Energy level statistics in weakly disordered systems: from quantum to diffusive regime

12 Pages·0.16 MB·English

by  Naohiro Mae

Energy level statistics in weakly disordered systems:
from quantum to diffusive regime
Naohiro Mae and Shinji Iida
Faculty of Science and Technology, Ryukoku University, Otsu 520-2194, Japan
(January 21, 2003)
3
Wecalculate two-point energy level correlation function in weakly disorderd metallic grain with
0
taking account of localization corrections to the universal random matrix result. Using supersym-
0
metric nonlinear σ model and exactly integrating out spatially homogeneous modes, we derive the
2
expression valid for arbitrary energy differences from quantum to diffusive regime for the system
n
with broken time reversal symmetry. Our result coincides with the one obtained by Andreev and
a
Altshuler [Phys. Rev. Lett. 72, 902 (1995)] where homogeneous modes are perturbatively treated.
J

1 PACS number(s): 05.40.-a, 72.15.Rn, 05.45.Mt
2

]
n
n I. INTRODUCTION
-
s
i Much research interest has been attracted by the universal properties of quantum systems with randomness; the
d
randomnessarisesfromvarioussourcessuchascomplexityofthe systemforthe caseof,e.g.,complex nuclei,stochas-
.
at ticity for disorderd conductors and instability of classical trajectories for chaotic systems [1]. If wave functions have
m enoughtime todiffuse throughoutthe system,the spatialstrucureofthe systembecomesimmaterial. Forsuchcases,
thesystemiscalledtobelongtotheergodicregimewherethestatisticalpropertieshavebeenextensivelystudied. For
-
d example, correlationfunctions of the energy levels [2] or the scattering matrices [3,4]are shown to have the universal
n forms where specific features of individual systems are represented by a few parameters [1]. These universal forms
o
can be calculated with use of standard random matrix models belonging to pertinent Gaussian ensembles.
c
If the relevant time scale is comparable with the diffusion time, we need to turn to spatial dependent parts of
[
the correlation functions which are not included in the universal random matrix results. These spatial dependent

2 parts for weaklydisorderedsystems arethe subject ofthis paper. The weaklydisorderdsystems are characterizedby
v
the condition that g 1 where g is the dimensionless conductance. The spatial dependent parts of the correlation
4 ≫
functions are expected to appear as higher order corrections with respect to 1/g to the universal random matrix
4
results. The two-point energy level correlation function R(s) (see Eq. (1.1)) with corrections up to 1/g2 has been
4
0 calculatedbyseveralauthors: Intheirseminalwork,usingdiagrammaticperturbationtheory,AltshulerandShklovskii
1 [5] obtained the smooth part of R(s) valid for large energy differences s 1. Deviations from the universalstatistics
≫
2 are determined by the diffusive classical dynamics of the system. However, diagrammatic perturbation does not give
0 an oscillatory part of R(s) characteristic of the universal random matrix result.
t/ The 1/g2 corrections including the oscillatory part were first obtained by Kravtsov and Mirlin (KM) [6] using the
a
supersymmetric σ model [7], the effective theory of a particle moving through a disorderd conductor. Their result is,
m
however, valid only for small energy differences s g [8]. Later, also using the supersymmetric σ model, Andreev
- and Altshuler (AA) [9] claimed that the oscillator≪y part can be obtained from perturbation around a novel saddle
d
n point in additonto the ordinaryone. Their result relies on perturbationand hence is restrictedto the regionof large
o energy differences s 1. The AA’s result was subsequently reproduced with use of the replica [10] and Keldysh
≫
c [11] σ model. In quantum chaotic systems, the analysis based on the semiclassical periodic orbit theory [12] and the
v: ballistic σ model [13] led to similar results to the AA’s.
i Inalltheseresults,deviationsfromtheuniversalformisexpressedintermsofeigenvaluesofacertainoperatororits
X
spectraldeterminant. This operatorisadiffusionoperatorforthe caseofdisorderdconductorsoraPerron-Frobenius
r operator for quantum chaotic systems [14]. We can surmise from these results that universal forms also exist in the
a
deviations from universal random matrix resuls where parameters representing system specific features are now the
spectra of this operator.
At present the available results are yet insufficient because KM’s result is valid for s g and AA’s one is valid for
≪
s 1. The method to calculate anexpressionvalid for entirerange ofenergydifferences is desirable [15]. This paper
pr≫esentsonesuchmethod. ExtendingtheKM’sprocedure,wecalculate1/g2 correctionstothetwo-pointenergylevel
correlation function
R(s)=∆2 ρ(E)ρ(E+ω) (1.1)
h i
1
validforarbitraryω,where s=ω/∆,∆isthe meanlevelspacing,ρ(E) isthe density ofstates atenergyE,and ...
h i
denotes averagingover realizationof the random potential. We consider disorderdsystems with brokentime reversal
symmetrybecausethepertinentrandommatrixensemble(unitaryensemble)isthesimplestamongthethreeclassical
ensembles (unitary, orthogonal, and symplectic). We follow the KM’s procedure but do not use ω expansion which
limits the range of their result’s validity to small ω. As a result, we verify that the AA’s expression gives the correct
g 1 asymptotic at arbitrary ω for unitary ensemble. For the unitary case, invoking a mathematical theorem, it
≫
is implicitly shown [16] that AA’s result is exact in spite of their use of perturbation. Here, we want to show it by
explicit calculation which we expect to be applicable in other ensembles with some modifications.
II. MODEL AND METHOD
The supersymmetric σ model is a field theory whose variables Q are supermatrices [7]. Statistical properties of a
particlemovingthroughrandompotentialscanbe derivedfromageneratingfunctionalofQ. TheexpressionforR(s)
reads
2
1
R(s)= Re dQe−S(Q) d~rstrkΛQ(~r) , (2.1)
16V2
Z (cid:20)Z (cid:21)
π D
S(Q)= d~rstr ( Q(~r))2+2is+ΛQ(~r) . (2.2)
4V ∆ ∇
Z (cid:20) (cid:21)
Here str denotes the supertrace, Q(~r)=T(~r)−1ΛT(~r) is a 4 4 supermatrix with T(~r) belonging to the coset space
×
U(1,12),Λ=diag(1,1, 1, 1),k =diag(1, 1,1, 1),V is the system volume, D is the classicaldiffusion constant,
s+ =s|+i0+, and 0+ is−posi−tive infinitesimal−. We u−se the notaions of Ref. [7] everywhere.
TheordinarysaddlepointisQ=ΛandAA’snoveloneisQ= kΛ. Theexpansionaroundthesesaddlepointsand
−
calculation of the functional integralby Gaussianapproximationgives AA’s result. The quadratic term of the action
S(Q) defines the diffusion propagator in terms of which a pertubation series is constructed by Wick’s contraction.
When s goes to zero, the diffusion propagatordiverges for spatially homogeneous modes (which we hereafter call the
zero-momentum modes, or zero modes for short, because their momentum is zero). Thus their approachis restricted
to the region s 1.
≫
KM have overcome the dfficulty of zero mode divergence by treating the zero modes and the other spatially
inhomogeneous modes with non-zero momenta (heareafter called non-zero modes) separately. The matrix Q(~r) is
decomposed in the follwing way [6]:
Q(~r)=T −1Q˜(~r)T , (2.3)
0 0
whereT isaspatiallyhomogeneousmatrixfromthecosetspaceandQ˜(~r)describesallnon-zeromodes(seeEqs.(2.4),
0
(2.5)). After perturbative calculation of the contributions by non-zero modes, there remains a defnite integral over a
fewzeromodevariables,whichcanbeevaluatedwithoutuseofperturbation. Thisamalgamationofbothperturbative
and non-perturbative calculation, in principle, could give the weak localizationcorrectionsvalid for entire range of s.
In the course of calculation, however, assuming small ω, KM expand the exponential of the energy term (the second
term of the action (2.2)). Thus their result is restricted to s g.
≪
The difficulty arising when the energy term is kept on the shoulder of exponent is that the propagators of non-
zero modes become dependent on zero mode variables. Then after eliminating non-zero modes by perturbation, the
resulting zero mode integral contains infinite sums and products of non-zero mode propagators (see Eq.(5.5)) as an
integrand. Hencethezeromodeintegral,thoughstillbeingdefniteoverafewvariables,doesnotseemfeasible. Inthis
paper, we show this is not the case: by replacing the order of integration, it turns out that we can first integrate out
the zero mode exactly. Then integration over the non-zero modes by perturbation can be done without any trouble.
We resume a summary of notations: When ∆ E (where E =D/L2 and L is the system size), the matrix Q˜(~r)
c c
≪
fluctuatesonlyweaklyaroundΛ. Thenthefluctuationscanbetreatedbyperturbationwiththe expansionparameter
1/g (where g =E /∆). A convinient parametrization is
c
Q˜(~r)=(1 W(~r)/2)Λ(1 W(~r)/2)−1, (2.4)
− −
where
0 W (~r)
W(~r)= 12 , (2.5)
W (~r) 0
21
(cid:18) (cid:19)
2
†
W (~r) = kW (~r) . W (~r) is expanded as W (~r) = φ (~r)W where φ is an eigenfunction of the diffusion
21 12 12 12 q~6=0 q~ 12q~ q~
operator D 2 with an eigenvalue D ~q 2. φ (~r) constitute a complete orthonormal set and φ (~r) = 1/√V. Thus
− ∇ | | { q~ } P 0
d~rW(~r) d~rW(~r)φ (~r) = 0. The Jacobian of the transformation Eqs. (2.3) and (2.4) from the variable Q to
0
∝
T ,W is
0
R{ } R
1 2
J(W)=1 d~rdr~′ strW (~r)W (r~′) +O(W5). (2.6)
− 16V2 12 21
Z (cid:16) (cid:17)
The derivation is given in Appendix A.
The spatially homogeneous supermatrix T is parametrized in a quasi-diagonalizedform
0
T =U−1Tˆ U (2.7)
0 0
with ‘eigenvalue’ matrix Tˆ given by
0
cosθˆ isinθˆeiϕˆ
Tˆ = 2 2 , (2.8)
0 isinθˆe−iϕˆ cosθˆ !
2 2
θ 0 ϕ 0
θˆ= F , ϕˆ= F , (2.9)
0 iθ 0 ϕ
B B
(cid:18) (cid:19) (cid:18) (cid:19)
where 0 θ < , 0 θ π, 0 ϕ 2π, 0 ϕ 2π. The ‘diagonalizing’ matrix U is given by
B F B F
≤ ∞ ≤ ≤ ≤ ≤ ≤ ≤
v 0
U = 1 , (2.10)
0 v
2
(cid:18) (cid:19)
0 ξ 0 ξ
v =exp 1 , v =expi 2 , (2.11)
1 ξ∗ 0 2 ξ∗ 0
(cid:18)− 1 (cid:19) (cid:18)− 2 (cid:19)
where ξ and ξ are anticommuting variables. For this parametrization, the measure dT is given by
1 2 0
sinhθ sinθ
dT = B F dθ dθ dϕ dϕ dξ dξ∗dξ dξ∗. (2.12)
0 (coshθ cosθ )2 B F B F 1 1 2 2
B F
−
III. CHANGING THE VARIABLES
Tosimplifythezeromodeintegration,wechangethevariables. Substitutingthedecomposition(2.3)intoEq.(2.1),
we obtain
2
str ( Q(~r))2 =str Q˜(~r) , (3.1)
∇ ∇
(cid:16) (cid:17)
d~rstrΛQ(~r)=strΛT −1YT , (3.2)
0 0
Z
d~rstrkΛQ(~r)=strkΛT −1YT , (3.3)
0 0
Z
where Y = d~rQ˜(~r). We make the supermatrix Y a block diagonal form as follows (for detail see Appendix B):
R eXYe−X =Qˆ, (3.4)
where X is a block off-diagonal supermatrix and
3
1 1
Qˆ =Λ V + d~rW2(~r)+ d~rW4(~r)+O(W6) . (3.5)
2 8
(cid:18) Z Z (cid:19)
Up to 1/g2 order calculated here, eXT can be replaced with T (see Appendix B). Thus in Eqs. (3.2) and (3.3), Y
0 0
can be replaced with Qˆ.
To eliminate the anticommuting variables of the zero modes in the action S(Q), we change the variables as
UWU−1 W. Eventually, we get
→
d~rstrΛQ(~r)=strΛTˆ−1QˆTˆ , (3.6)
0 0
Z
d~rstrkΛQ(~r)=strkΛU−1Tˆ−1QˆTˆ U. (3.7)
0 0
Z
IV. INTEGRATION OVER THE ZERO-MOMENTUM MODE VARIABLES
Expanding the pre-exponential term of Eq. (2.1) in the anticommuting variables of the zero modes, we obtain
2 2 2
strkΛU−1Tˆ−1QˆTˆ U = strkΛTˆ−1QˆTˆ 2ξ∗ξ ξ∗ξ strΛTˆ−1QˆTˆ +... (4.1)
0 0 0 0 − 1 1 2 2 0 0
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
where we have used that strQˆ = 0 and the dots indicate terms which vanish after integration of the zero mode
anticommuting variables. We write the contribution by the first (second) term of Eq. (4.1) to R(s) as R (s) (R (s)).
1 2
WithuseofParisi-Sourlas-Efetov-Wegnertheorem[17],integrationofzeromodevariablesinR (s)becomesthevalue
1
of its integrand at θ =θ =0:
B F
1 2
R (s)= Re dW J(W)e−S(W) strkΛQˆ +O(1/g3), (4.2)
1 16V2
Z (cid:16) (cid:17)
where
π D 2
S(W)= str d~r Q˜(~r) +2is+ΛQˆ . (4.3)
4V ∆ ∇
(cid:20) Z (cid:16) (cid:17) (cid:21)
Integrating the zero mode anticommuting variables, we obtain
R (s)= 1 Re dW dθ dθ J(W) sinhθBsinθF e−S(Tˆ0,W) strΛTˆ−1QˆTˆ 2+O(1/g3), (4.4)
2 −8V2 B F (coshθ cosθ )2 0 0
Z B− F (cid:16) (cid:17)
π D 2
S(Tˆ ,W)= str d~r Q˜(~r) +2is+ΛTˆ−1QˆTˆ . (4.5)
0 4V ∆ ∇ 0 0
(cid:20) Z (cid:16) (cid:17) (cid:21)
Here,
strΛTˆ−1QˆTˆ =coshθ strP(B)ΛQˆ+cosθ strP(F)ΛQˆ, (4.6)
0 0 B F
where P(B) = (1 k)/2 and P(F) = (1 + k)/2 and 1 is an unit matrix. Using coshθ = λ , cosθ = λ ,
B B F F
−
strP(B)ΛQˆ = 2V (1+A), and strP(F)ΛQˆ =2V (1+B), we obtain
−
1 πD 2
R (s)= Re dW J(W)exp d~rstr Q˜(~r) I(s,W)+O(1/g3), (4.7)
2
2 −4V∆ ∇
Z (cid:20) Z (cid:16) (cid:17) (cid:21)
where
∞ 1
I(s,W)= dλ dλ f(λ ,λ )g(λ ,λ ), (4.8)
B F B F B F
Z1 Z−1
4
f(λ ,λ )= 1 eiπs+(λB−λF), (4.9)
B F (λ λ )2
B F
−
g(λ ,λ )=eiπs+(λBA−λFB)[λ (1+A) λ (1+B)]2. (4.10)
B F B F
−
We introduce an indefinite integral of f(λ ,λ ): ∂2 F(λ ,λ )=f(λ ,λ ),
B F ∂λF∂λB B F B F
F(λ ,λ )= λF db ∞daeiπs+(a−b) 1
B F − (a b)2
Z−∞ ZλB −
= ∞dxeiπs+(λB−λF)x 1 1 , (4.11)
− x − x2
Z1 (cid:18) (cid:19)
where we have used the transformation (a b)−2 = ∞te−t(a−b)dt and iπs+ t = iπs+x. Then partial integration
− 0 −
gives
R
I(s,W)= [F(λ ,λ )g(λ ,λ )]λB=∞ λF=1 ∞dλ F(λ ,λ ) ∂ g(λ ,λ ) λF=1
h B F B F λB=1 iλF=−1−(cid:20)Z1 B B F ∂λB B F (cid:21)λF=−1
1 ∂ λB=∞ ∞ 1 ∂2
dλ F(λ ,λ ) g(λ ,λ ) + dλ dλ F(λ ,λ ) g(λ ,λ ). (4.12)
F B F B F B F B F B F
− ∂λ ∂λ ∂λ
(cid:20)Z−1 F (cid:21)λB=1 Z1 Z−1 B F
After λ-integration, we obtain
∞ λB=∞ λF=1
I(s,W)= eiπs+(λBA−λFB) dxeiπs+(λB−λF)x (λ ,λ ,x)
B F
−"(cid:20) Z1 I (cid:21)λB=1 #λF=−1
∞ ∞
=eiπs+(A−B) dx (1,1,x) eiπs+(A+B) dxe2iπs+x (1, 1,x), (4.13)
I − I −
Z1 Z1
where
1 1
(λ ,λ ,x)= Aˆ−1Bˆ−1[λ (1+A) λ (1+B)]2
I B F x − x2 B − F
(cid:18) (cid:19)
2 1 1
Aˆ−2Bˆ−2 Aˆ(1+B)+Bˆ(1+A) [λ (1+A) λ (1+B)]
−iπs+ x2 − x3 B − F
(cid:18) (cid:19) h i
2 1 1
+ Aˆ−3Bˆ−3 Aˆ2(1+B)2+AˆBˆ(1+A)(1+B)+Bˆ2(1+A)2 . (4.14)
(iπs+)2 x3 − x4
(cid:18) (cid:19) h i
Here,Aˆ=1+A andBˆ =1+B. FromEq.(4.13)wecanreadthatthepropagatordonotincludezeromodevariables.
x x
Therefore the difficulty of KM’s approach mentioned in Sec. II is resolved. We write the contribution by the first
(1,1) (1,−1)
(second) term of Eq. (4.13) to R (s) as R (s) (R (s)).
2 2 2
V. INTEGRATION OVER THE NON-ZERO-MOMENTUM MODE VARIABLES
In order to calculate the remaining integralover the non-zero mode variables W with use of the Wick theorem, we
need to know the contraction rules for the action
π D
S = d~rstr ( W(~r))2+is+(λ P(B)+λ P(F))W(~r)2 . (5.1)
0 B F
4V −∆ ∇
Z (cid:20) (cid:21)
The result is summarized as
1
str W(~r) W(r~′) = Π(~r,r~′)gg′[strP(g) strP(g′) strΛP(g) strΛP(g′) ], (5.2)
h X Y i −2 X Y − X Y
g,g′
X
5
1
str W(~r)str W(r~′) = Π(~r,r~′)gg′[strP(g) P(g′) strΛP(g) ΛP(g′) ], (5.3)
h X Y i −2 X Y − X Y
g,g′
X
where and are arbitrary 4 4 supermatrices, g and g′ are B or F,
X Y ×
... = −1(s,λ ,λ ) dW (...)e−S0, (5.4)
B F
h i D
Z
Π(q;λ ,λ )Π(q;λ ,λ )
B B F F
(s,λ ,λ )= , (5.5)
D B F Π(q;λ ,λ )2
q~Y6=~0 B F
−1
Π(q;λg,λg′)= 2V Dq2 is+λg +λg′ , (5.6)
π ∆ − 2
(cid:18) (cid:19)
Π(~r,r~′)gg′ = φq~(~r)φq∗~(r~′)Π(q;λg,λg′). (5.7)
Xq~6=~0
For R (s), using the contraction rules with λ =1 and λ =1 we obtain
1 B F
1 1 2
R (s)= Re 16V2+ d~rstrkW(~r)2 +O(1/g3)
1 16V2 4
* (cid:18)Z (cid:19) +(λB,λF)=(1,1)
1
=1+ Re Π(q;1,1)2+O(1/g3)
8V2
Xq~6=~0
1 d2
=1 ln (s,1, 1)+O(1/g3). (5.8)
− 4π2ds2 D −
In order to get R (s), we need to estimate the order of the each term of Eq. (4.14) with use of the fact that (i)
2
W2 O(1/g) for all s, (ii) the property of the exponential integral
∼
∞ e−ax
E (a)= dx (5.9)
n xn
Z1
where Rea>0 and n N, and (iii) KM’s result [6]: for s<1,
∈
∞
C (s)
n
R(s)= (5.10)
gn
n=0
X
where C (s) is O(1).
n
A. (λB,λF)=(1,1) point
Expanding (1,1,x) in W, there appears a term proportional to 1/x, (A B)2/x. The x-integration of this term,
I −
at first glance, seems to diverge. Actually it causes no problem becaue it vanishes after W-integration. In order to
obtain the corrections up to 1/g2, it is sufficient to expand the remaing terms up to the 4th order of W for entire
range of s [18];
1
(1,1,x)= (A B)2
I −x2 −
2 1 1 1 4 3
2 (A B)+ + (A2 B2)
−iπs+ x2 − x3 − x2 − x3 x4 −
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
2 1 1 1 3 2 1 9 18 10
+ 3 +3 + (A+B)+ + (A2+B2+AB)
(iπs+)2 x3 − x4 x3 − x4 x5 x3 − x4 x5 − x6
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
+O(1/g3). (5.11)
6
After x-integration, we obtain
∞ 2 1
(1,1,x)dx= (A B)2 (A B)+ +O(1/g3). (5.12)
I − − − iπs+ − (iπs+)2
Z1
After W-integration, we obtain
1
R(1,1)(s)= . (5.13)
2 −2π2s2
B. (λB,λF)=(1,−1) point
For s 1, E ( i2πs+) in Eq. (4.13) is O(1). Accordingly, by the same reason as the case of (λ ,λ ) = (1,1)
n B F
≥ −
point, it is sufficient to expand (1, 1,x) up to the 4th order of W for entire range of s again.
I −
1 1 4 1 1 6 1 1
(1, 1,x)=4 + +
I − x − x2 s˜ x2 − x3 s˜2 x3 − x4
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
1 2 1 2 2 5 3 6 1 3 2
+4 + + + + + (A+B)
x − x2 x3 s˜ x2 − x3 x4 s˜2 x3 − x4 x5
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
1 5 7 3 4 1 7 12 6 6 1 9 18 10
+ + + + + + (A+B)2
x − x2 x3 − x4 s˜ x2 − x3 x4 − x5 s˜2 x3 − x4 x5 − x6
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
1 1 4 1 3 2 2 1 9 18 10
+ + + + (A B)2
x3 − x4 − s˜ x3 − x4 x5 s˜2 x3 − x4 x5 − x6 −
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
+O(1/g3) (5.14)
where s˜= i2πs+. Using the relation E (s˜)+ nE (s˜)=e−s˜/s˜, we obtain
− n s˜ n+1
∞ e−s˜ 1 ∞ 1 3
dxe−s˜x (1, 1,x)=4 + e−s˜+2 dxe−s˜x (A B)2+O(1/g3), (5.15)
I − s˜2 s˜2 x3 − x4 −
Z1 (cid:20) Z1 (cid:18) (cid:19)(cid:21)
eiπs+(A+B) ∞dxei2πs+x (1, 1,x)=eiπs+(A(2)+B(2)) ei2πs+
I − (iπs+)2
Z1
+eiπs+(A(2)+B(2)) ei2πs+(A(4)+B(4))+ F(s) (A(2) B(2))2
" iπs+ 4(iπs+)2 − #
+O(1/g3), (5.16)
where (s) = ei2πs+ +2 ∞ dxei2πs+x 1 3 . The contribution by the second term of Eq. (5.16) is O(1/g3),
F 1 x3 − x4
because for s g it is obvious and for s<g it is found after W-integration.
≥ R (cid:0) (cid:1) 2
By the same way,the contributions by O(W4) terms emerging from Q˜(~r) andthe jacobianJ(W) turn out to
∇
be O(1/g3). Finally, we obtain (cid:16) (cid:17)
R(1,−1)(s)= 1Re dW J(W)exp πD d~rstr Q˜(~r) 2 eiπs+(A(2)+B(2)) ei2πs+ +O(1/g3)
2 −2 −4V∆ ∇ (iπs+)2
Z (cid:20) Z (cid:16) (cid:17) (cid:21)
cos2πs
= (s,1, 1)+O(1/g3). (5.17)
2π2s2 D −
VI. RESULT AND SUMMARY
Equations (5.8), (5.13), and (5.17) give
7
1 d2 1 cos2πs
R(s)=1 ln (s,1, 1) + (s,1, 1)+O(1/g3)
− 4π2ds2 D − − 2π2s2 2π2s2 D −
1 d2 cos2πs
=1 ln ˜(s)+ ˜(s)+O(1/g3), (6.1)
− 4π2ds2 D 2π2 D
where ˜(s) = (s,1, 1)/s2 is the spectral determinant of the classical diffusion operator, in agreement with
D D −
Refs. [9–11]. Here the contribution of the (λ ,λ )=(1,1)
B F
1 d2 1
ln ˜(s)= Re Π(~q,1,1)2 (6.2)
−4π2ds2 D 8V2
q~
X
is the ordinary perturbative result [5].
In summary, using KM’s separate treatment of the zero and non-zero modes, performing the integration over the
zeromodes exactlyandsubsequentintegrationoverthe non-zeromodes perturbatively,wefind the expressionfor the
two-pointenergylevelcorrelationfunctionvalidforarbitraryenergydifferencesupto1/g2 orderforunitaryensemble.
This expression is same as the one [9–11] obtained in a saddle-point approximation which in general valid only for
ω ∆ [19]. By explicit calculation, we verify that the exactness of the saddle-point answer for unitary ensemble,
≫
which is guaranteed in the ergodic regime due to the Duistermaat-Heckman theorem [16], holds even in the diffusive
regime. Since this is a specific feature of the unitary ensemble, it is very interesting to investigate the expressions for
the other (orthogonal, symplectic, etc.) ones.
The reasonwhy AA’s novelsaddle point appearsis as follows: The zero-mode integralis carriedout exactly. Then
the remainingexpressionfornon-zeromodesisevaluatedatterminalpointsoftheintegralregion,1 λ < , 1
B
≤ ∞ − ≤
λ < 1. These terminal points precisely correspond to the ordinary (λ = λ = 1) and AA’s novel saddle point
F B F
(λ =1,λ = 1). This correspondencebetweenthe terminalpoints ofzero-modevariablesandsaddle points seems
B F
−
to hold for the orthogonalensemble. Hence, for the orthogonalcase,a similar wayofcalculationmaywork: although
exact integration of zero mode variables is no longer probable, the iteration of partial integration enables one to
evaluate the integralonthe terminalpoints up to a necessaryorderof 1/g,if the additionalterms generatedby these
iterations are only higher order term of 1/g. For the symplectic case, the correspondence seems more subtle because
the novel saddle points are not isolated but consist of a saddle point maifold. Further investigation will be necessary
for the symplectic case.
ACKNOWLEDGMENTS
We would like to thank K. Takahashi for valuable comments. S.I. wish to acknowledge helpful discussions with
participantsinthe workshop“ quantumchaos,theoryandexperiment2002”heldatYukawainstitute fortheoretical
physics, Kyoto, Japan, where part of this work was presented.
APPENDIX A: CALCULATION OF THE JACOBIAN
WegivethecalculationoftheJacobianJ ofthetransformationQ W,T because,tothebestofourknowledge,
0
→{ }
there is no explicit expression of it in literature. It is calculated by the following Gaussian integral :
d(δW)d(δT )exp d~rstr (δQ(~r))2 =J−1. (A1)
0
−
Z (cid:20) Z (cid:21)
Here δQ(~r) is variation of Eq. (2.3) :
δQ(~r)=T−1 δQ˜(~r)+ Q˜(~r),δT′ T , (A2)
0 0 0
(cid:16) h i(cid:17)
where δT′ =δT T−1. Then,
0 0 0
2 2
d~rstr (δQ(~r))2 = d~rstr δQ˜(~r) +2 d~rstrδQ˜(~r) Q˜(~r),δT′ + d~rstr Q˜(~r),δT′ . (A3)
0 0
Z Z (cid:16) (cid:17) Z h i Z h i
The integrand of Eq. (A1) is expanded up to 4th order in W.
8
The δW dependent part of the integrand is
2
exp d~rstr δQ˜(~r) 2 d~rstrδQ˜(~r) Q˜(~r),δT′
− − 0
(cid:20) Z (cid:16) (cid:17) Z h i(cid:21)
=exp d~rstr(δW)2
(cid:20)Z (cid:21)
1 2
1+ d~rstrW2(δW)2+2 d~rstrX (δW)δT′
×( 2Z (cid:18)Z 1 0(cid:19)
+2 d~rstrX (δW)δT′ d~rstrX (δW)δT′
1 0 2 0
(cid:18)Z (cid:19)(cid:18)Z (cid:19)
1 1 1 2
+ d~rstrW4(δW)2+ d~rstr(W2δW)2+ d~rstrW2(δW)2
8 16 8
Z Z (cid:20)Z (cid:21)
+ d~rstrX (δW)δT′ d~rstrX (δW)δT′
1 0 3 0
(cid:18)Z (cid:19)(cid:18)Z (cid:19)
2 4
1 2
+ d~rstrX (δW)δT′ + d~rstrX (δW)δT′
2 2 0 3 1 0
(cid:18)Z (cid:19) (cid:18)Z (cid:19)
2
+ d~rstrW2(δW)2 d~rstrX (δW)δT′ +O(W5) , (A4)
1 0
(cid:20)Z (cid:21)(cid:18)Z (cid:19) )
where
n
X (δW)= ( 1)kWkδWWn−k. (A5)
n
−
k=0
X
The δW independent part is
2
exp d~rstr Q˜(~r),δT′
− 0
(cid:18) Z h i (cid:19)
=exp[2Vstr(δT′ ΛδT′Λ)δT′]
0− 0 0
1 2 d~rstrΛδT′Λ W2δT′ WδT′W
× − 0 0− 0
(cid:26) Z
(cid:0) (cid:1)
d~rstrΛδT′Λ W3δT′ 2W2δT′W
− 0 0− 0
Z
1 (cid:0) (cid:1)
d~rstrΛδT′Λ W4δT′ 2W3δT′W +W2δT′W2
−2 0 0− 0 0
Z
(cid:0) 2 (cid:1)
+2 d~rstrΛδT′Λ W2δT′ WδT′W +O(W5) . (A6)
(cid:20)Z 0 0− 0 (cid:21) )
(cid:0) (cid:1)
The integral of the δW dependent part over δW can be calculated by using the Wick theorem and the contraction
rules:
1 1
str δW(~r) δW(r~′) = δ(~r r~′) (str str strΛ strΛ ), (A7)
δW
h X Y i 4 V − − X Y − X Y
(cid:18) (cid:19)
1 1
str δW(~r)str δW(r~′) = δ(~r r~′) (str strΛ Λ ), (A8)
δW
h X Y i 4 V − − XY − X Y
(cid:18) (cid:19)
where and are arbitrary supermatrices and
X Y
... = d(δW)(...)exp d~rstr(δW)2 . (A9)
δW
h i
Z (cid:20)Z (cid:21)
9
The result is
2
d(δW) exp d~rstr δQ˜(~r) 2 d~rstrδQ˜(~r) Q˜(~r),δT′
− − 0
Z (cid:20) Z (cid:16) (cid:17) Z h i(cid:21)
=1+ d~rstr (δT′+ΛδT′Λ) W2δT′ WδT′W + d~rstrΛδT′Λ W3δT′ 2W2δT′W
0 0 0− 0 0 0− 0
Z Z
1 d~rstr W2δT′ 2+ 1 (cid:0) d~rstrΛδT′Λ 4W(cid:1)4δT′ 8W3δT′W +(cid:0) 5W2δT′W2 (cid:1)
−8 0 8 0 0− 0 0
Z Z
1 (cid:0) (cid:1) (cid:0) 2 (cid:1)
+ d~rstr (δT′+ΛδT′Λ) W2δT′ WδT′W
2 0 0 0− 0
(cid:20)Z (cid:21)
1 (cid:0) (cid:1)
+ d~rdr~′str (δT′ ΛδT′Λ) 2W(~r)2W(r~′)2δT′ 2W(~r)2W(r~′)δT′W(r~′)
8V 0− 0 0− 0
Z (cid:16)
2W(~r)W(r~′)2δT′W(r~′)+2W(~r)2δT′W(r~′)2+W(~r)W(r~′)δT′W(r~′)W(~r)
− 0 0 0
2 (cid:17)
1 d~r strW2 2+ 1 d~rstrW2 +O(W5). (A10)
−64V 64V2
Z (cid:18)Z (cid:19)
(cid:0) (cid:1)
Multiplying this by Eq. (A6), we find
d(δW) exp d~rstr (δQ(~r))2
−
Z (cid:20) Z (cid:21)
=exp[2Vstr(δT′ ΛδT′Λ)δT′]
0− 0 0
1+ d~rstr (δT′ ΛδT′Λ) W2δT′ WδT′W
× 0− 0 0− 0
(cid:26) Z
1 (cid:0) 2(cid:1) 1 2
+ d~rstr (δT′ ΛδT′Λ) W2δT′ WδT′W + d~rstrW2
2 0− 0 0− 0 64V2
(cid:20)Z (cid:21) (cid:18)Z (cid:19)
1 (cid:0) (cid:1)
+ d~rdr~′str (δT′ ΛδT′Λ) 2W(~r)2W(r~′)2δT′ 2W(~r)2W(r~′)δT′W(r~′)
8V 0− 0 0− 0
Z (cid:16)
2W(~r)W(r~′)2δT′W(r~′)+2W(~r)2δT′W(r~′)2+W(~r)W(r~′)δT′W(r~′)W(~r)
− 0 0 0
1 d~rstr (δT′ ΛδT′Λ)W2δT′W2 1 d~r strW2 2+O(W5) .(cid:17) (A11)
−8 0− 0 0 − 64V
Z Z (cid:27)
(cid:0) (cid:1)
The remaining δT integral can be also calculated with use of the Wick theorem and the contraction rules:
0
1
str (δT′) (δT′) = str str , (A12)
h X 0 12Y 0 21iδT0 −8V X Y
1
str (δT′) str (δT′) = str , (A13)
h X 0 12 Y 0 21iδT0 −8V XY
where and are arbitrary supermatrices and
X Y
... = d(δT )(...)exp[2Vstr(δT′ ΛδT′Λ)δT′]. (A14)
h iδT0 0 0− 0 0
Z
Since the zero mode variables enter into Eq. (A11) only through (δT′) and (δT′) , the Jacobian does not contain
0 12 0 12
the zero mode variables. We finally obtain the Jacobian (2.6) from the expression
1 2
d(δW)d(δT )exp d~rstr (δQ(~r))2 =1+ d~rdr~′ strW (~r)W (r~′) +O(W5). (A15)
0 − 16V2 12 21
Z (cid:20) Z (cid:21) Z (cid:16) (cid:17)
10