cid:10) Re trG (x+ω/2)(t(cid:11)r(cid:3)G (x ω/2) trG (x ω/2)) . (3.7)
2π2 + − − − + − 0
(cid:2)(cid:10) (cid:11)(cid:3)
Here,wehaveintroducedthenotation[...] fortheH –average. Theangularbrackets ... denote,asabove,averaging
0 0
h i
over H . In the sequel we will typically perform the H –average in a rather early step of the calculation while the
1 1
H –averageis performed in the final stages. We note that
0
1 1
ρ (x+ω/2,x ω/2)= X (r)+ δ(r) (3.8)
2 − D2 2 D2
defines the relationof the density–density correlatorto the spectral two–pointfunction (2.9) considered earlier. Here
r =ω/D is the dimensionless energy difference as in Sec. II.
B. One–point function and density of states
Toillustrate ourprocedureandtodefine relevantenergyscaleswe startwiththe averageone–pointfunction. With
the supersymmetric generating functional
Z (q )=detg−1([x± H] 1 +1 q L /2)
± ± 2 N ± g
− ⊗ ⊗
=exp( trgln(1 1 +G q L /2)) (3.9)
N 2 ± ± g
− ⊗ ⊗
we can write the Green function as
∂
trG (x)= Z (q ) , (3.10)
± ± ±
∂
q± (cid:12)q=0
(cid:12)
(cid:12)
where Lg = diag( 1,1) is a two–dimensional (super–) matrix. (cid:12)With standard techniques4,5 the H1–average of the
−
generating functional can be easily calculated to give
N
Z (J) = d[σ]exp trg(σ2) trgln([x+ H ] 1 +1 [γσ+J]) . (3.11)
+ 0 2 N
− 2 − − ⊗ ⊗
Z (cid:18) (cid:19)
(cid:10) (cid:11)
We have introduced the source matrix J =qL /2, and σ is a 2 2 supermatrix as in Eq. (2.5). In contradistinction
g
×
to the previous section the further evaluation of Eq. (3.11) relies on the saddle–point approximation. With the
saddle–point equation
γ 1
σ = (3.12)
−N x+ Hjj +γσ
Xj − 0
and with the ansatz γσ =(Γ +iΓ )D/2=ΓD/2 we obtain
0 1
2γ2 1
Γ=
−ND x+ Hjj +DΓ/2
Xj − 0
2γ2 p(0)(η)
dη . (3.13)
≈− D x+ η+DΓ/2
Z −
7
Equation (3.13) can be solved in the limit Γ B and we arrive at the approximate expressions
| |≪
2γ2 p(0)(η)
Γ P dη
0
≈− D x η
Z −
γ2 γ2
Γ 2π p(0)(0)=2π (Breit–Wigner width) . (3.14)
1
≈ D BD
As immediate consequences we find
1 NΓD
trG =tr =
+ x+ H +DΓ/2 − 2γ2
0
−
(cid:10) (cid:11) 1 NΓ D
ρ(x) = Im trG = 1 =Np(0)(x) . (3.15)
−π + 1 2πγ2
(cid:10) (cid:11) (cid:10) (cid:11)
Inparticular,thelastequationmeansthattheH averageofthemeanleveldensitycanbetriviallyperformed,giving
0
[ ρ(x) ] =Np(0)(x).
0
h i
We see fromEq.(3.7)that weneedthe averageofa G G andofaG G term. Itis wellknownthatinthe large
+ − + +
N limit the H –averageof the product of two Green functions with infinitesimal increments of equal sign factorizes,
1
1 1
trG (x+ω/2)trG (x ω/2) = trG trG =tr tr . (3.16)
+ + − + + x++ω/2 H +DΓ/2 x+ ω/2 H +DΓ/2
0 0
− − −
(cid:10) (cid:11) (cid:10) (cid:11)(cid:10) (cid:11)
The same turns out to be true after averaging over H . This average is easily performed by simply integrating over
0
the independent random entries of the diagonal matrix H ,
0
p(0)(η) p(0)(η)
trG trG N(N 1) dη dη
+ + 0 ≈ − x++ω/2 η+DΓ/2 x+ ω/2 η+DΓ/2
Z − Z − −
(cid:2)(cid:10) (cid:11)(cid:3) p(0)(η)
+N dη
(x++ω/2 η+DΓ/2)(x+ ω/2 η+DΓ/2)
Z − − −
N(N 1)
= − trG trG trG trG . (3.17)
N2 + 0 + 0 ≈ + 0 + 0
(cid:2)(cid:10) (cid:11)(cid:3) (cid:2)(cid:10) (cid:11)(cid:3) (cid:2)(cid:10) (cid:11)(cid:3) (cid:2)(cid:10) (cid:11)(cid:3)
The term N in Eq. (3.17) is seen to vanish upon closing the contour in the lower half plane. We conclude that in
∝
order to obtain the density–density correlator (3.7) we can simply replace trG = iπ/D.
±
h i ∓
C. Two–point function and nonlinear σ model
Our next goal is to calculate the remaining quantity trG trG , which leads to the familiar nonlinear σ model,
+ −
h i
and its H –average. First, we generalize the definition of the supersymmetric generating functional (3.11),
0
∂ ∂ ∂ ∂
trG (x+ω/2)trG (x ω/2) = Z(q ,q ) = d[Q]exp( (Q))
+ − + −
− ∂ ∂ ∂ ∂ −L
q+ q− (cid:12)q±=0 q+ q− Z (cid:12)q±=0
(cid:10) (cid:11) (cid:10) (cid:11)(cid:12) (cid:12)
(cid:12) ω D (cid:12)
(Q)= trgln [x Hjj](cid:12)1 + Λ+ ΓQ+J . (cid:12) (3.18)
L − 0 4 2 2
j (cid:18) (cid:19)
X
The quantity Q is a 4 4 supermatrix parameterizingthe saddle–pointmanifold4,5 to whichthe integrationhas been
×
restricted by the saddle–point approximation. Furthermore, J = diag(q L ,q L )/2 = q P +q P (this defines
+ g − g + + − −
the projectors P and P ), and Λ=diag(1 , 1 ). To perform the derivatives in Eq. (3.18) it is useful to expand
+ − 2 2
− L
up to second order in the source matrix J,
(Q)= (Q)+ (Q)+ (Q)+... (3.19)
0 1 2
L L L L
The three contributions can be written as
8
= trgln[(x Hjj)1 +(ωΛ+Γ D+iΓ DQ)/2]
L0 − 0 4 0 1
j
X
= trg[g J]
1 j
L
j
X
1
= trg[g Jg J] , (3.20)
2 j j
L −2
j
X
where we have introduced the abbreviation g = [x+(ω/2)Λ Hjj +(Γ D+iΓ DQ)/2]−1. After performing the
j − 0 0 1
source term derivatives in Eq. (3.18) we arrive at a central equation of this section,
trG (x+ω/2)trG (x ω/2) = d[Q](S (Q)+S (Q))exp( (Q))
+ − 1 2 0
− −L
Z
(cid:10) (cid:11)
S (Q)= trg[g P ]trg[g P ]
1 j + k −
j,k
X
S (Q)= trg[g P g P ] . (3.21)
2 j + j −
j
X
This is the particular form of the zero–dimensional nonlinear σ model describing the crossover between Poisson and
GUE statistics.
D. Evaluation of the nonlinear σ model
Asalreadymentionedabove,theQ–integrationinEq.(3.21)isrestrictedtothesaddle–pointmanifoldfamiliarfrom
numerous previous applications of the nonlinear σ model2. It belongs to the peculiar features of superanalysis that
integralsof the type (3.21), which are derivedby a supersymmetric change ofvariables,containan extra “boundary”
contribution (sometimes referred to as the Efetov–Wegner term). This boundary contribution is generically given by
the value of the integrand at Q=Λ. In our present case we have (Λ)=0, S (Λ)=0, and
0 2
L
1 1
S (Λ)= . (3.22)
1 Xj,k x−H0jj +DΓ0/2+(ω+iΓ1D)/2 x−H0kk+DΓ0/2−(ω+iΓ1D)/2
The H –averageis performed as in Eq. (3.17),
0
p(0)(η) p(0)(η)
S (Λ)=N(N 1) dη dη +
1
− x η+DΓ /2+(ω+iΓ D)/2 x η+DΓ /2 (ω+iΓ D)/2
Z − 0 1 Z − 0 − 1
p(0)(η)
+N dη
(x η+DΓ /2+(ω+iΓ D)/2)(x η+DΓ /2 (ω+iΓ D)/2)
Z − 0 1 − 0 − 1
1 1
trG (x+ω/2) trG (x ω/2) + dz
≈ + 0 − − 0 D z2 (ω+iΓ D)2/4
Z − 1
(cid:2)(cid:10) (cid:11)(cid:3) (cid:2)(cid:10) (cid:11)(cid:3) 2π 1
= trG (x+ω/2) trG (x ω/2) + . (3.23)
+ 0 − − 0 D2Γ ir
1
−
(cid:2)(cid:10) (cid:11)(cid:3) (cid:2)(cid:10) (cid:11)(cid:3)
We recall that r=ω/D. The first term in Eq. (3.23 together with Eq. (3.17) forms the disconnected contribution to
the two–point correlation function X (r). Upon inserting the results (3.14) and (3.15) one can easily show that the
2
sumofdisconnectedcontributionstoX (r)reducestounity(forΓ 0). ThesecondterminEq.(3.23)ispartofthe
2 0
≈
connected contributions. In the following we calculate the remaining connected terms by treating the saddle–point
integration perturbatively.
We startfromEq.(3.21). The generalstrategywillbe to expressboththe exponentandthe pre-exponentialterms
inEq.(3.21)intermsoftheindependentvariablesofthesaddle–pointmanifold. Itturnsoutthatwehavetoconsider
only quadraticterms in the exponentso that the integrationbecomes Gaussian andtherefore trivial. Concerning the
H –averageit is important to note that those terms involving correlations between the matrix elements of H in the
0 0
exponent and in the pre-exponentialterms can be neglected. In fact, it turns out that the pre-exponentialterms and
the exponentcanbe independentlyaveragedoverH . This amountstoatremendoussimplificationofourcalculation.
0
9
Amorethoroughdiscussionoftheseissuesaswellasanumberoftechnicalstepsomittedhereforclaritycanbefound
in Appendix B.
We express the deviation of Q from the diagonal value Λ in terms of the quantity ∆Q= Λ,Q /2 1, where .,.
{ } − { }
denotes the anticommutator. Then the H –averageof (Q) can be expressed as (see Appendix (B))
0 0
L
π rΓ
1
(Q) = i trg(∆Q) . (3.24)
L0 0 − 2Γ ir
1
−
(cid:2) (cid:3)
For our present purposes, a suitable parameterization of the saddle–point manifold is given by
Q=T−1ΛT =ΛT2
T = 1+R2+R
p0 t
R= , (3.25)
t 0
(cid:20) (cid:21)
where t and t are 2 2 supermatrices representing the unrestricted “free” variables of the saddle–point manifold. It
×
follows for that
0
L
rΓ
1
(Q) = 2πi trg(tt) . (3.26)
L0 0 − Γ ir
1
−
(cid:2) (cid:3)
Likewisewehavetoexpressthe H –averagesofS andS intermsoftandt. Again,thedetailedderivationhasbeen
0 1 2
deferred to Appendix B,
π2 Γ4
S =4 1 trg[ttP ]trg[ttP ]
1 0 D2(Γ ir)4 + −
1
−
(cid:2) (cid:3) π (iΓ )2
1
S = 4 trg[tP tP ] . (3.27)
2 0 − D2(Γ ir)3 + −
1
−
(cid:2) (cid:3)
Using Eqs. (3.21), (3.26), and (3.27) the remaining Gaussian integrations can be easily performed. To this end it is
usefultoemploycertainWick–typecontractionrulesasexplainedinRef.23. Alongtheselinesweobtainthefollowing
“diffusion” contribution from a perturbative treatment of the saddle–point integral,
trG trG diff = 1 Γ21 + 2iΓ1 = 1 1 . (3.28)
+ − 0 −(Dr)2(Γ ir)2 D2r(Γ ir)2 −(Dr)2 − D2(Γ ir)2
1 1 1
− − −
(cid:2)(cid:10) (cid:11)(cid:3)
The individual contributions of S and S to this result correspond to the first and the second term on the r.h.s. of
1 2
the first line, respectively.
In summary, the total (H and H ) average of trG trG consists of three terms, the disconnected part, the
0 1 + −
boundary term (3.23), and the diffusion contribution (3.28),
trG trG =C +C +C . (3.29)
+ − 0 disc bound diff
By definition, see Eqs. (3.7) and (3(cid:2).(cid:10)8), we have(cid:11)(cid:3)
D2
X (r)= Re(C +C ) . (3.30)
2 2π2 bound diff
If we insert our results for C and C and generalize the latter to arbitrary (GUE, GOE, GSE) symmetry by
bound diff
multiplying it with 2/β (β =1,2,4) we arrive at
1 1
X (r)=1 +X(1)(r)+X(2)(r)
2 − β (πr)2 2 2
1 Γ
(1) 1
X (r)=
2 πΓ2+r2
1
1 Γ2 r2
X(2)(r)= 1− . (3.31)
2 −π2β(Γ2+r2)2
1
The first term in this result is the perturbative (divergent) expression for the two–point correlation function of the
Gaussian ensembles. It can be replaced by the full non-perturbative result, i.e. by Eq. (2.18) in the case of the
GUE.Comparisonof the first andsecondordercontributionsto X (r), X(1)(r) andX(2)(r), obtainedso far with the
2 2 2
corresponding results of the gradedeigenvalue method in Eqs. (2.21) and (2.22), reveals missing terms in the present
calculation. We conclude this section with a discussion of the origin of this discrepancy.
10