Persistent currents in an ensemble of isolated mesoscopic rings

ANNALS

OF PHYSICS

219, 148-186

(1992)

Persistent Currents in an Ensemble of Isolated Mesoscopic Kings A. ALTLAND*,

S. IIDA+, A. MUELLER-GROELING, Max-Planck-Institut

ftir

Kerphysik,

AND H. A. WEIDENM~~LLER

Heidelberg,

Germany

Received May 1, 1992

We calculate the persistent current induced at zero temperature by an external, constant, and homogeneous magnetic field in an ensemble of isolated mesoscopic rings. In each ring, the electrons are assumed to move independently under the influence of a Gaussian white noise random impurity potential. We account for the magnetic field only in terms of the flux threading each ring, without considering the field present in the body of the ring. We pay particular attention to the constraint of integer particle number on each ring. We evaluate the persistent current non-perturbatively, using a generating functional involving Grassmann integration. The magnetic flux threading each ring breaks the orthogonal symmetry of the formalism; forcing us to calculate explicitly the orthogonal-unitary crossover. The resulting expression for the average persistent current is paramagnetic, periodic in the flux with a period of half the elementary flux quantum da, odd when the sign of the flux is reversed, and where d is the mean level spacing and E, the has a maximum value of about 0.3 &&a, Thouless energy. This is about two orders of magnitude below experiment. ci) 1992 Academic Press. Inc.

1. INTRODUCTION Quantum coherence is a central theme of mesoscopic physics. At sub-Kelvin temperatures, where the dephasing length L, due to inelastic scattering of electrons exceeds typical linear dimensions ( z pm) of microelectronic devices, electronic properties are determined by elastic scattering. And such scattering, even if due to many randomly distributed impurities, does not destroy wave coherence. This is shown by conductance fluctuations, and by Aharonov-Bohm type oscillations in devices with annular geometry. The most recent example is provided by persistent currents in isolated rings threaded by a magnetic flux 4. In each ring, the current is due to the phase factor exp( f2heq5/hc) picked up when an electron moves once around the ring, and is thus due directly to quantum coherence. In an experiment published two years ago, Levy et al. [l] demonstrated the existence of persistent currents in a system of about 10’ copper rings by measuring the magnetic dipole moment of the system induced by a slowly varying magnetic * Present address: Department of Nuclear Physics, Weizmann Institute of Science, Rehovot, Israel. t Present address: Max-Planck-Institut fiir Festkorperforschung, Stuttgart, Germany.

148 0003-4916/92

$9.00

Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.

PERSISTENT

CURRENTS

IN MESOSCOPIC

RINGS

149

held. A more recent experiment measured the current in single isolated rings [Z]. In the experiment of Levy et al. [ 11, each “ring” was actually rectangular, with a rectangular opening in the middle. Circumference L and width of the rings were 2.2 pm and 0.13 pm, respectively, and the elastic mean free path due to impurity scattering of electrons was 1z 10 nm $ L. The amplitude of the &dependent persistent current per ring was found to be of the order E,/&,. Here, & = he/e is the elementary flux quantum, and E, is the Thouless energy, given by x2 Dii/L2 with D the diffusion constant. Even larger values were reported in Ref. [2] for single rings. Theory actually preceded experiment, at least conceptually. After seminal work by Biittiker et aZ. [3], Gefen and collaborators investigated persistent currents in isolated “ideal” rings (no impurities) in a series of papers [4]. Two main findings were: (i) For a ring with a fixed number N of particles, the persistent current 1(d) at zero temperature is a sawtooth-shaped function of 4, jumping discontinuously from - 2ev,jL to +2evflL at 4 = 0, 4 = +&,, +_2&,, ... for N even, and at 4 = f ~~0, k $40, ... for N odd, decreasing linearly with increasing 4 between these points. Here, vF is the Fermi velocity. Here, vf is the Fermi velocity and the electron spin is accounted for by a factor of two. The discontinuities are due to degeneracies of the single-particle levels. (ii) The coefficients of the Fourier expansion of I(d) are smooth but oscillating functions of the particle number N. Averaging the current over a continuous distribution of N-values yields zero or very small values if the width of the distribution is larger than about unity. This second finding forecasts the difficulties encountered later in using the grand canonical ensemble. It is due to the fact that the persistent current originates from the last few levels below the Fermi surface. The influence of impurities was first investigated by numerical simulations [IS]. Impurity scattering was found to significantly reduce the amplitude of the induced current. The sensitive dependence on particle number N, and the importance of keeping N fixed, again emerged. This was also found in calculations using impurity perturbation theory. In the grand canonical ensemble with fixed chemical potential po, the average persistent current is exponentially small in L/Z [6], reflecting the fact that fixed p. implies averaging over a continuous distribution of N-values. In several papers [7-lo], a modified version of the grand canonical ensemble was therefore used. The chemical potential p = p. + 6~(4) contains a #-dependent small part all(d). Expanding the current to first order in 6~($) leads to an improved version of the theory. Evaluating the resulting expressions perturbatively, one encounters divergences which are removed by a cutoff. The cutoff is physically plausible because for the actual experiment, the temperature is larger than the mean level spacing in the ring. The persistent current calculated in this way has a maximum value which is about two orders of magnitude smaller than the experimental values [ 1,2]. This discrepancy is serious, and at present no convincing remedy for it is known. It is widely held that the interaction between electrons, so far neglected in the theory of all mesoscopic phenomena, is responsible for this failure of theory C8, 111.

150

ALTLAND

ET AL.

Being fully aware of the shortcomings of a theory without electron correlations, we calculate in this paper the average persistent current at zero temperature for a system of mesoscopic rings, the electrons in each ring moving independently of each other under the influence of a Gaussian random white-noise potential to simulate impurity scattering. Our motivation for doing so is twofold. First, we wish to impose strictly the condition of integer particle number on each ring. The results for ideal and disordered rings mentioned above justify our concern. And the expansion in powers of 6~ leaves several questions open. Terms of higher order than the first in 6~ for the persistent current involve correlation functions which have not been estimated, not to speak of calculated. Is the omission of these terms justified? Actually, the problem is a rather general one. It can be phrased as: How to perform an ensemble average for the canonical ensemble? In Section 2, we approach this problem from a novel point of view. We do keep the particle number on each ring fixed, but we allow it to differ from ring to ring. The resulting expression coincides with the first-order term of the fip-expansion, yielding a deeper understanding of the latter. Second, we address the divergences of perturbation theory. These divergences occur for half-integer values of d/&, including zero. We ask: What is the value of the average persistent current at zero temperature (when all cutoffs lose their physical significance) in the model of independent electrons? We show that the divergences are linked to the rotational invariance of the model. We identify their origin explicitly in zero modes which are massless precisely when &do is halfinteger. We perform the integration over these modes exactly (non-perturbatively). We show that for all other modes perturbation theory is adequate, and in this way we derive an expression for the average current at zero temperature which we believe to be the exact answer for the problem posed. In our calculation, we use the supersymmetric form of the generating functional as introduced by Efetov [12]. In this framework, the integration over the zero mode is of theoretical interest in its own right. Indeed, the magnetic flux breaks the orthogonal symmetry of the formalism and leads us to evaluate the crossover from orthogonal to unitary symmetry. It turns out that this is a rather non-trivial task. We are not aware of. any previous calculation of a crossover between symmetries in the framework of the generating functional. The orthogonal-unitary crossover has been calculated for the spectral two-level correlation function by Mehta and Pandey [ 131. These authors used the method of orthogonal polynomials developed by Mehta [14]. After deriving an expression for the current based on particle-number conservation on each ring in Section 2, we introduce the generating functional in Section 3. The integration over the zero mode is described in Section 4. Our result is presented and discussed in Section 5. A number of technical points is deferred to the appendices. While we prepared the manuscript for this paper, work [15] including electron interactions and giving the right order of magnitude for the current came to our attention. However, the calculations presented there are still under discussion and

PERSISTENT CURRENTS IN MESOSCOPIC RINGS

151

cannot be considered to be the final word. Some technical ideas originating from our work have been further developed [16] and used to calculate persistent currents through a dynamic response [17]. 2. CONSERVATION OF PARTICLE NUMBER In this section, we derive an expression for the persistent current, imposing the constraint of particle number conservation. We compare this expression with those of Refs. [7-91. We begin with some definitions. We consider an ensemble of rings (about 10’ in the case of Ref. [l]) with inner radius ri, outer radius r2, inner circumference L. We model each ring as two-dimensional. A static magnetic flux 4 threads each ring. We disregard the magnetic field in the body of each ring. The electrons move independently but subject to random elastic impurity scattering with elastic mean free path 1. We work at T= 0 and disregard inelastic scattering processes. Elastic scattering’is caused by an impurity potential V. Randomness is introduced by taking I’ to be a random white-noise potential. (In our notation, we do not distinguish between the ensemble and a realization of the ensemble.) With x and p two-dimensional position and momentum vectors and m the mass of the electron, the single-particle Hamiltonian H(d) is given by H(b)=&

(

p-;A

)

*+ V(x)=H,(q5)+

V(x).

(2.1)

In symmetric gauge, the vector potential A is given by 27c(4/L2)( - y, x). We use a right-handed coordinate system. The flux 4 is positive in the z-direction. The eigenvalues s,(d) of H,(4) are labelled by positive integers in increasing order. The normalized eigenfunctions i,(d; x) of H,(4) are periodic: Introducing planar polar coordinates r, 0 we have A = (d/L) e, and i,Jd; r, 0 $271) = fn(4; r, 0). Removing A from the Hamiltonian (2.1) by a gauge transformation leads, for the gaugetransformed eigenfunctions x,(4; x), to the condition that ~(4; r, 0 + 274= exp(2WV4,)

~(4; r, e),

(2.2)

where &, = he/e is the elementary flux quantum. Equation (2.2) implies that all observables are periodic functions of C$with period c&,. The ensemble of impurity potentials V(x) is rotationally invariant around the centres of the rings (while a single realization is not). For the calculation of the persistent current 1(d) induced by 4, one cannot use the grand canonical ensemble with fixed chemical potential p. The reasons were given in the Introduction. This is why in this paper, we introduce a different procedure. At the end of this section, we compare our results with those of Refs. [7-91. The problems of the grand canonical ensemble are obviously avoided when one uses the canonical ensemble. In this framework, the expression for the equilibrium

152

ALTLAND

persistent current at zero temperature mechanical formula I(#)=

-2c

ET AL.

coincides with

the simple quantum-

N aad) 1 n=l a4 ’

(2.3)

where N is the number of electrons on each ring. The sign is chosen such that I> 0 if the current flows counterclockwise in the (x, y) plane. Put differently, I> 0 for 4 >O means paramagnetism. The factor 2 is due to spin. Impurity averaging requires us to express Eq. (2.3) in terms of the Green functions; this is accomplished by writing Z(d)= -2c

=

----I

,fdEfE,

$$(E’-i.)]

c . E dE’tr 71 j 0

$[(E+-H))‘-

(E--H)-‘]

i

i

.

(2.4)

We have assumed sr > 0. A plus (minus) sign indicates an infinitesimal positive (negative) imaginary increment. In Eq. (2.4), the energy E must be chosen in such a way that (2.5)

E,
It is because of condition (2.5) that use of the canonical ensemble runs into difficulties. Indeed, ~~(4) and .sN+r(#) change, both with 4 and from one realization of the impurity potential to another, making it impossible to satisfy Eq. (2.5) by choosing E independent of 4 and of the realization of V(x). And we cannot weaken the constraint (2.5) light heartedly because calculations for ideal rings [4] have shown that here the main contribution to Z(d) stems from the last occupied level EN(b). We overcome this problem by introducing a generalization of the canonical ensemble: We demand the number of electrons N on each ring to be integers, but we allow N to vary from ring to ring about a mean value m within a range K$ N. We calculate the ensemble average Z(b) by averaging, not only over the impurity potential, but also over K. This generalization of the canonical ensemble is tailored to the experimental situation of Ref. Cl], where the approximately lo7 rings involved have inevitably tolerances in their geometries and volumes and, hence, in the number of electrons they carry. Guessing these tolerances to be a fraction of a percent, we find Kg lo7 . . . lo* % 1 with an estimated total number of 10” electrons per ring. Averaging the first expression (2.4) over a set of K neighbouring integers with associated eigenvalues ~~(4) in an interval S, we find Z(b)= -2cK-’

j,dE

IfdE’

ff,

2&E’--E.,)][

f N=l

&E-E,,,-@

11 ,

(2.6)

PERSISTENT

CURRENTS

IN MESOSCOPIC

RINGS

153

where 6 is positive infinitesimal. The condition (2.5) on E translates (for K even) into the following condition on the endpoints El, E2 of the interval S:

Both E, and E2 are again dependent on C$and change from one realization of the impurity potential to another, and it seems that we have gained nothing in going from Eq. (2.4) to Eq. (2.6). But this is not so: For Kg 1, we are allowed to disregard the constraints (2.7), i.e., to keep S fixed and independent of 4 and of the realization of V(x). This enables us to calculate 1(d) by ensemble-averaging only the curly bracket in Eq. (2.6). To justify this approximation, it is necessary to discuss the dependence of E, and E, in the constraint (2.7) on both 4 and the realization of the impurity potential V(x). We consider first the dependence of E, and E, on 4 for a fixed realization of the impurity potential. Numerical calculations [S] of the &dependence of the eigenvalues ~~(4) in such a case show oscillations. The amplitude of the oscillations is of the order d, the mean level spacing at the Fermi surface for fixed spin. Of the K eigenvalues contained in S = [E,, E,] for 4 = 0, say, only a few will leave the interval as C$varies from 0 to &, and some few others will enter S. The error incurred in keeping S fixed in Eq. (2.6) is of the order K-l. With Kg 10’. . .108 for the experiment Cl], this error is negligible. We now consider for fixed 4 the change of the constraints (2.7) as we go from one realization of the impurity potential to another. Al’tshuler and Shklovskii [lS] have estimated the change 6N with disorder realization of the number of eigenvalues in an interval of length E. For E > E, (the Thouless energy, given by E, = rr2 DA/L2, where D is the diffusion constant), they find 6N- (E/E,)3’4. To keep the number of eigenvalues in [0, E] fixed, E would have to be changed by a corresponding amount, AE- d(E/E,)““. (This estimate may be too pessimistic since we consider quasi one-dimensional rings. For strictly one-dimensional systems, the exponent i is replaced by a). For the actual experiment, we have E/d% 10” and EC/d 2 300, so that A&% 104d is large compared to d, and the neglect of the constraints (2.7) cannot be justified as before because K is large. In this case, however, the justification can be given a posteriori: The ensemble average of the curly bracket in Eq. (2.6) depends explicitly only on (E-E’). The dependence on the variable (E + E’) is implicit through the average level spacing d. But d does not change significantly as the energy changes by a few thousand mean spacings. In this sense, the average over the curly bracket is independent of E, justifying the neglect of the constraints (2.7). Similar arguments apply to the change of length of the interval S with impurity realization. Our procedure for calculating r(4) thus consists of a double average, over both N and all realizations of the impurity potential V(x). For a fixed realization of k’(x), we average I(d) over K rings, each ring carrying another number N of electrons, the mean value of N over these rings being Rr 10”. For K 9 1, we are sure

154

ALTLAND

ET AL.

that variations of K with 4 are negligible. When we change the realization of the impurity potential and keep S fixed, we effectively change N by as much as 6N g (E/E,)“” z 104. However, the result is independent of the choice of N, permitting us to do so. It might seem that our generalization of the canonical ensemble is useful also for calculating the persistent current on a single ring. Indeed, neither the actual form of the impurity potential nor the precise number N of electrons on the ring are known, and averaging over both therefore appears highly suitable. Unfortunately, this is not the case. Whatever its value, the number of electrons on the ring is either even or odd, and calculations [4] for ideal rings show that the persistent current is very different in both cases. Averaging over N smears out this difference, and the result cannot be compared meaningfully with an experiment on a single ring. We have not found a way to overcome this difficulty. Returning to Eq. (2.6), we keep S fixed and of length Kd. We show in Section 4 that the average over the curly bracket in Eq. (2.6) which we denote by g(E, E’) is a function only of E = E- E’ with the property that g(E)= -d-E).

(2.8)

Moreover, g(s) falls off rapidly as IsI increases. Hence, I(d) = -2cK-’

j

s

dE s ddE’g(E-

E’)

0

! s E

3 -2cK-’ z -2cK-’ = -2cd

s

ss

dE

0

dE

d& g(E)

s 0* d& g(E)

(2.9)

ao g(E) d&. s0

Here, g(s) is given by (we use the steps leading from Eq. (2.3) to Eq. (2.4)) g(s)=tr{b(E-H)}

tr

II

C.C.

. (2.10)

e=E-E’

The bar denotes the ensemble average. In writing the second line of Eq. (2.10), we have omitted terms containing a product of two advanced or two retarded Green functions. Such terms yield exponentially small (in L/I) contributions. It is of interest to compare our expression for r(b) as given in Eqs. (2.9) and (2.10) with the expression obtained in Refs. [7-91 in the framework of the grand

PERSISTENT CURRENTSIN MESOSCOPIC RINGS

155

canonical ensemble by introducing the first-order variation ‘6~(4). To be definite, we use the expression of Ref. [7]. With s = 2 for spin degeneracy and A = id for the total average level spacing, we have fAAG,(q5)= cd; Carrying out the differentiation, symmetry (2.8), we find

(1: dE tr(G(E-

using the definition

Lx = -2cd = 2cd

ZZ)})‘.

(2.11)

(2.10) of g(s), and the anti-

’ dE’g(E’-p) i0 g(E) dEg 2cd lrn g(E) de.

(2.12)

0

Except for a sign (which we believe to be erroneous, see Appendix A) the expression of Al’tshuler et al. coincides with ours, and the same statement holds for Refs. [8,9]. This is an interesting result because it shows that the higher-order terms not considered in Refs. [7-91 become small upon averaging over particle number N. It also shows that the approach of Refs. [7-91 is unsuitable for calculating the persistent current on single rings; the diffkulty of separating even and odd N-values is the same there as in our approach.

3. GENERATINGFUNCTIONAL AND ENSEMBLEAVERAGE

We describe the construction of a generating functional for the persistent current Z(d) introduced in Section 2. Using Eq. (2.6) and the subsequent discussion, and expressing the delta functions by Green functions, we find, putting !i = c = 1 throughout, I($)=

$dEJEdE’g(E, 0

E’),

(3.1)

where g is defined in Eq. (2.10). We employ Efetov’s supersymmetry method [12] but use the conventions and notation of Ref. [19]. After ensemble averaging and the Hubbard-Stratonovitch transformation, we approximate the resulting functional by the saddle-point method. This yields a non-linear sigma model for the persistent current. While we do not reproduce standard steps in this derivation, we are explicit about those points where the presence of the magnetic field requires special attention. 595-/219/l-11

156

ALTLAND

ET AL.

3.1. The Generating Functional We start from Eq. (3.1) for 1(d), with H given by Eq. (2.1). The impurity potential is a Gaussian random white-noise potential, V(x) =o;

V(x) v(Y)=&x-Y).

(3.2)

Here, r is the elastic scattering time and v is the density of states per spin direction at the Fermi energy. With V denoting the area of a single ring and d the mean level spacing per spin, we have v = (dV)-‘. We proceed by constructing a generating functional separately for each of the two traces in the last line of Eq. (2.10). We define a supervector cpl(x) with the help of commuting complex (S,(x)) and anticommuting (xl(x)) fields: cp1=

S,(x) ( ) Xl(X)

(3.3)

.

The vector x denotes some position in the body of the ring. The field cpl(x) must be periodic. In the notation used in Section 2, it obeys cpl(r, @+ 2~) = cpl(r, 0 The Hamiltonian

is given a structure corresponding

(3.4)

to Eq. (3.3) by writing (3.5)

where we distinguish explicitly between the flux arguments @sand dX. Later, these arguments will play the role of source fields. We introduce the notation &=E’+@l,,

&=&-A,

(3.6)

and we define the generating functional Z, as Z, = 1 d[rp,] exp is d*xrpi(x) fi The symbol d[q,] denotes the measure associated over the superfield q,(x). We have detg=exp(trg involves a spatial integration over x in addition Z, Because of the supersymmetric formulation,

(3.7)

with the functional integration In), and the supertrace (trg) to the standard graded trace. is automatically normalized,

PERSISTENT

CURRENTS

IN MESOSCOPIC

(E’+

1dH

157

RINGS

We have

= -tr

i

-H)-

(3.8)

z 1.

Similarly,

Therefore,

=~(a,-;,)z,C),,~,llo,=m,=,.

(3.10)

A functional .Z2 representing the second trace in Eq. (2.10) is constructed analogously. With the definitions

&=fW)Ql,,

(3.11)

&=E-Q12, L32=&A2, we have

-GUI = j dCe1exp where j is a commuting

d2X43$)(fi2 + jl) cp2(x)

scalar auxiliary variable, where Z[j] tr{(E-

-H)-‘}

= $a,Z,[j]ljco.

(3.12)

= 1, and where (3.13)

We observe that the energy increments in E * serve as convergence-generating factors for the functional integrations in Eqs. (3.7) and (3.12). Hence, the signs in the exponents of both functionals are fixed and are not at our disposal. We combine Z, and Z2 into a single functional Z,

Z[4,, 4, A= Z,Cds,4x1z2m.

(3.14)

158

ALTLAND

Writing d[@] = d[q,]

ET AL.

d[qp2] and

Q(x)=( f-P,(x) (P2(x) >’

“=(‘d -9,>~ 8,), z2=(i g,

d$

(3.15)

we obtain for Z the expression z[+$,, bx, j] = 1 d[@] exp i 1 d2x Q+(x) A”‘(b

+ j12) A”‘@(X)

(3.16)

Obviously, Z[d, 4, 01 = 1 and -H)-l$}

tr{(E-

-H)-‘} (3.17)

Our task is to calculate the average Z over the ensemble (3.2). In doing so, we encounter the following subtle point. We are modelling a system which, for ~,=~,=qLO, is invariant under time reversal. This symmetry is broken by the external field and its magnetic flux 4 which eventually change the universality class of the system from orthogonal to unitary. It turns out that the graded space spanned by the supervectors Q(x) defined in Eq. (3.15) is insufficient to describe the full symmetry of our problem and that it is necessary to reexpress the generating functional Z in Eq. (3.16) in such a way that this symmetry becomes manifest. This drawback of the form of Eq. (3.16) is not obvious at the present stage. It is true that the supervector Q(x) does not have an internal structure relating to time reversal, although for $ = 0 it is meant to describe a time-reversal invariant system. This fact is a hint of the problem. The deficiency of the present formulation is more obvious in models [20,21] of mesoscopic systems which are based on a discretized disordered region and on random-matrix theory. There, progress is impossible after ensemble averaging without having endowed Q(x) with an additional structure. In the present continuum formulation (which we have adopted for the benefit of readers who are familiar with perturbative calculations), this difficulty is more hidden, but it leads to manifestly incorrect results. In any case, the solution of this problem is wellknown [ 121 and follows. We use the identity @+n”*(8+jz2)

A”2@= $D+A”‘(B+jl,)n”‘@ + f [ @+Al’*(b + jZ2) A “2@]T =

4 lput/fl’l’(~/

+

jr;)

,‘1/2

y,

(3.18)

PERSISTENT

CURRENTS

IN MESOSCOPIC

159

RINGS

where with p = 1, 2,

(3.19)

A’ = A 0 l,,

z;=z,o1,.

The vector Y(x) does have the internal structure missing in Q(x). Indeed, we have, for p = 1, 2, 0 0 1 0 0 0 0 1 Y; = CYp, where C= (3.20) o-1

0

0

This symmetry property is related to the time-reversal invariance of the field-free system (4 = 0). We expect the external magnetic field, represented by the magnetic flux 4, to distinguish .between the original graded space (S,, x,) and its “doubling image” (S,*, I,*). The definition of 6’ in Eqs. (3.18) shows that this is indeed the case: The operator acting on (S,*, x,*) is not fip itself but the transpose l?$ of BP. With V real and symmetric and pT = -p for the momentum operator, we have from Eq. (2.1) that

(3.21) We collect our results after introducing

&=E-E',

some further definitions. With E * = E) id, we have

~=~(E+E’).18=Eo.ls,

6’ = g + ihA’ - i&A’ - Q,

(3.22)

where A,

0

0

0 fi= i 00

0 0 AT

I?, 00

0 0 . A;0 i

(3.23)

160

ALTLAND

ET AL.

We introduce the matrix r,=diag(l,

1, -1, -1, 1, 1, -1, -1)

(3.24)

and have fi=&(p@l,-A@r,)‘+

V(x)

01,.

(3.25)

The generating functional has the form Z[q5,, $x, j] = [ d[ Y] exp i 1 d2x !PT(x) LI”/~(I {

The average persistent current is given by

3.2. Ensemble Average and Nonlinear Sigma Model

To calculate the ensemble average Z of the generating functional (3.26), we use the statistical properties (3.2) of V(x). We perform the average over V(x), eliminate the resulting quartic term in P(x) in the exponent of Z by means of a HubbardStratonovitch transformation, thereby introducing a graded matrix Q(x) with the same symmetry properties as the dyadic product !PYt regarding commuting and anticommuting variables, and perform the remaining integration over d[ Y]. Details are given in [12]. The result is ~[~~,),,~]=~d[Q]exp{

-~trgQ2-ftrgln[db-~Q+il,]~.

(3.28)

As in Section 3.1, the supertrace (trg) includes an integration over x, and Bb is given by L?’ with V(x) = 0, see Eq. (3.22). Further evaluation of the average functional (3.28) relies on the saddle-point approximation. This amounts to computing leading-order terms of an asymptotic expansion in (k&‘, where k, is the Fermi momentum [12]. The diagonal solution of the saddle-point equation reads QD = -i/i’.

The saddle-point manifold (the totality tion) is obtained by transforming Q,,

Qc = T-‘(x)

(3.29)

of solutions to the saddle-point

condi-

Q, WI.

(3.30)

Here, T(x) is a graded matrix field subject to certain symmetry requirements

PERSISTENT

CURRENTS

IN MESOSCOPIC

[12, 19 J. All matrices Q, of the saddle-point linear constraint Q”, = - 1, and, therefore,

161

RINGS

manifold

obviously obey the non(3.31)

trg( Qi) = 0.

The general Q-field can be decomposed in the form Q(x) = Q&x) + de(x). Here, 6Q(x) denotes contributions which do not lie in the saddle-point manifold and therefore constitute “massive modes.” These are integrated out in Gaussian approximation. With a suitable decomposition of the measure into the invariant measure for QG and a measure for SQ [12, 193, the Gaussian integrals yield unity. We are left with a functional of the Goldstone modes Q, only. Because of Eq. (3.31), only the trg In term in Eq. (3.28) survives. We expand this term to lowest non-vanishing order in (k,l)-‘, E, and j. This is a standard but lengthy procedure. We obtain (3.32)

S(Qc)=ytrg[(VQ,)‘]+ytrg

iQcn’-jQcl*

1 ,

(3.33)

where Q= -id+e[A;]

(3.34)

is the covariant derivative, where A=t-‘diag(~,,(6,,~,~)Odiag(l,

-l)e@

(3.35)

breaks the symmetry embodied in the representation (3.30) of Q, and thereby incorporates the violation of time-reversal symmetry by the external magnetic field and where the diffusion constant is given by (m is the electron mass)

*=lkZ,z

(3.36)

2 m*'

Equations (3.32) to (3.36) together define a nonlinear sigma model. Inserting Eq. (3.32) into Eq. (3.27), we obtain for the persistent current the expression

4QGl trgCQ&l trgCVQ,CKQJI x exp ytrg[(VQG)*]+ytrg

,

(3.37)

where K=diag(-l,l,

1, -l,O,O,O,O)

(3.38)

162

ALTLAND

ET AL.

and where V= -id+?

[r,, .] e,.

As before, trg implies an integration over position. To simplify the notation we consider strictly one-dimensional rings with y1 = r2 = r, so that x = (L/2rr) 0 is periodic in 8 with period 271. This step is also physically meaningful because for 1< L diffusion takes place only in the 6’ direction. At the end of this section we indicate the necessary changes for the quasi one-dimensional case (r2 - r, g I4 L) which is appropriate for the experimental situation. Anticipating the result that the overwhelming contribution to the persistent current (2.3) in the one-dimensional case stems from the immediate vicinity of the Fermi surface E,, we expect these changes to be quite small. We turn to the symmetry properties of I(#) as displayed by the generating functional (3.37). We consider the gauge transformation

QG+ exp -it,rx (

49 )

Q,exp

(

+iz,-xe&m 2L >

(3.40)

with rnE Z. This is a diagonal unitary transformation; it obviously leaves unchanged the invariant measure d[Q]. With e& = 271 (we recall +i= c = l), we have e&m exP +ir,y(x+L) (

>

=exp

(

e&m +iz,yx

>

exp(imn)

(3.41)

so that the transformation (3.40) leaves invariant the property Q&x + 15) = Q&x). (This would not be the case for m not an integer). Therefore, I(d) as given by Eq. (3.37) does not change under the transformation (3.40). On the other hand, inserting (3.40) into Eq. (3.37) is equivalent to replacing 4 by (4 - (m/2) do). This shows that I(d) is periodic in 4 with period (m/2) q$,, i(y+~~$,)=I((),

maninteger.

From the study of ideal rings it is known that, while rings of fixed particle number have a persistent current which is periodic with period m&, averaging over even and odd particle numbers yields the periodicity (m/2) #O. We ascribe the periodicity (3.42) to the same origin.

PERSISTENT CURRENTS IN MESOSCOPIC RINGS

Another property of 1(d) emerges when we use the periodicity Q&x + L) T Q&l t o write Q(x) as a Fourier series,’ QG(x)=

z Q,,,exp(2irrmz). m= -02

163 property

(3.43)

Using this expansion in the exponent of Eq. (3.37) we find

trg{(VQG)‘} = Tt%{CLT3, Qol~‘} +...,

(3.44)

where the dots indicate contributions with m # 0 and where gg does not imply integration over position. Equation (3.44) shows that the mode Q, becomes massless for 4 + 0, so that a perturbative evaluation of the integral (3.37) diverges at 4 = 0. Hence, Q, is a Goldstone mode, and the integration over Q, must be performed non-perturbatively for small 141. Because of the periodicity property (3.42) the same situation arises for d = (m/2) do with m integer. The existence of this pattern of zero modes is a mirror of the rotational invariance of our problem after ensemble-averaging. Because of the property (3.42), we are permitted to confine ourselves to the interval -&/4 < 4 < +&,/4, where Q0 is the only zero mode. For (b in this interval, we perform the integration over Q0 exactly and that over the Qm with m # 0 perturbatively. We show in Section 5 that a perturbative calculation is justified for the Q, with m # 0. A third important property of I(&) is its antisymmetry about 4 = 0. This property is obvious at the level of Eq. (2.3). Indeed, the Hamiltonian (2.1) does not change if we simultaneously take the complex conjugate and replace 4 by ( -4). Assuming all eigenvalues s,(d) to be non-degenerate, we see that each s,(QI) is even in 4, so that I(d) is odd. This property must survive ensemble-averaging, and we expect 1((b) = -I( -4).

(3.45)

To verify Eq. (3.45) directly from Eq. (3.37), we recall the definition (3.20) for C, define c = CO I,, and consider the transformation of variables

Q&J + ~Q,@) CT.

(3.46)

With cc’ = 1, this transformation leaves unchanged the invariant measure d[Qc], so that I(d) is invariant under the substitution (3.46). On the other hand, i Strictly speaking, Eq. (3.43) does not hold as it stands. The property Qi = - 1, used below is not consistent with the definition Q,=(l/L){ &Q&x) implied by (3.43) with Q’,= -Is. Instead, the Fourier expansion must be carried out at the level of independent variables which parametrize Q(X) in such a way that Q’(x) = -1s is fulfilled. The Q, with m #O in Eq. (3.43) and further below must be understood as a convenient shorthand notation for terms which arise in this way, and which in fact are rather complicated. Since we calculate non-perturbatively only the contribution from Q,, no confusion should arise.

164

ALTLANDETAL.

substituting (3.46) into Eq. (3.37) explicitly, we note that e’/i’e= A’ and that eTZ2 e= Z2, so that the terms trg(/i’QG) and trg(QGZ2) remain unaltered. But CTz,e= -z3 and c’Kc= -K, so that trg(VQG)’ remains unchanged if we simultaneously apply the transformation (3.46) and replace C$ by (-d), while trg{VQG [K, Qc] } changes sign under the same operations. Hence Z(4) is odd in 4. Using the related (but not identical) transformation QG(x) + Q:(x), we can also show directly that Z(d) is odd in E, as claimed in Section 2. We also note that the only dependence on E, = $(E + E’) resides in the diffusion constant D, cf. Eq. (3.36), and in the density-of-states factor v, and in both cases this dependence is weak. The property (3.45) permits us to confine ourselves to the interval O<@&&l.

(3.47)

For ideal rings at T= 0, the persistent current in the interval (3.47) has the shape of a single saw-tooth: It jumps discontinuously to the positive value 2eu,/L at 4 = 0 and decreases linearly with increasing 4, reaching zero at 4 = &,/4. We expect disorder to smooth the discontinuity at 4 = 0 and to reduce the amplitude 2eu,/L, but not to affect the overall features. This is indeed what we find below. In concluding this section, we comment on the changes which result when we consider two- or three-dimensional rings. The zero mode Q0 is constant throughout the volume of the ring. The other modes, Q, with m # 0, are calculated perturbatively. This calculation leads to the introduction of the diffusion propagator. Here, only the lowest of the transverse modes (a constant) is taken into account, and effectively nothing changes as we go from one to two or three dimensions. This is true as long as the transverse dimensions are of the order I< L.

4. INTEGRATION

OVER THE ZERO MODE Q,

In this section, we calculate the contribution I,(#) of the zero mode Q, to the average persistent current Z(4) of Eq. (3.37), neglecting the contribution of all modes Qm with m # 0. We demonstrate in Section 5 that this latter contribution can be calculated perturbatively; such a perturbative calculation yields the same result as diagrammatic perturbation theory. At the end of Section 3, it was shown that g(E, E’) of Eq. (2.10) depends only weakly on E. = $(E + E’). Moreover, we indicated why g(E, E’) is antisymmetric in E = E-E’, as claimed in Eq. (2.8). Anticipating that g(s) falls off rapidly with IsI for IsI % d, we use the steps leading to Eq. (2.9) in Eq. (3.37), reduced to the integration over Q, as defined in Eq. (3.43). This yields

L(d)= -

(4.1)

PERSISTENT

CURRENTS

IN MESOSCOPIC

RINGS

165

where

K(Y)= yl’* lomdx 1 4Qol exp{ y trg[QoZ3QoT31 +~trs[Qdl] x trsCQJJ . (trgCQo~3QoJ4 -4)

+ C.C.

(4.2)

and

(4.3)

We have carried out the space integrals which yield a trivial factor L (the symbol trg therefore does not imply spatial integration in the present section); we have used &, = 2n/e and E, = n*D/L*. Aside from the integration over energy (x), the expression (4.2) involves 16 integrations, eight over commuting and eight over anticommuting variables. In comparison with previous calculations of this type [12, 191, we face the additional difficulty that the matrix z3 appearing in Eq. (4.2) breaks the time-reversal symmetry of the matrix field Q,, as is obvious from the relation e7:e = - z3. This feature poses a novel problem not encountered in Refs. [12, 191. We show which steps are taken to overcome this problem, but do not repeat other details which are analogous to Refs. [12, 191. The crucial step in the calculation of K(y) consists in the integration over the Grassmann variables. For reasons given below, only the term of highest (eighth) order in the expansion of the integrand yields a non-vanishing contribution. The construction of this term is highly non-trivial since it receives contributions from trg(Q,Z,), from trg(Q,z,Q,K), and from exp(y trg(Qo73Q,,73)). This construction is described in Subsection 4.1. The remaining integration over commuting variables entails an integration over the group SU(2). This is described in Subsection 4.2. The remaining integrals over the “eigenvalues” of QO are described in Subsection 4.3. A perturbative evaluation of K(y) is given in Subsection 4.4. We repeatedly simplify our expressions using Berezin’s theorem [21a]; we describe some implications of this theorem in Appendix B. We find it useful to employ for Q, Efetov’s parametrization rather than the one introduced in Section 3. For this reason, we recall Efetov’s parametrization in Appendix C. In introducing Efetov’s parametrization into the expression (4.2) for K(y), we use Berezin’s theorem. Indeed, this parametrization can be reached by a transformation of the variables introduced in Section 3, and the theorem states that no matter how involved this transformation may be (and to what extent it mixes commuting and anticommuting variables), it nevertheless operates under the same rules as for ordinary integration variables, provided only the transformation is non-singular.

166

ALTLAND

ET AL.

But the origin is a singular point and must therefore receive special attention. This is detailed in Appendix B. By virtue of the same theorem, of all the terms arising in the expansion of the integrand in Eq. (4.2) in powers of anticommuting variables, only the term of highest (eighth) order survives (and perhaps the term of zeroth order, see Appendix B). This is why we evaluate this term only. In calculating the expression (4.2), we replace [trg(Q,,r3Q,r,) - 41 by trg(QOt,QOr,). The omitted term does not contain a contribution of eighth order in the anticommuting variables. This term is, however, of importance when we consider possible contributions from the singular point at the origin (which may be of lower order). 4.1. Term of Highest Order We observe that trg[QJ’] therefore cousider

does not contain anticommuting

variables, and we

M= ewb trg(Qor3Qor3))trg(Q,~,Qd) trg(QJ2).

(4.4)

Our procedure is in principle straightforward: We expand each of the three terms on the r.h.s. of Eq. (4.4) separately and look for combinations which yield a contribution to the term of eighth order. All steps in this subsection have been supported and checked by the computer-algebraic system epicGRASS [22], without which this calculation could not have been done. In Efetov’s parametrization, all anticommuting variables are located in the matrices ui and ui , see Eq. (C.6). In calculating the trace in the exponent of Eq. (4.4), we find that U, and u, appear always and only in the form

f=tl;z,u,,

g=iqz,u,.

Explicit calculation of the expression (4.5) shows that f=(

exp(d) o

0 I,)3([exp~)‘+

(4.5)

f can be written as p> 2

(4.6)

where (4.7)

Here, 3 has the form

(4.8 1

167

PERSISTENT CURRENTS IN MESOSCOPIC RINGS

and similarly for g. We observe that f contains only two anticommuting variables q2, qz. If it were possible to replace in exp{y trg(Q,,r3Q0r3)} the expression f by f (and similarly for g), the expansion in Grassmann variables would obviously yield only terms up to fourth order. This replacement off by f and of g by g can be achieved as follows. Writing Eq. (4.6) in the form f = U,fU-J and similarly g = Vi gU,, we define u; = UJ u2 Ui and parametrize u; by new commuting variables in the same way as u2 is parametrized in Eqs. (C.6) and (C.8) by old commuting variables. We substitute in the integral over M the new commuting variables for the old ones. This is permitted by Berezin’s theorem; the invariant measure is unchanged. The matrices U,-( UJ) and U,( Vi) generated in this way as additional factors exactly cancel the corresponding matrices U$(Uf) and UL(U,) arising from Eq. (4.6) and from g = Ui gU, in trg(Q0z3Q0t3). (We note that U, is effectively of dimension two and commutes with sin 0 and cos 6). Formally, the upshot of this procedure is that we replace f by f, g by 2, putting the matrices U and Ut (and their counterparts for g) equal to unity. This is the essential simplifying step in the calculation of the term of eighth order. For readers who wish to follow our calculation in detail, we give intermediate results in Appendix D. Here, we proceed to write the final result involving the eighth-order term as

lsin3(8)1 )sinh(8,)1 /sinh( ' (c0sh(e,+e,)-~0~(8))*

(c0sh(e,+e2)-c0s(e))2

(1 +d+

jm11*)3

where M, is defined as the coefficient of the term of eighth order in the Grassmann expansion of M in Eq. (4.4). We cannot give here the complete expression for M,; the computer printout covers many pages. 4.2. Integration

over the Unitary Group

Of the remaining eight integrations over commuting variables, those over the “phase variables” x and 4 (see Eq. (C.8)) are trivial: The phase factors exp(iXr,) and exp(i&,) appear in complex conjugate pairs only and therefore cancel. A similar statement holds for the variables m, m, , and rn: contained in F, , see Eq. (C.8). Indeed, F, appears in Ms only in the form T= tr(F,z,F,z,).

(4.10)

Writing F, as (4.11)

168

ALTLAND

(where U, u are implicitly (uj2+ 1u12= l), we have

defined by comparison T=2

This and the unitarity integration variable,

ET AL.

condition

with Eq. (C.8) and where

Ju12-2 (~1’.

(4.12)

allow us to choose T as the only non-trivial

lul’=$(2+T);

lulz=$(2-T).

(4.13)

WithRem,=m~,Imm,=m~,and-oo~mm,m~,m~~oO,theinvariantmeasureof the group SU(2) has the form d,u(F,)=-$(l+m2+m~2+m~2)~3dmdm~dm~.

(4.14)

The variable T appears in Ms in the form M, = exp(aT) i

(4.15)

aiT’,

1=0

where c1= 2y sin2 0 and where cli are functions of 8, 0,) and 8,. We therefore need to calculate for i = 0, 1, 2, 3, Ri = j dp(F,) exp(aT) T’.

(4.16)

Since for j= 1,2, 3 we have Rj= aLRo, it suffices to work out R,. We show in Appendix E that

Jdp(F,)

T’=s.

1 o’ 9

n even n odd.

(4.17)

Expanding the exponential in R, and using Eq. (4.17), we obtain R _ sinh(2cc) 02a

(4.18)

and, therefore, Ri = d cosh(2a) -$

sinh(2a),

sinh(2a) - $ cosh(2a),

(4.19)

PERSISTENT

CURRENTS

IN MESOSCOPIC

169

RINGS

We have used the computer-algebraic system REDUCE to replace in the expression for Ms as given by epicGRASS all integrals of the type Ri, Eq. (4.16), by the expressions (4.18), (4.19). The resulting expression is sufficiently simple to be displayed here. For the remaining three integrations over 8, O,, e2, we define new integration variables by A = cos 8, (4.20)

A, = cash O,, A., = cash 02, write exp(ftrg(&n’))=exp(ix(i-1,1,)), and have

X

(16yi12+1)(1-12)-2(1-e8~“-‘2’)~;(-8y(;1;-1)+1) ~:+I;+11*-2&~2-l

+2(12-IzI.,i,)(l-12)-(1-e + C.C.

8y(l--12))(~:-l)(;1:+~2-IZ:(Iz:+$+12-2J11,1,-1)2

1) > (4.22)

4.3. Eigenvalue and Energy Integration To calculate the last four integrals in Eq. (4.22), we follow two alternative strategies. First, exploiting the simple exponential dependence of the integrand on x, we perform a Fourier transformation of the integrand. With t the variable of the Fourier transform, the term exp(ix(l - AIL,)) becomes 2mS(t - ,I+ 1,R,), rendering trivial the integration over 1. We are left with the integrations over ,I,, 12, t, and x. These were done numerically. The Fourier transform does not reduce the number of integrations, but it eliminates from the integrand the oscillatory factor exp(ix(l - ,I, I,)) which causes difficulties numerically. The advantage of this approach lies in the fact that we can perform the x-integration as the last step. Looking at the numerical output before this last step is taken, we obtain information about the range of x-values contributing to I(4), i.e., about the number of states below the Fermi surface which contribute to the persistent current. The disadvantage is that the numerical procedure is rather involved. Second, we start by integrating over x with an infinitesimal cutoff factor in the exponent. The result of this first integration is a rational function of 1,. Therefore,

170

ALTLAND

ET AL.

the integration over II, can also be carried out analytically, integration over 1 and A,,

leaving us with the

K(~)= +25y1/2 i', dAjlmd&-*1'+@) (b,+e(y,

A)-12)4J,

4Y,

A,)-

2) 4 + 1 + C.C., A,E,

(4.23)

where ~,=(l6y~2+1)(1-12)-2(1-~e8J.~‘-“2~)~~(-~y(;l~-l)+1), e( y, 2) = 1 - e8Y(l- A2),

44

h)

= (142)W l(1; _

1)1/2 [‘T-q 2

(4.24)

,1~~2;;;;~-~,l,2)].

We did not succeed in performing the remaining integrals analytically and used numerical procedures. The result agrees with that of the first procedure, giving us confidence in the interchange of energy (x) and eigenvalue integrations used in the second procedure. 4.4. Perturbation

Theory for QO

It is useful to compare the exact result for 1(b) obtained by the steps described in Subsections 4.1 to 4.3 with perturbation theory. This is why we describe the latter approach in the present subsection. We start from the expression for I(#) in Eqs. (4.1), (4.2) and consider y to be a large number. With EC/d N 100 for the actual experiment, the validity of this assumption (and of perturbation theory) depends on the value of the magnetic flux 4, as would be expected. To parametrize the saddle-point manifold of Q-matrices we employ Eq. (3.30) with T(x) given by

Cl91 T(x) =

Jl+ab - ib

ia J-1l+ba

’

(4.25)

The matrices a, b represent the independent variables of the parametrization. Their explicit form may be found in [ 191. We decompose these matrices according to (4.26) where the indices refer to the space in which the symmetry is broken by the r3 -matrices. For lowest order perturbation theory it suffices to expand all graded traces to

PERSISTENT CURRENTS IN MESOSCOPICRINGS

171

lowest order in the matrices a and b and perform the resulting Gaussian integrals by means of a generalized Wick theorem. We have y trg[Q,r3Q,t,]

-+ - 16~ 1 trg[a”‘bjY](l ii’

- Sjj,)

2 trgCQo~l -+ -ix trg[ab] (4.27) trg[QoKQorj]

+ 4 - 8 c trg[a”‘Z@Z](

1 -S,,,)

Jj ’

trg[Q,Z,]

+ -4i+

2i trg[baZ].

This gives for K(y) K(y) -+ - 16iy’j2 Ix)dx dp(t) s0 5 x exp

1

-i

c [32y( 1 - Sjj, ) + 2ix] trg[&Pl) JJ ’

x 1 trg[u*‘VZ](

1 -S,?,) trg[buZ] + C.C.

(4.28)

Jj’

Using Wick-type contraction rules [21, 231 for the a, b-matrices in the preexponential terms we finally arrive at 1 K(y) -+ - 256iy’j2 * dx (32 y + 2i~)~ + ‘J-L 0

5

= -256~“~

jam dx ((32;;:T4x2)2

8 = -3. Y For y + 0, K(y) diverges as expected. This is in contrast to the exact evaluation; inspection of Eqs. (4.23) and (4.24) shows that K(y) + 0 for y + 0 as is expected on physical grounds. Comparing in Section 5 the perturbative result (4.29) with the numerical integration of Eqs. (4.23), (4.24), we obtain a criterion for the validity of perturbation theory. This criterion is then used to show that in the interval 0 < 4 < $bo, perturbation theory for all modes but Q. is justified. 5. RESULTS AND DISCUSSION We present the result of the exact integration over the zero mode, compare it with perturbation theory, and give the full persistent current corresponding to the model of electrons moving independently in a random potential.

172

ALTLAND

ET AL.

We first discuss the contribution (4.1) of the zero mode Q, to the persistent current and return below to the contribution of the modes Q, with m # 0. Instead of Eq. (4.1), we write l,,(d) = (m/d,) K(y) with K(y) = -K(y)/(871)“‘. The solid line in Fig. 1 shows K(y) versus y I’* . To an excellent approximation, this curve is parametrized as

f =JE,d KIY”* 0

(5.1)

--&-l+lc,’

where rcl z 2.25 and IC* z 12.40. Therefore, the current behaves [24] for small values of y as lo % 4E,~j/&. At y”’ = K; I/* z 0.28, the current reaches its maximum value [S] I,,,,, z 0.3 m/d, and decreases monotonically for y > K; ‘. Our nonperturbative calculation does not show the divergence typical for perturbative approaches, cf. Eq. (4.29). The dash-dotted line in Fig. 1 gives the result (4.29) of a perturbative calculation of the contribution of the zero mode Q. to the persistent current. We note that the perturbative calculation agrees very closely with the exact result for y > 1, while it deviates ever more strongly from it as y is decreased below unity, until it diverges at y = 0. The close agreement of both curves for y > 1 is very satisfactory: It shows that only in the immediate vicinity of the divergence at y = 0, is perturbation theory inadequate. We use this result below to argue that in the interval 0 < 4 < bbo, we are permitted to evaluate the contribution of all other modes Q, with m #O perturbatively. The zero-mode contribution (full line in Fig. 1) does not display the periodicity of the full current discussed at the end of Section 2, nor does it show any of the zeroes expected at I$ = f $jo, + ido, ... . These features result only when all other 0.3

0.2

.3 /y

01

-01

’ 0.0

FIG.

1.

I

I

1.5

3.0 Y v2

The function

K(y)

as explained

I

I

4.5

6.0

in the text

versus

y”‘.

PERSISTENT

CURRENTS

IN MESOSCOPIC

RINGS

173

modes Qm with m # 0 are included in the calculation. It was pointed out earlier that strictly massless modes in the functional integral occur not only at q5= 0 (here, it is Q, which becomes massless), but also at 4 = f (m/2) &,, m = 1,2, 3, ... . At each of these points, a calculation completely analogous to the one in Section 4 for Q, has to be done for the corresponding zero mode. The tails of the corresponding curves (analogous to the full curve in Fig. 1, but shifted in 4) have to be added to that full curve in the interval 0 d q5< iq$, to obtain the complete zero mode current. The dashed line in Fig. 1 gives the result of a perturbative evaluation of the full functional (3.37) including Q0 to lowest order in the disorder parameter (k,l)-‘. This curve coincides with the zero-temperature limit of the result of Ref. [7]. Since the result of this calculation cannot be written as a universal function of y (in contrast to the contribution of Q0 alone), we have used EC/d= 100 in the figure. This seems a realistic value for the experiment [l]. We know from the comparison of exact calculation and of perturbation theory for Q, that in the interval 0 < 4 < $J$,, the use of perturbation theory for all modes Q, with m # 0 is legitimate. We observe, on the other hand, that both perturbative results (dashed and dash-dotted curves) nearly coincide for y < 1. This demonstrates the dominance of the zero mode (Q,) contribution in the interval 0 d y < 1. In this interval, modes Qm with m #O give a negligible contribution. It follows that to an excellent approximation, the complete answer to our problem is obtained by taking the full curve in the interval 0 d y Q 1, and by continuing it beyond this interval and up to the point q5= &,/4 by the result of perturbation theory for all modes (dashed curve). Outside the interval 0 d q5< &,, the average persistent current is given by the arguments of antisymmetry and periodicity. The resulting form for 1(d) is easily understood: For ideal rings [4], the persistent current at T= 0 has a sawtooth shape, jumping discontinuously from - 2ev,/L to + 2ev,/L at I$ = 0 and decreasing linearly as 4 increases, with a zero at 4 = &,/4. Here, vF is the Fermi velocity. Impurity scattering has two effects: It softens the sawtooth function into a smooth curve, and it reduces the maximum value of the current (without changing the sign) by the expected [S] factor where M is the number of transverse channels. a (WW)“‘, It is of interest to determine the number of single-particle states at the Fermi energy which contribute to the averaged persistent current 1 In the vicinity of the maximum value at JJ* Z 0.28 (where the zero mode approximation is still meaningful) we find that significant contributions to the integral over AE stem only from the domain lAE/ Q 10d or so, i.e., from the last few levels below the Fermi surface. In summary, we have presented an (almost) exact calculation of the persistent current at zero temperature for an ensemble of rings, assuming the electrons in each ring to move independently in a random white-noise potential. This calculation keeps the particle number on each ring an integer and proceeds non-perturbatively. We have thereby solved the problems formulated in the Introduction. We have not removed the discrepancy with experiment-theory is too small by roughly two orders of magnitude. As stated in the Introduction, it is widely held that the interaction between electrons is responsible for this discrepancy. But the methods

174

ALTLAND

ET AL.

employed in this paper to calculate the effects of disorder, suitable for a one-particle problem, are not easily extended to a many-body system. The combination of many-body physics and the theory of disordered systems poses a great challenge.

APPENDIX

A: A

DIFFERENCE

IN SIGN?

Referring to Ref. [7] as I, we trace the difference in sign between Eqs. (2.9), (2.10), and (2.12). We indicate the arguments of all functions explicitly, as this is important. Inserting p = p0 + 6~ in Eq. (1.1) yields (c = 1)

(A.11 From the thermodynamic

identity N = -LJQ/ap, we have with N = N, + 6N that

For fixed pO, N, is still a function of 4. This is physically obvious because for T= 0, N, is the number of eigenvalues below pO, and this number depends on 4, because the spectrum does (although pLgdoes not by definition). Averaging Eq. (A.1 ) over the ensemble, using that &2/@ is exponentially small, and using Eqs. (A.2), we obtain

(A.31

This coincides with Eqs. (Il), (1.5) in more explicit notation. replacing (aN/ap)&, 4) by A-‘, the total mean level density,

f(4)=

+A6N%.

aN0

We follow I in

(A.41

To compare with our explicit formulas in Section2, we observe that at T= 0, with a factor of two for spin, N=2

a(9) dE tr{G(E-

s0

H)}

(A.51

PERSISTENT CURRENTS IN MESOSCOPIC RINGS

175

so that N,=2[%r{&-H)J, 0

(3)/= 2 {pa+611dE tr{G(E-

H)},

(‘4.6)

PO

aN0 T=(-)2tr

1

c!?(P~-H)$J.

Inserting the last two expressions in Eq. (A.4) and using Eq. (2.10), we do obtain Eq. (2.9), including the sign, if we assume that 6~ is large in comparison with the correlation width which determines the fall-off of g(s). As detailed in Sections 2 and 4, this last assumption is justified: 6~ is of the order 103d while the correlation width of g(s) is of the order 1Od. The difference in sign arises when aNo/@ is replaced by &?N/&z5. But the starting equation of I is I= -(aF/&$)(& N) with N and 4 independent variables. Hence, aN/@ = 0, and N = No + 6N then yields aN,/d# = -a~?N/&p, i.e., the correct sign.

APPENDIX

B: IMPLICATIONS OF BEREZIN'S THEOREM

Let ui(cli) and ui(pi) with i= 1, .... n be two sets of commuting (anticommuting) variables, and let us consider a non-singular transformation (f, y) between these two sets,

with i = 1, .... n. It is assumed that, with regard to the commuting variables ok, the functions fi and yi have compact support. Berezin’s theorem [21a] states that the integration measures for the two sets of variables are related by fi du, fi duj=detg i=l

(B.2)

j=l

in complete formal analogy to the commuting case. The graded determinant in Eq. (B.2) is called the “Berezinian.” Equation (B.2) is nontrivial because the transformation (B.l) is allowed to mix commuting and anticommuting variables. The new commuting variables may be given as functions of the old ones and an additional term, which is an even polynomial in the old Grassmann variables with coefficients which are also functions of the old commuting variables. It is far from obvious that such a mixture may be considered as an ordinary commuting variable. We have used this aspect of the

176

ALTLAND ETAL.

theorem to absorb the matrices exp($ and (exp(q))f appearing in Eq. (4.6), and corresponding matrices appearing in the calculation of g, into the commuting variables parametrizing the matrices u2 and v2. The Berezinian of this transformation is unity. We have also used Eq. (B.2) to transform our original parametrization of Q(x) in Section 3 into Efetov’s form, see Appendix C. This transformation again mixes commuting and anticommuting variables. But an additional difficulty arises: The transformation is singular at 8 = 8, = 8, = 0, and it does not have compact support because the range of 8,, e2, m, Re(m,), and Im(m,) extends to infinity. The last point is easily remedied by truncating the ranges of integration; the final result is not altered significantly because all integrals converge. The first point (singularity at the origin) is dealt with as follows. We remove from the integration volume a sphere centered at the origin with radius 6 = (ez+e:+e,)2 “* . Cutting off the integrand at radius 26, we fulfill the condition of compact support, can apply Berezin’s theorem, and introduce Efetov’s parametrization. In letting 6 -+ 0, we observe that the integrand in Eq. (4.1) vanishes at the origin. This is why there is no contribution from the singular point of the transformation. Otherwise, the term of zeroth order in the Grassmann variables might survive. In a singularity-free domain of integration with compact support, it is a trivial consequence of the definition of Grassmann integration that only the eighth-order term survives. Terms of lower order are always due to singularities.

APPENDIXC:

EFETOV'SPARAMETRIZATION[ 123

Our matrix Q, is related to Efetov’s matrix QE by Q, = - iQE.

(C-1)

Here, QE is written as (C.2)

where 8 is a graded matrix

(C.3)

and (C.4)

PERSISTENT

The conjugation

CURRENTS

IN MESOSCOPIC

177

RINGS

of a graded matrix, indicated by a bar in Eq. (C.4), is defined by ;=u+ V= kvtk,

k=diag(l,

cc.51

1, -1, -1).

The graded matrices u and v have the following structure:

u* =

VI=

(C.6)

1 + 2pP + 6(pp)* -2i(l+2pp)p

2ip( 1 + 2@) 1+2pp+6(pp)’

>’

v2=( 41 42) . The quantities q and p are two-by-two variables rll, vz, ply p2, v=

P=

‘11 ( -1:

q2 -vi+ > ’

Pl

( -p:

P2

-p:

) ’

matrices consisting of anticommuting

qq+=

P-Pt=

while F,, F2, dl, and d2 contain only commuting

F,=D-’

(1-im)2-lm,12 -2im,

;;

;I

( (

>

(C.7)

P: p2*

P2

p1

>

variables:

-2im: (1 +im)2- lrn,12 > ’

D=1+m2+lm,12

F2 = exp(ib3), (C.8) 41=

12

4* = ev(ix~3)y x Et-0,xl. In Eq. (C.8), m is real, while m, is complex. Here and in the main body of the paper, we denote by z3 a matrix of dimension 2, 4, or 8, as the case may be. We believe that the context always uniquely defines which of these possibilities is chosen. The integration measure of Efetov’s parametrization reads

178

ALTLAND

ETAL.

/sin3 191lsinh O,l lsinh f3,1

4QEl = j&i de, do, de (cosh(e,

+ e,) - cos e)* (cosh(e,

- e,) - cos e)2

1 Xdmdm~dm:(1+m2+/m,12)3

APPENDIXD: Explicit calculation Appendix C yields

SOME STEPS IN THE CALCULATION

of the exponent

OF M,

in M in Eq. (4.4) with

the help of

S= trgCQo~3Qo~31 = - trg [cos 0 G fu2 cos 0 i& fu2 + 2 sin 0 6 gv, sin 4 i& jiu2 +cos~v,gv2cosi9~gv2]

= -trf3C~,+~II+~,r,l,

(D.1)

where

f =iq3u1= ;: i’* ) ( > 22

g=v,z,v,

=

g11

03.2)

g12

( g21

g22 >

and

fil= -812~3~+exprj7, (evr?Jt, f22= 73

-

W51273,

fi2=4evr?

*

(

f2,=4(vT

>

9

(expr?)+=fT2,

.:‘)”

P.3) gll

=

8~2~:

g,,

=

73 +

+ 8~;

(exp

73 exp

P,

~2~3,

g,, =4i(exp P)+ g2,=4i(pf

P)’

(

*

-“iexpp=

‘*

>

, -g12.

PERSISTENT

CURRENTS

IN MESOSCOPIC

RINGS

179

The matrix rj is given in Eq. (4.7); the matrix j has the form PC-4

O ( PTP,*

--PIP2

0

) .

(D.4)

It is easily checked that f has the form (4.6) and that a similar statement holds for g. This causes us to proceed as in Subsection 4.1. The matrices cos 4 and sin 4 have the explicit form

@,I cos8=( (cos (cos O),, 1’ sin 0 =

(sin d),,

(

(D.5)

(sin Q,, >

with (cos d),, = cos 8.12,

(sin f?),, = sin 8. l,,

(cos d),,= (sin @22 = i

cash l3r cash 8, sinh t12sinh 6r

sinh Or sinh 8, cash 6, cash 8, >

P.6)

cash t& sinh Or sinh 8, cash 8, sinh O2cash 8, cash 0, sinh 8, > ’

We treat the terms abbreviated by A,, A,, and A,,,, respectively, in turn and begin with

- (~0soh2fims

Q,, fbi

= cos2 0 tr[ - 16q21]:r3 exp q r,(exp q)+ + r:]

X

cash 8, cash 8, sinh 8, sinh O,e-‘*+

- (1 - 8~:r/~)~ tr

sinh 8, sinh 02eizd cash 8, cash 8, >

cash Or cash 8, sinh 0, sinh 8,

sinh Or sinh 8, cash tI1 cash O2

(D.7)

180

ALTLAND ETAL.

Noting the relations (valid for both q and p) exp f(exp q)+ = 1, CD.81

z3 exp ii z3 = (exp f)+,

we obtain -trg[A,]

= 2(cos2 8 - cosh2 0r - cosh2 8, + 1) + m&( - cash’

--OS2 8 + 2 COS8 COSh8, COShe2

8, - cosh2 e2+ 1).

(D.9)

We come to the second term in Eq. (D.l),

-Q&M

= 2 trCfil(~2h (sin hI Gh gII(u2hI (sin hI (Uzh + 2f12b2)22 (sinb2, (u2)22 g2du2h(sinh, (tT;h,

(sinh2, (u2)22 g22(u2)22 (sinh2, = 2 trCA, + A, + &J. -f22b2)22

Again, we treat the individual 2 tr[AII,]

contributions

= 2 sin* 8 tr[( -8r],q3,

&)22

(D.lO)

separately: + exp q z,(exp f)+) exp rj F, exp fi

x (8~~~3~ + (ev bit z3 expP)(expPI+f’f(expr?)+l = 2 sin2 0( 1 - 8~2v/;)( l + 8~2~;) T,

(D.ll)

where we defined T as in Eq. (4.10). Next, we have

= 2% sin

8tr [ exp q (,,:

x (exp p”)+Fl(exp q)+ = -26 sin 8 x tr

[( q: p; sinh 8, cash 0, eic9+ X,

-q2p2 sinh 8, cash e,e?(@+x) - q:p2 cash 8, sinh 0r e’(bx)

-31

= -26 sin 0(q2p;u* qf@%

-p2)evP41

1

y12p$ cash f32sinh OIePi(Cx)

x(i: -

v2)F2Wnb24i(pz

cash 8, sinh f?,ePi(~-X)-~2p,u*

sinh 8, cash elePi(#+X)

sinh e2 cash OIei(“+X)- I]:P~U cash t12sinh 0Iei((-X)).

(D.12)

PERSISTENT

CURRENTS

IN MESOSCOPIC

Here, we have used a shorthand representation in Eq. (D.lO) can be simplified to give

for F,, Eq. (4.11). The third form

(r3 - 8r,*vlzz,) e’@r3( cash 0, sinh 8, sinh 8 2cash e ,

tr

= -2i*

181

RINGS

sinh 8, cash 8, e - i,p) cash e2 sinh 8,

>-i&K, 1

cash ez sinh 8, sinh e2 cash 8, e sinh e 2cash e 1 cash e2 sinh 8, - ~$3~) - 256v]:~,p,*p2)(coshZ 8, -cash’ 0,).

x (T3 + 8pTp273) eiXT3 = (4 + 32(p;p,

(D.13)

Finally, we have to deal with the last term in Eq. (D.l): -trg[A,]

= tr[(cos 0),, g,,(cos 0),, g,, + 2(cos @,, g12eiX’3(cosd),, e-‘X’3g2, - (cos d),, e -ix~3g22eix~3(cos(j)22 e-i~r3g22ei~T3] =cos2 8 tr[(8p,&,

+ (exp fi)+ 73 exp p)‘]

+2cos8(-16)trg[(expP)t(pz

e2 X cash 8,. .cash .

“‘) . ..

cash

e1cash e2>

cash 8, cash 8, sinh e2 sinh 8, =

sinh 8, sinh 8, cash 8, cash 8,

0 - cash* t12- sinh’ 0,) + 32p:p,

~(COS* x (-~0s~

e

+ 2 cos 6 c0sh 8,c0sh8,-

c0sh2 e2- sinh2 e,).

(D.14)

Collecting the results from Eqs. (D.9), (D.l l), (D.12), (4.7), (D.14) and proceeding analogously for trg[QJ2], we arrive at the following expressions: trg[Q0r3Q0ts]

= -2((2-

T) sin’ @+ 4 sinh2 0,)

+ 24(v]2r/2*-p2pz*)[(2

- T) sin2 e

+4c0sec0she,c0she2-4c0sh2e2] -26 sin 8(q&u*

cash 9, sinh Ole-i(d-x)

-~,p,~*sinh8,cosh0,e-~(@+~)+c.c.) - 27r/2qTp2p:(Tsin2

0 + 2(cosh2 0, - cosh2 (3,))

182

ALTLAND

trg[Q,J*]

ETAL.

= -2(cos 0 + cash 8, cash (3,) + 16(cosh 8, cash 8,-cos

8)(p,p:

+p2p:

+8p,p:p,p:)

+ 16 sinh 8, sinh 02(P1P;efZiY -P:Pze-2iX).

(D.15)

The corresponding expression for trg[Q,KQ,t,] is rather involved. Therefore, we restrict ourselves to those terms in trg[Q&Q,r,] which actually contribute to the highest-order term in M and indicate the suppressed parts by dots:

twCQo~Qo~J = C- W

COS’ 8 + Tsin* e) + 16(cosh2 0, + sinh2 0,) ] qrq:

+ [ 128 T sin2 8 + 384(c0s2 8 - cos 8 c0sh e1 c0sh e,) + 128(cosh2 8, +cosh2 e,-cos

ecosh 8, cash e2 -l)}

qrq1*r/&

+ [128(cosh2 8, - cosh2 0,) - 64Tsin2 01 ~r~:p2p; - 256 sin 8 cash 0, sinh 8, VV, qj+q~p~e""+

x)

- 256 sin tI sinh 8, cash tI,uq, q~q~p2ei(4-x' + 256 sin 8 sinh 8, cash 82u*~,tl:~2p:ePi+X) -256 sin 8 cash f31sinh t?2u*q,~:~2p2e-iC~+X) + 1024Tsin* e ~r~:n2~:p2p:

+ ... .

(D.16)

The simplifications due to the transformation u2 + u; = Uu, which absorbs the matrix exp(rj), and of the corresponding transformation for u2, is that trg(Q,r,Q0t3) depends on the four anticommuting variables with index two only, so that subsequent calculations become much simpler. To proceed, we introduce a symbolic notation. We define

S= trgCQo~3Qo~31; E = exp( yS) J= trgCQ&Qo~J

(D.17)

D = trgCQJ21 and associate with the Grassmann variables ql, VT, q2, rl:, pr, p:, p2, and p: the numbers from 1 to 8, respectively. A certain term of the Grassmann expansion of one of the expressions in Eq. (D.17) (S, say) is then identified by the corresponding lowercase letter and certain indices indicating the Grassmann variables involved h‘l denotes a term containing the product q2q$ which contributes to the Grassmann expansion of S). Uppercase letters with indices (like Si) just denote the collection of all ith order terms in the expansion.

PERSISTENT

CURRENTS

IN

MESOSCOPIC

183

RINGS

Now, Eqs. (D.15) and (D.16) can be reexpressed in the following way:

s=s,+s,+s, =’ + + + + + + + s34

s78

s38

s47

s37

s48

s3478

J=J,+J,+J,+J,

(D.18)

= . . + j,2 +

+ j1278 + j12,, + j,,,, + j1247+ j,,,, + j123478 + ..

j123,

D=D,+D,+D4

= d + 4, + 48 + 47 + 4s + d,,,, And for E = exp(yS) we have E=Eo+ES+E4

(D.19)

= exp(ys)( 1 + yS2 + 5 y2S: + yS4). Finally, we obtain for the eight-order term in M = E. J . D, M,=E,(J,D,+J4D4)+E,(J,Do+J4D2+JzD4)+E4(J4Do+J2D2+J0D4)

=exp(ys)(j

123478

d 56 +

11234

d 5678

+ Y(s34j1278+ s7sj1234+ s37j1248+ s3sj1247+ s47j1238+ s4sj1237) d,, + Ys34jndm

+

b’s3478

+

Y*(S,,S,,

+ S47S3g))j,&f.

+S37S48

(D-20)

This result for the term of maximal order in the anticommuting variables would be far more complicated if it were not for the transformation u2 + u;: The Grassmann expansion for S terminates at fourth order. Without this modification we would have to deal with all eight Grassmann variables in S which complicates matters considerably.

APPENDIX Comparing

E:

PROOF

OF

EQ. (4.17)

Eqs. (C.8), (4.7), and (4.13), we have T=4

l~l*-2=2(1-8D~~

lm,l*).

(E.1)

To calculate the integral of Eq. (4.17), we first evaluate for I> k + k; + k;, Z(k, k;, k;, I) = lrn dm dm; dm;’ D-3p’m2k(m;)2k; --m

(my)*“;.

03.2)

184

ALTLAND

ET AL.

This integral can be rewritten and solved to give

X

00 s

dm dm; dm; + a;(m;)*

--oo (b + ai(

+ am2)’

l)!!.

x (2(Z-k-k;-k;‘)+

(E.3)

From Eq. (E.l ), we have T”z2”

i

i

;

i=lJ

j-0

(>O

;

(-g)i~-*i(~;)*i

(mf)*(i-jl

(E.4)

and, therefore (we recall the definition of dp(F,), Eq. (4.14)),

=2”+2~o~o(;)(;)

=2

n+l

(2j-

(-8Y

I)!! (2(i- j)-

l)!! (2i+ l)!!

2(2i + 2) ! 2*’

(-4)’ ____ cjci-j(2i+ i! (2is 2)!

l)!!.

(E.5)

We note that the coefficients c =(2&l)!! k

WI

2kk !

appear in the Taylor expansion of (1 -x)-‘I*: (l-x)-l”=

f k=O

ckXk.

057)

PERSISTENT

CURRENTS

IN MESOSCOPIC

RINGS

185

Observing that

= f k=O

=

f i-0

f

CkC,Xk’l

I=0

i j=*

CjCipjX’,

(E.8)

1.

(E.9)

we are led to the identity i

cjci-,=

j=O

Inserting this into Eq. (ES) we obtain

s

dp(F,)T"=2"

i-0’ i

0

':

(n even) (n odd).

(E.lO)

This is Eq. (4.17).

ACKNOWLEDGMENTS We are grateful to V. Kravtsov, I. Lerner, A. D. Stone, F. Wegner, and M. R. Zimbauer for helpful discussions. One of us (A.M.G.) acknowledges financial support by the Studienstiftung des deutschen Volkes.

REFERENCES 1. L. 2. V.

P.

LEVY, G. DYLAN, J. DUNSMUIR, AND H. BOUCHIAT, Phys. Rev. Lat. 64 (1990), 2074. CHANDRASEKHAR, R. A. WEBB, M. J. BRADY, M. B. KETCHEN, W. J. GALLAGHER, A. KLEINSASSER, Phys. Rev. Lett. 67 (1991), 3578. 3. M. BOTTIKER, Y. IMRY, AND R. LANDACJER, Phys. Left. A 96 (1983), 365.

AND

186 4. H.

ALTLAND

F.

CHEUNG,

Y.

GEFEN,

E. K.

RIEDEL,

ET AL.

AND

W.-H.

SHIH,

Whys.

Rev.

B 37

0. ENTIN-WOHLMANN AND Y. GEFEN, Europhys. Lert. 8 (1989), 477; H. F. CHEUNG, E. K. RIEDEL, IBM J. Res. Deuel. 32 (1988), 379. 5. H. BOUCHIAT AND G. MONTAMBAUX, J. Whys. (Paris) 50 (1989), 2695; G.

6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16.

17. 18. 19. 20. 21. 21a. 22. 23. 24.

6050; AND

MONTAMBAUX,

SIGETI, AND R. FRIESNER, Phys. Rev. B 42 (1990), 7647; H. BOUCHIAT, G. MONTAMBAUX, AND D. SIGETI, Whys. Rev. B 44 (1991), 1682. H. F. CHEUNG, E. K. RIEDEL, AND Y. GEFEN, Phys. Rev. Left. 62 (1989), 587. B. L. ALTSHLJLER, Y. GEFEN, AND Y. IMRY, Phys. Rev. Letr. 66 (1991), 88. A. SCHMID, Phys. Rev. Lett. 66 (1991), 80. F. VON OPPEN AND E. K. RIEDEL, Phys. Rev. Lett. 66 (1991), 84. K. B. EFETOV, Phys. Rev. Letr. 66 (1991), 2794. V. AMBEGAOKAR AND U. ECKERN, Phys. Rev. Left. 65 (1990), 381; 67 (1991), 3192. K. B. EFETOV, Ado. in Phys. 32 (1983), 53. A. PANDEY AND M. L. MEHTA, Commun. Math. Phvs. 87 (1983), 449; M. L. MEHTA AND A. PANDEY, .Z. Phys. A 16 (1983), 2655; L601. M. L. MEHTA, “Random Matrices,” Academic Press, New York, 1967. U. &KERN AND A. SCHMID, Europhys. Left. 18 (1992), 457. A. ALTLAND, S. IIDA, AND K. B. EFETOV, preprint, 1992. K. B. EFETOV AND S. IIDA, preprint, 1992. B. L. ALTSHULER AND B. I. SHKLOVSKII, Zh. Eksp. Teor. Fiz. 91 (1986), 220 (Sou. Phys. JETP 64 (1986), 127). J. J. M. VERBAARSCHOT, H. A. WEIDENM~~LLER, AND M. R. ZIRNBAUER, Phys. Rep. 129 (1985), 367. S. IIDA, H. A. WEIDENM~~LLER, AND J. ZUK, Ann. Phys. (N.Y.) 200 (1990), 219. A. MUELLER-GROELING, Ph.D. thesis, University of Heidelberg, 1992. F. A. BEREZIN, “Introduction to Superanalysis,” D. Reidel, Dordrecht, 1987. U. HARTMANN AND E. D. DAVIS, Compuf. Phys. Commun. 54 (1989), 353. J. ZUK, Phys. Rev. B 45 (1992), 8952. Y. IMRY. in “Quantum Coherence in Mesoscopic Systems” (B. Kramer, Ed.), Plenum, New York, 1990. H.

BOUCHIAT,

D.

(1988), Y. GEFEN,