Statistical scattering theory, the supersymmetry method and universal conductance fluctuations

ANNALS

OF PHYSICS

Statistical

200, 219-270 (1990)

Scattering Theory, the Supersymmetry and Universal Conductance Fluctuations S. IIDA, H. A. WEIDEN~LLER,

Ma.u-Planck-Institut

fir

Kerphysik,

Heidelberg,

Method

AND J. A. ZUK* Federal

Republic

of German)

Received November 20. 1989 This paper describes a novel analytical approach to the problem of conductance fluctuations in mesoscopic systems which, in particular. gives account of the influence of the coupling to external leads. We consider the case of a linear disordered sample in the metallic regime. which is coupled to two ideally conducting external leads. Using the many-channel approximation to Landauer’s formula, we relate the conductance to the total transmission probability through the sample. The microscopic Hamiltonian of the quasi-one-dimensional disordered sample is formulated in terms of a random matrix, and the elements of the associated scattering matrix which determine the transmission are constructed from statistical scattering theory. We show that in addition to the Thouless energy, E,, and the mean level spacing, d, there exists in the theory, a third energy scale, r, determined by the number of channels in the leads and the strength of the coupling between disordered sample and leads. Related to r, is a new length scale, L,. We find that for sample lengths L 7 L,, the properties of the conductance depend only weakly on the coupling to the external leads and, for very large L, become identical with those of quasi-one-dimensional conductors in the weak localization limit. On the other hand, for L < L,, the coupling to the leads strongly affects the behaviour of both the average and the variance of the conductance. The magnitude of L,, is typically several magnitudes of ten times the elastic mean free path and thus comparable to the sizes of experimental devices. A further novel aspect of our work is the demonstration that the assumption of GOE statistics for the Hamiltonian is suflicient to yield universal conductance fluctuations. c 1990 Academx Press. Inc.

1. INTRODUCTION Recent advances in micro-electronics, especially techniques such as contamination lithography using transmission electron microscopes, have enabled the fabrication of sub-micron experimental devices [ 11. This has led to the observation of new phenomena like that of universal conductance fluctuations which may be described as follows: At low temperatures, the conductance of small metallic wires exhibits fluctuations as a function of external magnetic field, Fermi energy, or impurity configuration with an rms amplitude always of order unity in units of e2/h independent of the sample size or its average conductance. These fluctuations are not timedependent noise, but are reproducible for a given sample. Clearly, classical reasoning, which implies that conductance should be a self-averaging quantity, predicts much smaller fluctuations. The decisive requirement is that the temperature be suf* Present address: Department of Physics, University of Manitoba, Canada.

219 0003-4916/90 $7.50 CopyrIght t,~’ 1990 by Academic Press, Inc All rqhts of reproductmn m anv form reserved

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ficiently low so that the inelastic scattering length exceeds the sample size, in which case conductance must be understood as the coherent quantum-mechanical diffusion of the electron through a disordered region. Such systems are said to lie in the mesoscopic regime, which separates the microscopic (or atomic) regime, dominated by the quantum mechanical properties of the individual constituents, from the macroscopic regime where classical statistical mechanics is valid. In a quantummechanically coherent system, it is the transfer matrix which is a central quantity since one should now consider the conductance as being proportional to the total transmission probability of the electron across the sample. However, if one supposes the effect of disorder such that the eigenvalues of the transfer matrix become Poisson-distributed random variables (i.e., uncorrelated), then the rms amplitude of the fluctuations would be of the order of the mean and hence much too large. This leads one to infer the existence of correlations (i.e., eigenvalue repulsion). The theoretical challenge is to demonstrate the origin of such correlations. Although most experimental work involves measuring the conductance of just a few samples as functions of a varying externally applied magnetic field, this is expected to be equivalent, by an ergodic argument [2], to single measurements on a large ensemble of different but macroscopically identical samples. It is this latter situation which is most readily amenable to theoretical investigation, as it can be formulated in terms of the ensemble average over a random Hamiltonian akin to Wegner’s N-orbital generalization of the Anderson tight binding model. The problem has prompted considerable theoretical activity and has been studied from various points of view, including diagrammatic perturbation theory [2, 33, numerical simulation [4], renormalization group analysis [5], and the statistics of the transfer matrix [4,6,7]. Because of the approximations used, the above-mentioned analytical methods are essentially all confined to sample lengths very much larger than the elastic mean free path. The present work formulates an approach which enables the calculation of the average conductance and its variance analytically and non-perturbatively (in the impurity potential) for all length scales between a few elastic mean free paths and the localization length. In a novel approach to the problem, the many-channel approximation to Landauer’s formula and a statistical scattering theory, recently developed in the context of nuclear physics, are applied to a disordered sample coupled to two ideally conducting leads, at zero temperature. The disorder is modelled microscopically in terms of a random Hamiltonian which is defined on a chain of sites, connected by nearest-neighbour random hopping, each of which carries the random electron states within a sample slice whose thickness is taken to be of the order of the elastic mean free path. In this way, the original multi-dimensional problem is reduced to a multi-channel one-dimensional one. Physically, the sites characterize successive scatterings of the electron as it moves through the sample. Most importantly, realistic account is taken of the coupling between the disordered sample and the two leads-necessary to define and measure the conductance. This procedure yields a formal expression for the conductance in terms of the underlying microscopic random Hamiltonian.

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To calculate the average conductance and its variance, the formalism of the generating functional involving both commuting and Grassmann variables, first introduced in a similar context by Efetov, has been employed. Apart from the fact that, unlike the replica-trick, it is mathematically well-founded, the main advantage of this method is that ensemble averaging (over the random impurity conligurations) can be performed directly on the generating functional. After ensembleaveraging and the introduction of composite variables, the original theory is seen to be equivalent to a theory of iteracting graded matrices belonging to a certain supersymmetric coset space, which is of the non-linear a-model type. This resulting theory is analysed by performing an asymptotic expansion of the averaged generating functional in inverse powers of the number of open channels. The asymptotic expansion is valid as long as one is far from the localization transition. Indeed, it is found that the sample size limiting the asymptotic expansion is the localization length itself. This is the maximal length scale for which one could have hoped. The results for the mean conductance and the variance show that the coupling to the leads, treated only cursorily in most previous work, is essential for a complete understanding of universal conductance fluctuations. Although irrelevant for sufficiently large sample sizes, it constitutes an important factor in the behaviour of both the mean conductance and the variance for values of the sample length up to several magnitudes of ten times the elastic mean free path, which is comparable with the sizes of experimental devices. It introduces into the problem (which is commonly discussed entirely in terms of the Thouless energy) a new energy scale-the width for electron emission from the sample into the leads. Any finite system coupled to open channels is characterized by such a decay width. It is found that whenever this width is less than or of the order of the Thouless energy-a condition satisfied for samples of sufficiently short length-the contribution to the conductance fluctuations due to the coupling with the leads is significant in maintaining their universal character. In this regime, which is also characterized by the fact that the electron emission time must be larger than or comparable with the time for diffusion through the sample, the average conductance attains its maximal value and is nearly independent of sample length, i.e., non-Ohmic. The development of the formalism so far has been limited to studying samples whose transverse dimensions do not significantly exceed the mean free path for elastic scattering, so that there is essentially no diffusive motion of the electron in the transverse directions. It is possible, however, to extend the model to samples with larger transverse dimensions. The analytical calculations associated with this case are quite involved, though certainly feasible. On the other hand, the approach can easily be generalized to different quasi-one-dimensional (i.e., multi-channel ) sample geometries. This includes the two-lead ring experiment as well as four-lead and multi-probe devices. It would be particularly interesting to study the four-lead case since present experiments actually utilize a four-lead measurement, and also, one expects the observed (non-local) voltage fluctuations to depend crucially on the coupling to the open channels. Another development which is foreseen involves the

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inclusion of the effects of an external magnetic field, which can be done via the Peierls’ substitution. This will enable the calculation of Aharonov-Bohm oscillations in the ring geometry, and of the coherence energy of the “magnetofingerprints,” as well as a comparison of the theoretically calculated variance with experimental data [ 11. In essence, the present formalism is not restricted to electron diffusion through disordered wires: it applies equally well to the propagation of waves through pipes randomly filled with non-absorptive elastic scatterers. In other words, the multiplescattering of classical waves such as light and sound in a disordered medium is also amenable to investigation.

2. MODEL FOR DISORDER AND ITS HAMILTONIAN

The total system consists of a disordered sample of finite length, L, connected at either end to infinite perfectly conducting leads, all taken to lie along the x-axis. With appropriate boundary conditions imposed on the wavefunction in the transverse directions, electrons propagate along the leads as plane waves in the +x-direction, populating one of a finite number, /1, of transverse modes. To estimate L! we use the linear dimensions of typical disordered samples. In Section 5.1, we show that this estimate is valid even when the transverse dimensions of the leads exceed those of the disordered sample: in such a case, the additional channels do not couple to the disordered sample. Metallic samples have a two-dimensional cross-section with transverse lengths L, and L,, so that LI is given by k2,L,L,/n, where kF is the Fermi wavenumber. Semiconductor samples (i.e., films) have a one-dimensional cross-section with transverse length L, and n given by k,L,/n. Typical values are L, = 0 . 1 pm, L, = 0 .02 pm for metallic samples and L, = 5 pm for semiconductor samples. With kg1 N 5A, this yields .4 N 2500 and /1 N 3000, respectively. We note that /1>> 1 in both cases. We therefore calculate C and var(G) in terms of an asymptotic expansion in powers of /i -‘. We consider a disordered sample whose transverse dimensions do not significantly exceed Z, the mean free path for elastic scattering. We recall that in metallic samples, 1 is of the order of several hundred angstrom, while in semiconductor films, 1 is at least an order of magnitude greater. We divide the disordered region along the x-axis into K equal slices of length 1. (The fact that we thereby allow only for discrete values, L = KZ, K integer, of the length is irrelevant for what follows.) With each slice, we associate a site at which there are N possible states generated by a random Hamiltonian. Because the number N must accommodate the finite length, 1, of each slice, it is reasonable to assert that N N /i . (k,l). We shall work in the weak localization limit, kF1 9 1, so that N 9 A. Depending on the symmetry of the system, the random Hamiltonian is assumed to be a member of the Gaussian unitary ensemble (GUE), the Gaussian orthogonal ensemble (GOE), or the Gaussian symplectic ensemble (GSE). The GOE (GUE) exhibits the symmetries appropriate for systems with (without) time-reversal invariance and

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‘23 I _

absence of spin-orbit coupling, whereas the GSE takes account of the presence of such coupling. In this paper, we focus attention on the GOE, and defer the treatment of the GUE and the GSE to future publications [22]. Physically, the sites characterize successive elastic scatterings of the electron as it moves through the sample. Propagation of the electron through the disordered region is then provided by nearest-neighbour hopping between sites--either fixed or random-and by hopping between the left (right) lead and the first (last) site, respectively. Our preceding construction has reduced the original multi-dimensional problem to a multi-channel one-dimensional problem which comprises a left lead extending from x = -cc to x = uL < 0, followed by a chain of K discrete sites, and then the right lead extending from x = uR = uL + L. > 0 to x = + cc. In order to define the Hamiltonian, H, and also for the purpose of the subsequent scattering-theory treatment, we find it convenient to regard it as split into a free part, H,, and an interaction term, V, so that N= H,, + V. From the point of view of time-dependent scattering theory, V is the component adiabatically switched on at time t = -cc and switched off as t -+ + co, whence the eigenstates of H,, correspond to the asymptotic states of the theory and form a complete orthonormal basis for the quantum-mechanical Hilbert space of states. We choose V to be the site-site and site-lead hopping. Thus H, describes a totally disconnected system of two semi-infinite leads and K sites. Its eigenbasis contains both continuum (i.e.. scattering) and bound states, viz. the plane waves in the leads and the states localized on the sites, respectively. The eigenstates of the Hamiltonian H, consist of two classes; namely, those whose wavefunctions have support in the left and right leads, and those with support on precisely one of the discrete sites. To construct H,, we first consider its projection onto the right (R) and the left (L) lead. In each of these, we define transverse modes by appropriate boundary conditions, and we denote the associated orthonormal eigenvectors and eigenvalues by 1~:) and E,, respectively, with a = 1, 2, 3, .... c = L, R, and with the E, ordered in such a way that E, I cb for a 2 h. For .v~a, (or for x2 a,), the motion of the electron is governed by the operator of kinetic energy. The wavefunctions are composed of linear combinations of functions which are non-zero only for x I aL (or for x 2 uR) and are of the form exp( f ik,x) xt;bri there; but because of the need for boundary conditions at the endpoints of the leads, not all linear combinations are admissible wavefunctions. Here, blj is the electron spinor wavefunction with ci = + 4, and k, is defined by (2.1 ) where E is the energy of the electron, and m its mass. We shall refer to the modules labelled a as channels. At energy E, a channel is open (closed) if the corresponding solution, k,, of Eq. (2.1) is real (imaginary). The number of open channels at the Fermi energy, E = E,, is denoted by A. At x = a,, c = L, R, we impose arbitrary local boundary conditions, subject only

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to the constraint that the Hamiltonian HO, thereby defined independently for x < uL and for x > uR, be self-adjoint. In other words, given the absence of transmission, flux conservation implies that the allowed eigenstates in the leads correspond to waves totally refected at the endpoints. The scattering eigenfunctions of H,, Y&(x) = (x ( Y ‘&), supported on the interval x I aL for c = L and on the interval x 2 aR for c = R, can be written in several forms: (i) As functions Yz,i j(x) obeying incoming-wave boundary conditions in channel a and for spin direction d; (ii) as functions Ys,& ‘(x) obeying outgoing-wave boundary conditions in channel a and for spin direction ci; (iii) as “real” functions Y;,,(x). These are normalized to obey the following relations: fY’6,,Sa,6(E-E’)= (Yy!;:‘l Y$‘d’)

=(y~;‘Iy$;)) 5. = (Yug,l

Y$.

(2.2)

In the foregoing discussion, we have adopted the multi-index notation e = (a, 4) where a labels the channel, and ci = + $ is the associated spin direction. The inner product of Y(+ ) and Y’ ~ ) defines the unitary S-matrix, (Y$‘I

Y~(~‘)=S~)‘(E)6(E-E’)6”“,

We also display the asymptotic yL(+) .%a + ($'

(2.3)

form of Y “,!d ) for x + - co:

[ k;‘/‘exp(ik,x)

xbdri+ 1 S::)“(E) b,b

k;‘12 exp( -ik,x)

-~

1

xbdb .

(2.5)

The asymptotic form of Y kc;’ then follows from Eqs. (2.2)-(2.5). The asymptotic behaviour of the functions Y, R(+ ~ ) for x -+ + cc is obtained from relation (2.5) via the substitutions L -+ R and ii, + - ikb in all exponentials. The S-matrices S(O)’ carry the upper index zero to indicate that they correspond to the unperturbed Hamiltonian, Ho. The “real” functions Y’&(x) are defined in terms of the squareroot matrices [SC”‘] ‘I2 through the relations (Y’E,J Y$>‘)

= [s(Oqg

6(x5-E’).

(2.6)

Equations (2.1)-(2.6) contain results well known from non-relativistic many-channel scattering. We feel justified in reproducing’them here because of the presence of some misleading statements in the current literature. Let 47:’ denote the projection of Ho onto the subspace of states localized at site i. By hypothesis, there exists for each site i= 1, 2, .... K, a basis of 2N states

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FLUCTUATIONS

(the factor of two being due to the spin of the electron) such that the reduced matrix elements Hj,!J, defined by (2.7)

H&! 6, = (ipl Ho I&),

have a given statistical distribution whose form can be specified by expanding Hi;’ in spin space in terms of the Pauli matrices g-C,oJ, gZ. We again use the multi-index notation ,D= ( ,LL,fi) where p labels the state, except for its spin direction ,C= 2 4. Since we confine ourselves, in the present paper, to spin-independent interactions and the GOE, we state only the formulae relevant for this case. Then Hi:,’ is independent of spin,

with H(i)

= V”’

HliL

= L’LJ

H(i!*, I”

The independent matrix elements are Gaussian centred at zero, with second moments given by

(2.9)

distributed

random

variables

(2.10) The operator V can now be defined in terms of its matrix elements between the states Y;,, and lip). We assume that V connects only neighbouring sites, (ipI V ljl,) = HI:;.‘? 1 - 6,,)

(2.1 I)

with Hi;.” = 0 unless sites i and j are neighbours, i.e., have one side in common. To specify -V further, we consider two models [S]. (i) In the model with site-d~qonal disorder, the matrix elements of V for neighbouring sites are multiples of the unit matrix and form a fixed hopping matrix, H”.‘)= y

~(8i,r+ 1+ 6,+ 1,,) 81” 6pi

(2.12)

for i# j. Here, v is real. (ii) In the model with locally gauge-invariant disorder. the phases of the eigenfunctions are uncorrelated from site to site, and the elements, H”,“. of V connecting neighbouring sites are Gaussian distributed random variables centred at zero. They are uncorrelated for different pairs of neighbours (i, j) and are not correlated with the H(‘). Their distribution is restricted by the symmetry properties of the GOE. Using spin-independence once again, we have H;‘,” = H;;,” S,;,

(2.13)

H”,” = H”j.il = HILlI* it 1’ in’ ’ V,C

(2.141

and

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with

The parameter ratio v2/,12 measures the strength of the hopping-matrix elements in relation to the site-diagonal matrix elements, both in the site-diagonal and in the gauge-invariant disorder models. This can be seen by comparing CrV IH$j)l* in both models, We turn to hopping between leads and their adjacent sites. The associated matrix elements are given by (2.16) Inspection reveals that the quantities W’,,,(E) are real, provided V is Hermitian. We assume that their dependence on energy, E, is so small as to be negligible. This is justified because the energy scale relevant to our problem is the Thouless energy, E,, (introduced below) which obeys E, 6 EF. Therefore, most of the wavenumbers, k,, defined in Eq. (2.1), and consequently the matrix elements W(E), change very little as E moves through the interval [EF -EC, EF + E,]. The precise values of the Wz+ are left unspecified, except for the fact that they satisfy an orthogonality relation (See Eq. (5.3).) which is assumed for the sake of technical simplicity. We complete the definition of V by observing that all matrix elements between continuum states must vanish,

Having now formulated our model, we note that it does not allow for the presence of ballistic electrons, as there is no way for an electron to pass between the left and right leads without hopping. Since the relative intensity of ballistic electrons in comparison with diffusively transmitted electrons is given by exp( -2K), we expect our model to be quantitatively reliable for K> 3-5, depending on the desired accuracy.

3. CONDUCTANCE

AND S-MATRIX

We seek the connection between the Hamiltonian, H, defined in Section 2 and the conductance, G, or rather its dimensionless counterpart, g = (h/e*)G. This connection is furnished by the many-channel approximation to the Laundauer formula,

g=,c,(IS;‘;,R(E,)l*+ IS:~(E,)I*),

(3.1)

which expresses g in terms of the elements, SkF(E,), of the S-matrix for the Hamiltonian H. Here, S$(E) is the element for a particle coming in via the lead

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labelled c’ in transverse channel b, and going out via the lead labelled c in transverse channel a. A factor of two has been included to take account of the spin degeneracy of the electron. The Landauer formula, its derivation from linear response theory, and range of validity have been extensively discussed in the literature [9-121; a recent review is available [13]. Thus we do not wish to re-open the discussion here, and we shall simply take the validity of Eq. (3.1) for granted; at least in the case of many open channels, /i 9 1. In order to construct S”“, we use the Lippmann-Schwinger equation with H, determining the “free” resolvent and V acting as the perturbing potential. Since the projection of the V onto the subspace of continuum states, !P&, vanishes (viz. Eq. (2.17)) the kernel of the associated Lippman-Schwinger equation is of finite rank, and thus the Fredholm problem can be solved algebraically. We suppress all detail (which can be found in Ref. [14]) and proceed directly to the resulting closed-form expression for the S-matrix. It takes the form y; = (yC’SCO)C [s’““]~;! W&[D‘I;;;,’ w;:,,,[s’“““];~;. (3.2 uh - 2i7c 1 ‘I’.W./l.I’

The matrix D has dimension

K. N and is given by

Div = E dI,,, JG - Hi;‘)

+ in c W:,@ W~,Jdl, u.c

S,., + hi, d,.,) cY/,

(3.3

where i, j = 1, 2, .... K and p. v = 1, 2, .... N. The upper indices (c, c’) on D ’ in Eq. (3.2) stand for (1, 1) if (c, c’) = (L, L), for (1, K) if (c, c’) = (L, R) etc., in an obvious fashion. In the definition of Dz,, given by Eq. (3.3), we have omitted from the right-hand side a principal-value integral over the term C, Wi+ Wz,,,/(EE’). We have done so because the calculations become less transparent if we carry this term along. Moreover, its inclusion only modifies the formal definition of the “sticking probabilities” introduced in Eq. (5.42) without otherwise affecting the results. Since we do not attempt to calculate the sticking probabilities from a microscopic model, this modification is immaterial. Note also that we have introduced the notation Hi;‘) - Hf,!. It is easy to verify that S”“’ is unitary (One has to use the fact that [S’““]i” has this property.) and that the solutions of the Lippmann-Schwinger equation satisfy relations analogous to Eqs. (2.2)-(2.6), with S(O)’ replaced by 5’“” and the appropriate boundary conditions for the incoming and outgoing waves satisfied in each of the leads. On substituting Eq. (3.2) into Eq. (3.1), we find that the resulting expression for g does not depend on S (‘jr . This is physically reasonable as it demonstrates that g does not depend on the boundary conditions imposed at the end of either lead. Moreover, this finding implies that we may set S(“)C equal to the unit matrix without loss of generality. We do so, and use instead of Eq. (3.2) the simplified form S$‘=6cc’d,h-2in

c W:,,[D-‘1;;;: Il.Y

W;Iv.

(3.4)

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Our aim is to calculate the ensemble average of g and its variance (denoted g and var( g), respectively) from Eqs. (3.1), (3.3), and (3.4) using the statistical properties of the Hamiltonian H defined in Section 2. For this purpose, we introduce the notation gp:,(i)

=

7C

1

“Z.p

wZ,v(6cL

6il

+

6cR

hiK)3

(3.5)

a,‘

q,(c)

= 6”(6,, hi, + 6,. 6,,) 9)“(i),

for c = L, R, i, j= 1, 2, .... K, and write g in the form g = 4[tr Q(L) D-‘&?(R)(DP’)+

+ tr Q(R) D-‘Q(L)(D-‘)+I.

(3.6)

Since we shall focus on the model with locally gauge-invariant disorder in the next section, it is convenient to introduce a short-hand notation for the variances of the HE;” by writing H”~“H$‘$ P”

= (6,,, 6,,,, 6;,. ~5,~. + 6,,, 6,,, 6,s 6ji@4,

(3.7)

instead of Eqs. (2.10) and (2.15). Thus

M,=~[i2~,+u’(6,j+,+s,+,,j)].

(3.8)

4. GRADED FORMALISM

To calculate 2 and var(g), we employ the formalism of the generating functional involving both commuting and Grassmann variables. This formalism, first introduced by Efetov [15], has been described in detail in two recent papers [ 16,171 in a manner suitable for direct application to the present problem. It is for this reason that we present here only the essentials of the method. We do so for the model with gauge-invariant disorder; for the problem with site-diagonal disorder, we shall only cite the results. As in Refs. [16, 171, we define a generating functional for products of resolvents as the inverse graded determinant Z(E) = detggl(D

+ J[e]),

(4.1)

where E represents the quadruplet E= (E:, E:, E;, E;). The matrix D is defined as an extension of the matrix in (3.3), with slightly modified notation: D$(ip,

jv)=6”“‘6,~[E6,,

6”-

Hz;-‘) + i( -l)p+l

(O:,(L)

+QE,(R))].

(4.2)

The indices p, p’ = 1, 2 are block indices which denote the advanced and retarded Green function, respectively, while the indices c(, a’ = 0, 1 govern the grading of the

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matrix elements, with c1+a’ even for commuting variables [ 161. The source matrix, J, has elements J~~:(i~,jv)=6”“‘(S,,,,+,

229

and odd for anti-commuting

+s,+,,,.)(e”,SZ~,,(L)+~~521:,,(R)).

(4.3)

Then it is easy to verify that

for cx= 0, 1, and

The method of Refs. [ 17, 151 enables the ensemble average to be performed directly on the generating functional. After ensemble-averaging and the introduction of composite variables, the original theory is seen to be equivalent to a theory of K interacting graded matrices, o(i), i= 1, 2, .... K, viz. trgln(El+J(j)+iQ(j)-I:)

, (4.6)

where X,,,,(j)=S,,o(j) and g= AC’ (cf. Eq. (3.8)). Due to the orthogonal nature of the random ensemble, there is a doubling of the formerly four-dimensional graded (CI, p)-space, such that the matrices a(i) have dimension eight and the source matrix acquires the additional block structure J(i) = diag(J(i), J(i)). Likewise, Q(i) = diag( Q(i), Q(i)) where

Let us assemble all the indices now required to label the matrices o(i) into a collective index denoted by letters from the set A, B, .... For each site, i, the graded matrices (~,~~(i) span a topologically non-trivial domain as described in Ref. [ 171. In order for the expression (4.6) to be well defined, we require that M be positive definite. This requirement is always met for the physically relevant case, where r2 4 L’ (cf. Section 7). In the limit of large N, the integral in Eq. (4.6) is dominated by its saddle-points. which are given as the solutions of the equation

1 gjta(i)= -‘,f

a(j)-$-iQ(j)]

(4.8)

I

As already shown in Ref. [lS], due to the breaking of local graded symmetry by the site dependence inherent in g,,, and the breaking of global graded symmetry by

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the presence of the coupling to external channels, embodied in Q(j), point solution, o,(j), is diagonal and unique, having the form

the saddle-

a,(j)=rjl-idjL,

(4.9) The diagonal matrix L has entries solely determined by the p-index via ( - l)p+ I. A parametrization of a(j), suitable for its expansion about the saddle-point, a,(j), has the form o(j)= (Tj)-’ (o,(j>+W)Tj (4.10) with the structure (4.11) .

and 6P1=dlag(6p:,, 6 pz2 j ). The matrices a(j) and Tj have dimension eight. We expand the exponent in Eq. (4.6) in powers of the 6Pj up to second order and carry out the resulting Gaussian integrals. This approximation amounts to retaining only the leading term of an asymptotic expansion in N-‘. The same approximation is used in calculating the Jacobian of the transformation of variables a(j) --) (6Pj, t{,). Below, we shall evaluate Z(E) by using an asymptotic expansion in .&’ and keeping only the leading terms. Since N//i N (k,l), retaining terms of order n -’ and neglecting terms of order N -’ is tantamount to making an asymptotic expansion in powers of (kFf )-I and keeping only the leading term [2, 33. The present expansion is therefore valid in the metallic regime. After integration over the 6P/ the generating functional reduces to Z(E)=!

fi j=

dp(t:‘,)exp 1

C

- f c go trg adi)

a&)

‘.I

- f z trg ln(El+ J(j) + ~Q(A - GW)],

(4.12)

J

where (W,”

(A = ~,“d~)

(4.13)

(T’)-’

(4.14)

and a,(j)=

a,(j)Tj.

An expression for the invariant measure, dp(ti2), is presented in Section 7.1, and the matrices ti2 are discussed in Appendix A. Let us also mention that for the model with site diagonal disorder, the analogue of Eq. (4.12) for the ensemble-averaged generating functional takes the form

(4.15)

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where V = diag( V, V) with (4.16) which is a non-diagonal (hopping) matrix on the space of sites i,j= 1, 2, .... K. One should note that when used together with the logarithm in Eqs. (4.6), (4.12), and (4.15), the symbol for the graded trace, trg, implies a summation also over the level indices ,u, v, whereas this is not the case otherwise.

5. STICKING PROBABILITIES,

SOURCE TERMS, EXPANSION

AROUND THE SADDLE POINT

In this section, we expand Z(E), as given by Eq. (4.12), in powers of J(j). and keep only the terms which contribute to the calculation of 2 and 2. We then show that both the effective Lagrangian remaining in the exponential, and the source terms which we retain depend on the coupling matrix elements W:,,, only through specific coefticients, Tz, known as the “sticking probabilities,” which we interpret physically. This statement is proved to hold exactly. Finally, we expand the effective Lagrangian, the invariant measure and the source terms in powers of t{? and ri, around the saddle point, keeping only terms up to second order. In this way, we generate the lowest-order term for 2 of an asymptotic expansion in inverse powers of /1. The calculation of var( g) is described in Section 7.2. 5.1. The Trace over the Level Index p

In Eq. (4.12), the logarithmic term in the exponent is to be traced over the level index p, To perform this operation, we first observe that by virtue of Eqs. (4.9) and (4.14), we have trg ln(E-

o,(j))

= 0.

(5.1 1

Furthermore, both J(j) and Q(j) vanish unless j = 1 or K. Hence, the logarithmic term differs from zero only for j= 1 or K. We also observe that J(j) and Q(,j) are both the direct product of a graded 8 x 8 matrix and .!Z$,,(j), so that for ,j = 1, K. J(j) + iQ( j) = al’@ P(j).

(5.2)

After developing the logarithm in a Taylor expansion around the point .d i = 0, we can perform the trace over ,U in each term of the series by assuming the orthogonality relation

with j = 1, K when c = L, R, respectively, since it implies that tr S’(j) = 1 (Xi)‘, LJ Cl

(5.4)

232

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AND

ZUK

for I= 1, 2, .... In general, the quantities Wz,p can always be transformed to ones which satisfy Eq. (5.3) by means of the orthogonal transformation on the channel indices, a, b, .... which serves to make the average S-matrix diagonal. Since such a transformation does not affect the conductance, we can assume relation (5.3) from the outset without loss of generality. On summing the expression which results from implementing Eq. (5.4), we find that the logarithmic term in Eq. (4.12) can be expressed as -ij=;K

; trgln[E-o,(j)+

iX~L+Xi,(E$B+c$F)],

(5.5)

where the “trg” now extends only over the graded space. The 8 x 8 graded matrices R,, EF? which arise from the source term, J(j), are given by (b),,

=@,,.*+1

(5.6)

+bp+,)b,F,

where k,, k, are the 4 x 4 graded matrices (5.7)

We find it convenient to sometimes adopt the shorthand notation E.E=EgkB+&J&, E

(5.8)

. k E Esks + cFkF,

We have also made the identification (E;, E;, &,

E;)

E

(E;,

E” R’

$7

(5.9)

8;).

Let us rewrite Eq. (4.12) as Z(E)=1

fi i=

such that .Y&~(O)

dlr(ti,)ex~{-~~~,,,,,,(E)j

(5.10)

expW%J

1

= 0. Defining the matrix

A(tj)=

( U&(1

*:2t41

+ t;zt$,)“2

it{,( 1 + f& t{2)1/2 -

t:1

t:2

(5.11) I

so that oG(j) = go(j) - 2i d-‘d( t’),

(5.12)

and using the relations L2 = 1, trg In L = 0, we see that (5.13)

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with

=%I

=

$

c

gi;

Q

OG(~)

a&)

1. I

= -1

g, d’d’[trg

.A(?) .kY(t’) + trg LK(r’)]

(5.14)

i. I and, recalling Eq. (5.5), % = t C trg ln(E1 - C,(j) I

+ iQ(j))

=- ’ 1 Ctrgln[l+A,+x,+lj(E-r,)r”AiL.“(“)]. 2 /=l.K 0 0

(5.16

Furthermore.

El- E,(:,+jQ(,j) =i

C ,=l.K

1 trgln(1 a

J(i)]

t-x:).

(5.17)

where we have defined - ix: X:=AJ+X:-i(E-rJ)L+2A’L.X(f’)

L(e’

R).

We retain the term Y&r in the exponent and expand Z(E) in powers generate the terms needed for the calculation of 2 and z. 5.2. Sticking

(5.18) of (c/.x’)

to

Probabilities

We consider the term YI and show that it depends on the parameters XL, ~3‘, r’, and E only in the form of particular combinations, Tz, which have a simple physical interpretation. For this purpose, we focus attention on the argument of the logarithm involved in Eq. (5.15), omitting the indices j, a. Defining the quantities p3 6, by A P=((E-r)2+(A+X)‘)‘~2’ IHL e =cos%+isin%L A+X+i(r-E)L = ((E-r)’ + (A +X)‘)“”

15.19)

(5.20)

234

IIDA,

the trace of the logarithm

in Eq. (5.15) can be written in the form

WEIDENMijLLER,

AND

ZUK

trg ln( 1 + 2peieLLd).

(5.21)

LJi? = eieLLd 9

(5.22)

After setting

we work out Eq. (5.21) by expanding in powers of L&?. This expansion is greatly simplified by observing that (L.R;e)’ = A( 1 + 2 cos B(zz)),

(5.23)

where

A=( t1bf21 ,2;t,j,

(5.24)

[A, LJtl/] = 0.

(5.25)

and that

These relations allow us to consider a = A and z = Ld’ as c-numbers, express higher powers of z in terms of a and a term linar in z. Using Eq. (5.23) in the identity z”+~ = z’z”, we obtain the relation z”-2acosBz”-’

-az

n-2-

-0

and to

(5.26)

for n = 2, 3, .... We read this as a recursion relation subject to the initial conditions z” = 1, z’ = z. The characteristic equation has the roots ik=acos8+,/Z

which, in conjunction

with the initial

(5.27)

conditions,

leads to the solution

(5.28)

The coefficients c, can be expanded as a power series in a. This yields

C(n-

=

z.

1)/7-1

(2 cos t!ler (“-:-‘>

(2cos Oa)n--r-l,

(5.29)

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FLUCTUATIONS

where [p] denotes the largest integer less than or equal to p. It is useful to note that the powers of a appearing in the sum (5.29) extend from $I-1) in

for for

to n-l to n

n odd, n even.

(5.30)

Now, for an arbitrary function, F(z) = f

a,?,

(5.31)

fl=O

we have trg F(D)

= f

a, trg zn

n=O

Since c, is a polynomial

(5.32)

~(,trg(c,z+c~~,a).

=“!,

in a and a is block-diagonal,

we have (5.33)

trg(c,z) = trg(cner@‘u), Using also the fact that trg(@)

=0 for all integer k>O,

trgWJO = &

it follows that

f a,, trgC(2 cos Ba)(c,- , + cos &,,)I. n=l

(5.34)

We insert Eq. (5.29) into Eq. (5.34), and find, after some simple algebra, that

trgF(LA/)=&p=of Fp trg(2

cos 8~)~ + I,

(5.35)

where

Fp=n=O i 0*n (~,+p+~f~o~~~,+p+,).

(5.36)

Writing p = j? cos 8, we use the results (5.35) (5.36) in the power-series expansion of the expression (5.21), for which we have

a,= _ t-v

cos n

w ’

(5.37)

Then, with the aid of the identity 1 -= 4

co s0

dt e -q’,

(5.38)

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AND

ZUK

applied to the denominator of Eq. (5.37), and after some straightforward tions involving the summation of a binomial expansion, we obtain trgln(l+2~cos8L~)=trgln(l+4~(1-~)cos28t,,t2,).

manipula(5.39)

Note that the graded trace on the rhs extends only over matrices of dimension four. Recalling the definition p = j cos 9, and using Eqs. (5.19), (5.20) for p and /I yields the result 1 %=i,=;,,

T trgWl+W,%).

This shows that the effective Lagrangian W& only through the coefficients

(5.40)

depends on the coupling matrix elements

T;dljTf;+6,T,R, j= 1, K, which are the sticking probabilities,

(5.41) given by

4Aj.Y’

(5.42)

T’=(E-$)2+(ii+~;)2.

We observe that 0 < T’, < 1. Let us now give a physical interpretation of these coefficients. To this end, we remark that a calculation similar to the one described in Section 4 and the present I section, though considerably simpler, can also be performed to yield SG’,--the ensemble-averaged S-matrix. This is presented in Appendix B. As a result, one finds I sz = iv 6&,~, (5.43) where E

is given by [c = L, R t+ j = 1, K] F=r’,-E+i(Ai-X;), aa r’-E+i(A’+X’,)’

(5.44)

from which it follows that T; = 1 - jE,c’.

(5.45)

Equation (5.43) shows that the average S-matrix is diagonal. In view of the statistical assumptions formulated in Section 2, and Eq. (5.3), this fact I is not surprising. While the full S-matrix, S$, is unitary, the ensemble-average, S; , is not; the coef1 licients TE measure the unitary deficit of Ss . Equating the ensemble average with the running average over energy for a single member of the ensemble, via an 7 ergodic hypothesis, we see that Ss describes the fast part of the scattering process. (We can use a Fourier transformation to go from the energy representation to a

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time-dependent picture. Since, by delinition, St;’ is energy-independent, it yields the fast part of the reaction.) Consequently, TT, measures that part of the flux incident in channel (c, a) which is not re-emitted instantaneously into that same channel. Hence the name “sticking probability.” The fraction TE of the incident flux is absorbed on site 1 (or K), and is either re-emitted diffusively into some channel (c, h), or moves diffusively into the disordered sample. It is only this part which can contribute to the conductance. In the metallic regime (kFI$ l), where the electrons penetrate deeply into the disordered sample before they undergo the first scattering process, we expect the T: to be close to unity.’ Indeed, in the metallic regime and for sufficiently small sample sizes, we expect g to be close to its maximum value, /1. We show below that, in the framework of our model, g is actually given by ix,,, Tz under these circumstances, and is therefore nearly equal to ,4 only if Tf, 2~ 1 for most a, c. In any case, the asymptotic expansion used below to calculate 2 and 2 is one in inverse powers of &,,, T; rather than in inverse powers of /1. As a side remark, let us mention that a significant reduction of all the Tz to values far below unity can be expected in the case where the electrons have to tunnel through a barrier on their way from the leads into the disordered sample, or vice versa. We believe that it would be of interest to explore this possibility experimentally, and to compare the results with our theory. At the beginning of Section 2, it was indicated that the number, A, of open channels in which flux can be absorbed is given in terms of the cross-sectional area, A DS? of the disordered sample, rather than in terms of the cross-sectional area, A LD, of the lead, even if A,, b A,,. Let us make this comment more quantitative. In physical terms, given a pipe of area A,,, closed at one end but with an opening of area ADS in the closing lid, it should be obvious that the flux which leaves the pipe through the lid is bounded by Ans, independent of the size of A,,, as long as A LD 2 A,,. In formal terms, the argument is not rigorous. It is based on the observation that for a lead-site hopping interaction, V, which has a range small in comparison with I, Xi N C, ( IV:,,)’ is proportional to the square of the overlap of the channel wavefunction, Y& (taken at x 21 a,) with the area of the site. Therefore, Xl; scales roughly with the ratio ADS/A,,. Since T: is non-linear in XI;, the argument is only an appproximate one. 5.3. Source Terms

In the previous section, it was shown that the effective Lagrangian depends on the lead-site coupling matrix elements, WE,,, only via the sticking probabilities, T:;. In the present section, we show that the same statement applies to the source terms needed to calculate g and var(g). The source terms arise from the expansion of the expression exp{ - -Y)source(&) 1 in ’ We emphasize that this expectation is based on physical model. where r’, A’ and Xi are free parameters.

reasoning

and cannot

be deritled

within

our

238

IIDA,

WEIDENMijLLER,

AND

ZUK

powers of E . R. Noting that x ‘,, defined in Eqs. (5.17) (5.18) is linear in E-R, and indicating by dots, terms which do not contribute to 2 or to 2, we have exp { - Zource )=exp{-i

,C J=~,K

~trgln(l+~~)] a

= exp

trg x’, - i trg(X’,)’ + . . .

11

This shows that we need to evaluate the two traces, trg x’, and trg(Xi,)*. Dropping the indices j, a as before, using the notation (5.19), (5.20) with p = j cos 8, and writing LA! = edieLLdH, (5.47) we can express x in the form x= -i(l

-P)cos

8

1

-=L(&.

1+2jIcoseL?e

R).

(5.48)

We now perform a calculation that is completely analogous to the one which leads from expression (5.21) to Eq. (5.39). The result is x= -i(l-P)cos8

l[

2p cos 8 1 + 4/?( 1 -/I) cos* ea

From Eq. (5.6), we see that in block notation,

(Z

- 2p

cos

ea)

I

e-ie’L(c

.R).

(5.49)

E. R is off-diagonal, (5.50)

Therefore, only the off-diagonal

trg 2 = 2$( 1 -/I)

part, zoff, of z = Ly

cos2 8 trg

[

1 i + 4p( 1 - p)

contributes to trg x. Hence,

COS* ea

‘owe

1

-‘eLL(& .R) ; (5.51)

which, recalling that (5.52) can be written as T -8LL(&. k) . 1 + Tu ‘OtTe

1

(5.53)

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Using the definitions that

CONDUCTANCE

of z,~ (cf. Eq. (5.11)) and e-“‘.,

/q = zoRe --iBLL(c.

(

239

FLUCTUATIONS

as well as Eq . (5.50) we find

t,,(l + t*,~I*)“*(E.k) 0

0 t,,( 1 + t,,t,,)‘!’

(E .k) >

(5.54)

This and the diagonal structure of a yield the result trgn=itrg

T 1+

tr2(1 + t,, t,,)“’

(e.k)

1

Tfnt21

+ (tlz~

t,,).

(5.55)

This completes the proof for trg x. The expression for trg(X’) can be simplified an entirely analogous way. Here, it is helpful to use the fact that e- yE. Keeping only those contributions 1 T trg x2 = ; trg i i 1 + 7312f2, + i (1 - T) trg

Q = (E. I;)eieL.

(5.56)

relevant for calculating

g and 2, we find

1

t,,(l +t,,t,,)“*(~.k)

12

*+(t,y+~,)

(E. k)

1 TFtf2: 21

in

Tr**

1+

fl2 Tt2,

(E.k) t12

+ .. ..

1

(5.57)

Equations (5.46) (5.55), (5.57) and the definition (5.45) of T’, constitute our general result for the source terms. To summarize this section, let us assemble all the various components by writing

(5.58)

where the dots indicate terms which do not contribute 5.4. Asymptotic

to 2 or 2.

Expansion

Before generating this expansion, we simplify expression (5.14) for the term W,, which appears in the generating functional, Z(E), by assuming that v2/A2- (k,I)-’

G 1.

(5.59)

240

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WEIDENMtiLLER,

AND

ZUK

(This approximation will be further discussed in Section 6.) We accordingly expand g, = (M-‘),j as a power series in u2/A2, keeping only terms up to first order. This yields (5.60) With the help of Eq. (5.1 l), the term 9, then reduces to

- trg ti2(1 + tiI ti2)lj2 ti,(l

+ fj2t&)1/2]

+ [t:, tf t&l,

(5.61)

where we have assumed that A’ = A is independent of the site label j, which implies that the average level density, p = N/(nA), is the same at all sites; and we have defined 4N A2v2 x=34.

(5.62)

b

Thus Y0 depends on the characteristics of the saddle point via the combination x. The asymptotic expansion is now generated by expanding nent of the generating function in powers of ti2, t’;r , and by to second order in 9& and lowest possible order in L&,,. [trg(t’,,t’,,)Expanding

the logarithmic

trg(t;2t&)]

and the disorder only the terms in the expokeeping only terms up For &, this yields

+ [ti2t-t

term in Eq. (5.40) is straightforward

t’,,] + ....

(5.63)

and gives

1 =% =2 j=F,a F trg W + W:,G,) =k JJ, ,

(C T&) trg(t{,t&)+ a

(5.64)

....

With regard to the source terms, we recall Eq. (5.46) which implies that in x(x’), we need to keep only terms linear (quadratic) in ti2 and fir. Thus, trgX’,=$

T’, trg[(f{2+f&)si.k]

trg(X’,)2=~(Tj,)2trg[(f(2si.k)2+(t{,.si.k)2]+

+ ...,

(5.65) ....

Finally, to lowest order in t,, and t,, , the expression for the measure &( t,,) is simply given by the product of the differentials of the variables comprising the

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matrix f,,. This is the order needed to generate the leading term of the asymptotic expansion. We abbreviate the product as dt,,. Collecting everything, we find Z(c)=/

fi &exp /=I

1-k

=
c (n-‘1,

trg(f;,lj,)}

exp(-2&,urce(i:)) (5.66)

i,’

where ev f - =Kour,, > is to be replaced by Eqs. (5.46) (5.65). The matrix

dimension

I7

I is of

K and has the form

(z7 -‘)ri =.X[2bd-6i~i+,

=.f+

-6,,i+,]+6ij[S,I(TL--.Y)+firK(TR--~)]

1:

-:..-:

:

,;;J

(5.67)

where we have introduced T”= C, T; and defined y“ by T”= sy’, c= L, R. We note that by construction, the integration over the t I2 has become Gaussian. Hence the source terms can now be simply evaluated with the help of Wick’s theorem and the contractions given in Appendix A. The leading-order result is

where the dots indicate terms that are of order /1’ or smaller. We shall consider the evaluation of such higher-order terms in Section 7. 6. THE MATRIX

Ii’ ~~’

To evaluate Eqs. (5.68), it is necessary to calculate the element IJIK in terms of the matrix ZIP’ defined in Eq. (5.67). At the same time, it is necessary to attain a deeper physical understanding of the properties of 17 _ ‘, since this matrix obviously determines the behaviour of the mean conductance and, as shown in Section 7. of var(g). Indeed, we note that the last three factors on the rhs of Eq. (5.68) can be interpreted as (i) the probability (C, T:) for the electron “to get stuck” at the first site, having arrived in any of the channels of the left lead; (ii) the probability ( nlK) to diffuse from site 1 to site K; (iii) the probability (Cb TF) for the electron to leave site K and enter the right lead, populating any channel. By time-reversal symmetry, this last probability is equal to the probability for the electron, arriving in the right lead in any given channel, “to get stuck” at site K.

242

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ZUK

Our purpose in the present section is to display the most important properties of 17 -I, and to work out IZIK. Owing to the tridiagonal structure of l7-‘, all of this can be done analytically. More details will be supplied in a separate paper. 6.1. Expansion in Powers of v2JA2 We recall that the inverse of the matrix M, in Eq. (5.60) was constructed via a series expansion in powers of v2JA2, terminated after the terms of first order. This step is an essential ingredient in the construction of I7 -’ and requires justification. We now show that v2/A2 is of order (k,l)-‘. The omission of terms of higher order is therefore consistent with the condition N $ A, which we used in Section 5. For simplicity, we consider a strictly one-dimensional sample; the extension to higher dimensionality is obvious. A fully microscopic model of disorder would treat each of the K sites (“slices”) introduced in Section 2 as consisting of a linear array of lattice sites; the number of lattice sites in each of the big sites (“slices”) being roughly equal to N N k,l. Electronic motion within each slice would be modelled as nearest-neighbour random hopping between sites, whence the corresponding Hamiltonian matrix, X”, would be tridiagonal with the diagonal and non-diagonal elements denoted D, and ND,, respectively. To relate this microscopic model with our model in Section 2, we equate the mean square elements,

Coupling between neighbouring slices in a microscopic model is due to the coupling element, ND, connecting the last and the first lattice sites within the two slices. Hence,

Therefore, v* 1 ND’ -N-N i2-N$+ND2

(k,l)-‘.

(6.3)

This result agrees with our later finding (See Section 6.4) which relates v*/A* with the diffusion constant D, and again shows that v*/A* is of order (k,Z)-‘. 6.2. Calculation of IIlK To calculate nIK, we recall the abbreviations T’=x

T;,

T’ = xy”,

(6.4)

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and define the N-dimensional Eq. (5.67) implies that

243

FLUCTUATIONS

tridiagonal

matrices MCN’(y)

by [ 1, y, 1IN. Then

( > --i

Kdet(n-L)=(yL+l)(yR+1)det(M(K-2)(-2)) +(yL+yR+2)det(M(K-3)(-2))+det(M(K~~4’(

-2)).

(6.5)

Using Kramer’s rule, we obtain IT,, =x”-‘[det(flP’)lP’.

(6.6 )

It remains to calculate det(MCN’( -2)). Using induction, det(MCN)( -2)) = (- l)N (N+ 1). Therefore,

one easily finds that

1

U,K = x(yL + yR + (K- 1) ,IL”r,R)’ We simplify this expression by making

the physically plausible assumption

that

TL = TR = T. We also recall that T: 2 1 for all a, c, so that Tg A. Then,

XY g= 1 + i(K-

(6.8)

1)y’

Eq. (6.8) implies that 2 = xy 2: A 2.x

g-F

for

(K-

l)yL2,

for

(K-

1)~ % 2.

(6.9)

The latter of relations (6.9) shows that for sufficiently large lengths, 2 is Ohmic and independent of the coupling to the leads, while the former of these relations shows that for short sample lengths, S is non-Ohmic and entirely determined by the coupling to the leads. We define a critical length L, = &I, which characterizes the transition between the two regimes, by the condition Lo/l = K. = 2/y.

(6.10)

6.3. Eigenvalues of II - I

We calculate the eigenvalues, Pi, and the normalized eigenvectors, ufn’, of the matrix n -I, where n = 0, 1, .... K- 1. For pedagogical reasons, we begin with the case of an isolated system without coupling to the leads, i.e., ‘Jo = yR = 7 = 0. Then

(6.12)

244

IIDA,

WEIDENMtiLLER,

AND

ZUK

where i= 1, 2, .... K. We observe that in this case the lowest eigenvalue is zero, and the associated eigenfunction is a constant. Non-zero coupling to the external channels can be taken into account by first-order perturbation theory in y under the assumption y + 2/K, which serves to ensure that the shift due to the perturbation is much smaller than the unperturbed level spacing. The result for the shifted eigenvalues is then 1 --&(K-

l)(K-2) (6.13)

n = 1, 2, ...) K- 1. If we retain only leading-order large-K limits, then we obtain

corrections

n=l,2

i&=x(s+g)+O(Ky’),

provided we restrict our attention to the low-lying Also, the lowest perturbed eigenvector is given by iYi”)-i[l+(f(K+l)-i)~]. JK

,...,

in the small-y and

(6.14)

excited eigenvalues, 1
(6.15)

Consequently, K-l

=L-E+O(K’y),

2xy

4X

(6.16)

consistent with Eq. (6.7). This confirms that the cross-over to Ohmic behaviour of g coincides with the limit of validity of the perturbative form (6.16). We have also worked out the eigenvalues of 17 -i in the non-perturbative region. This analysis will be published elsewhere; suffice it to say that all eigenvalues are positive, and that one of them increases (essentially linearly) with y. 6.4. Diffusion

Model

This model is most easily discussed under the assumption A’= A for all j= 1, 2, .... K, which was introduced in Eq. (5.63) and used subsequently. The saddlepoint equations for the aO(j) differ from K uncoupled equations by (i) the coupling between the a,,(j) pertaining to neighbouring sites, which are due to g,j and are of order u2/A2 z (krl))’ 6 1; (ii) the coupling to the channels. These latter terms are of order A/N z (k,l) - ’ G 1. It is therefore probably a reasonable procedure to

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FLUCTUATIONS

approximate the solutions, A’, of the saddle-point equation by omitting the coupling terms (i) and (ii). Then, AJ = A is given by the solution of the saddle-point equation for a single GOE-we have A = E, if we put the energy in the centre of the GOE semicircle, and the mean level density, pj, at this value of the energy is given by p,i = p = N/(rcA) for each site j. Let us use a heat kernel representation to write Z7,, and, more generally, ITI,, in terms of (7 ~~*,

1s0I’dtP,(t), IT,,(?)=-

(6.17)

2nhp

whence P;(t) = [exp( -Zi-‘(2nhp)-’

satisfies the differential equation

t)lli

(6.18)

P= -zr’(27rhp)~‘P,

subject to the initial condition Pi(O) =cSJ1. Upon interpreting 17,j as the total occupation probability of site j (for a particle which starts out at time t = 0 at site l), the explicit continuum form of the heat equation for P,(t) will allow us to identify an expression for the diffusion constant, D. Insertion of Eq. (5. 67) into Eq. (6.18) yields the master equation ~,=~v(P;+,fP;~~1)-w(2-6i,-~iK)P,-h~1(TL6i,+TR~iK)P,,

involving

the transition

rates per unit time u’, _, -+ , = u’ = x/(27&p), f L.R = TL.R/(2np)

(6.19)

and widths (6.20)

for decay (i.e., electron emission) from the first (last) site into the leads. We shall discuss the physical significance of these quantities in Section 6.5. Omission of the coupling to the external channels (T= 0) and transition to the continuum limit P,(t) + P( t, y) where y = jZ, via K% 1, yields the diffusion equation (6.21 )

Thus we see immediately

that the diffusion constant is given by DC----

Xl2

2dip.

Using Eq. (5.62) A = A, p ‘v N/E,, and D N vFI, where E,, vF are the Fermi energy and velocity, respectively, this again yields v2/E.’ 2: (kFI) ‘, consistent with the discussion of Section 6.1. Solution of Eq. (6.21) subject to the boundary condition of no transport across the ends of the sample, d!P(t, Y)I~=~,~ = 0, leads to the energy eigenvalues E.,=hD$$,

n=o,

1,~ ,....

(6.23)

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AND

ZUK

For T=O, this result agrees with Eq. (6.14). We observe that El coincides with the Thouless energy, E,. The condition y $2/K, which ensures that g is independent of L, can now be written as rL+rR r decay = (6.24) %E,. K Conversely, the Ohmic behaviour of g holds if

r decay The condition

(6.25)

3. Ec.

(6.10), defining the critical length Lo, takes the form rdecay(LO)

= -$

EAL,).

(6.26)

We also comment that the eigenfunctions for the case (6.24) agree with those of Eq. (6.12) for Kg 1. We therefore see that, in this case, the coupling to the open channels can be taken into account through the replacement EO= rdecay. In Appendix C, we confirm the validity of this replacement by directly constructing the continuum limit of the matrix Zi-‘, with the coupling y included from the outset. Since the appropriate initial condition for Eq. (6.21) is given by P(0, y) = 6( y - 0 + ), the full solution becomes P(r,v)=i[exp(

-F)+iJ,

exp( -$f)ca(~)],

(6.27)

which clearly shows that, subject to the inequality (6.24), P(t, y) equilibrates (i.e., reaches the limiting constant equilibrium distribution) before it decays; thus maximizing the flux in the rightward direction, and hence S. In the opposite limit (6.25), decay occurs much more quickly than diffusion through the sample; and this reduces 2. We do not display the analytical solution, P(y, t), for this case. 6.5. Discussion The analysis of the matrix n-i and the calculation of ZZ,, have given us a complete understanding of the leading-order contribution (5.68) to g. Aside from the factors of 2 (arising from spin degeneracy), TL and TR (the probabilities for the electron to enter and leave the disordered region, respectively), the average conductance, g, is given by LrlK of Eq. (6.7), i.e., by the probability for the electron to diffuse from site 1 to site K. This quantity is length-dependent and can be expressed as 17,,=(TL+TR)-‘(I+$$’ (6.28)

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As already shown in Eq. (6.9), g becomes Ohmic and independent of the coupling to the leads when LBLO. In fact, in this limit it attains the standard form, 4npfiD/(IL) (cf. Eq. (6.22)). These facts do not require any interpretation. Rather, it is the fact that for L 5 LO, 2 is nearly constant (and therefore non-Ohmic) which must be explained. The latter form in (6.28) suggests that this behaviour is linked with the appearance of the new energy scale rdecay. We thus turn to an interpretation of this quantity, and to an estimate of L,. A first hint of the physical significance of rdecay was obtained from Eq. (6.27) with E,, = rdecay. After equilibrating within the disordered sample, the probability P(t, v) continues decaying exponentially with a decay width rdecay because it continues to feed the open channels. This interpretation of rdecay as the (mean) decay width is corroborated by the defining equations, viz. (6.24) (6.20) and (5.42), which together read v TL

r decay -La,r L (1 2lTpK

(6.29)

Following Ref. [19], we recall that p is the mean level density of each site, so that (pK)-’ is the mean level spacing, d, of the entire disordered sample. Therefore, h/d is the Poincare recurrence time of the sample, and d/(27c) is fi times the average frequency with which a time-dependent solution of the Schrodinger equation returns to some fixed spatial volume, say site 1 (or site K). The probability that it escapes from here to one of the open channels, a, is given by TA (or T,“), whence the associated partial decay width is Ty(27cpK) (or Tf/(27q1K)). It follows that (6.29) gives the total width as a sum of the partial widths for decay into any one of the open channels. The origin of the width rdecay can be traced all the way back to Eq. (3.3) in which the coupling to the open channels causes the eigenvalues of the matrix D-‘, and the poles of the S-matrix, (3.2) to lie below (rather than on) the real energy axis. Ultimately therefore, the appearance of rdecay is a consequence of our description of electron transport as a scattering problem, and rdecay can be interpreted as the mean decay width of the levels (quasi-bound states) in the disordered solid due to their coupling with the open channels. Equation (6.29), when written as rdecay = d( TL + TR)/(27r), implies that r$ d. Indeed, the mean decay width is typically two or three orders of magnitude larger than the mean level spacing-in other words, the levels overlap very strongly. This statement holds regardless of the length of the sample since T/d does not depend on L. The behaviour of g is strongly influenced by rdecay as long as rdecay5 E,,, or as markedly influences the lowest eigenvalue of 17-l. Now, since d long as rdecay decreases like L - ‘, so does rdecay; while E, falls off like Lp*. Thus, there exists a cross-over length, viz. LO, which we can estimate by appealing to Eq. (6.10), which we write as L,=4rcptiD/(lA). (We have assumed T’~L A here.) We use D = u,I where vF = fikF/m* is the Fermi velocity and m* is the effective mass of the electron; this relation holds for one-dimensional conductors. We recall that in the centre of the GOE, p = N/(7cl), and we also use N= kiL,.LZl/n, A = kiL,.L,/z. Then, on equating 21 (i.e., the radius of the GOE semicircle).with the Fermi energy,

248

IIDA, WEIDENtiLLER,

AND ZUK

EF = A2k2,/(2m*),

we obtain Lo = 161. All geometrical and momentum-dependent factors have cancelled out exactly. Although the numerical value of 16 may be model-dependent, and is therefore uncertain, nonetheless, L, characterizes a physically distinct length scale, as is shown by the fact that L, 9 1 if Tz 4 1 for all a, c. In metallic samples, we typically have I z 300 A, so that L, N 0.5 pm. Hence, the cross-over length is of the same order as typical experimental sample sizes. The behaviour of 2 is now easily interpreted in a time-dependent framework. The time ~diU= h/E, measures the diffusion time through the disordered sample, while =ckcay= Wdecay is the decay time for emission of electrons into the leads. For L 5 Lo, we have rdecay5 E, and tdiff 5 rdecay. In other words, after the incident electron has become “stuck” at site 1, the equilibration of occupation probability throughout the disordered sample is faster than emission of the electron into either lead. Therefore, emission into either lead is ultimately equally probable, in which case g attains nearly half the maximum possible value of 24 and is nearly independent of L. (See the first of Eqs. (6.9).) In the opposite limit, L % L, or Ec + rciecay9 the diffusion time through the sample is the longest time scale in the problem, and it determines the behaviour of g. (See the second of Eqs. (6.9).) In summary, we see that for L 5 L,, the value of g depends in an essential way on the coupling to the channels. In the next section, we shall show that the same statement applies also to var(g).

7. TERMS OF HIGHER ORDER

Equation (5.68) gives the leading-order term of an asymptotic inverse powers of n (or, rather, of T and/or x). Symbolically, g=g,A+g,A”+g-,n-l+ A corresponding

expanion of g in

. ...

(7.1)

series can be generated for the mean value of g*, &&4*+&!+&4°+g~1‘-1+

. ...

(7.2)

The variance of g is the difference between z and g2, and is given by var(g)=(z’-((g,)*)/1*+(2-22glgO)/i +(&(go)*-2glg~&fo+

.. ..

(7.3)

In demonstrating the existence and calculating the magnitude of universal conductance fluctuations, we have to show that z-- (gi)* and z- 2g, 8, vanish, so that var(g) is of order no, and we have to evaluate the coefficient of no in Eq. (7.3). Obviously, this task necessitates the calculation of terms beyond those displayed in Eq. (5.68). The calculation is straightforward but rather involved. Therefore, we proceed as follows. In Section 7.1, we display the details of the calculation of go, and we discuss the resulting expression for g. In Section 7.2 and Appendix D, we

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give the essential steps in the calculation of the first three coefficients of the series (7.2), and in Section 7.3, we discuss the resulting expression for var(g) and the influence of the channel coupling on the behaviour of this quantity. 7.1. Calculation

of go

To calculate g, we must evaluate the first term on the right-hand side of Eq. (5.46) which contains a double summation over channels, yielding /i”. Since, to lowest order in ti2 and t;,, we have Eq. (5.65) for trg x, the evaluation of (trg x)’ gives a term quadratic in t:,, ti,, tf2, tz and, hence, a factor ZIIK k n ‘. Altogether , this provides the leading-order term g, n displayed in Eq. (5.68). Terms of order no arise when we carry the expansion in powers of ti2, t;, to the next order. This must be done systematically for the source term itself, for the logarithmic term of Eq. (5.40), for the expansion of the “kinetic term” of Eq. (5.63 ), and for the expansion of the invariant measure dp(t& We proceed to list the terms multiplying the exponential of - i xi,, (ZI7P’)V trg ti2 t:, in the argument of Z(E) in Eq. (5.66), which arise from expanding these various contributions. In doing so, we also present terms of next order in tf2 t{, . These go beyond the requirements for calculating 2 up to order no, but will be needed in the calculation of var(g). It is convenient to introduce the notation T; = c (T:)“,

so that T{ = 6,jTL + 6,TR, (a)

(7.4)

with T”, c = L, R, as defined below Eq. (5.67).

Expansion of the relevant source term, as given by Eq. (5.55) yields 1 trgX:= ,.0

c [T’, trgt:Z(&J.k)-(T~-~T:)trgt!zt!-,t:?(E’.k) ,= I.K +(TI,-~Ti2-dT’,)trgt12t:,t.lZti,t:2(~’.k)]+

. ...

(7.5 1

In presenting this expansion (as well as those which follow), we have made use of relations such as (A.9) and (A.lO) of Appendix A, which arise from the mutual dependence of the matrices t,2 and tzl in the GOE. (b) When expanding the logarithm, 9, , of Eq. (5.40), we must remember that the terms quadratic in the t-matrices are kept in the exponential. Accordingly, let us define 9, to be the result of removing the quadratic part from 2,. Then we make the expansion exp(-~,}=l--~‘-~16’+~(~~4))2+

= 1 +a

c J=

. .. T$ trg(t{2t&)2-i.

T; trg(t;,t;,)3

1

1.K

/=I.1 2

+

Tj, trg(t{,t;,)’ 1

595!200/2-3

..

(7.6)

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We note that second term on the rhs of Eq. (7.6) is of order A-‘, while the following two terms are of order n -‘. (c) Upon carrying the expansion (5.63) of the “kinetic term,” YO, to higher order, we must similarly remember to retain the quadratic terms in the exponent. Letting PO denote the result of subtracting these terms from YO, we obtain exp{-8,}=1--dPO

(4) _

=l+:yf’

yr)

+

f(pb492

+

. . .

trgt;2(ti2:1-fll)til:1(tq:1-f:l) r=l

-a

yi’

trg ti,,(t::‘tf:‘-ti,ti,,)

t::‘(tf:‘t::‘-tf,ti,,)

*=l 2

+$

[

yi’

1

’ -t~,)tf~‘(t~~‘-~~,)

trg ti,(t$

2+ . . . .

r=l

(7.7)

Terms of fourth order in the t-matrices are of order n -‘; the remaining terms are of order /i -‘. (d) The expansion of the invariant measure uses the identifies [18] fl d~(t{~) = n detgg1j2( 1 + r{2t{1) dt(, i i = exp

-ic

trgln(l+li,t&)}fl

dt:, i

J

=exp{-q:,,}exp

(7.8)

where dri2 is the product of differentials. Here, 4,” denotes the result of subtracting the component quadratic in the t-matrices from the part, q,,, of the effective Lagrangian which arises from the invariant measure. We have exp{ -S&}

= 1 +t

,f trg(ti2t<,)2+

.. ..

(7.9)

J=l

The second term is of order ,4 -‘. The retention of the quadratic term in the exponential, here, leads to the replacement in Eq. (5.66) of the matrix IJ by a modified matrix, fi, such that I? - ’ = 17 - ’ + 1, which we temporarily consider to be formally also of order A-‘. At the end of the calculation, we expand fi=n-n’+

. ...

(7.10)

It is now straightforward to collect the contributions to g of order /1’. In evaluating these contributions, it is advantageous to realize that, since the integration over the r-matrices is to be performed with respect to the Gaussian distribution of

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Eq. (5.66) (with l7-t fi), we can use Wick’s theorem to carry out the calculation. Details are provided in Appendix D. For clarity of presentation, we quote final results only for the left-right symmetric problem, Tf; = Tf; whence we write T,, = T’,, j = 1, K. Defining y,, by T,, = xyn, so that y1 = y, we find 2XY

2 l+i n=, (2+$(

g=2+(K-l)y-3

1))” > +“(A-1)3

(7.11 )

where a, = -371, a2 = 2y,‘(yi a3 = -2y;‘(2yf

+ 3y: - 3y, + 3yz),

(7.12)

- 2’11+ 31)2).

In the special case of T; = 1 for all a, c, the coefficients a, reduce to a, = - 37, x2 = 2y(y + 3) a3 = -2(2y2 + l), and, as shown in Section 6.5, y attains the value y = $. The first term on the rhs of Eq. (7.11) coincides with the expression (5.68), and the second term yields, for yK$ 1, the well known weak localization correction to g of -$ [7]. Equation (7.11) furnishes an estimate for the validity of our asymptotic expansion. For this purpose, we compare the magnitudes of the two terms. Assuming that yK% 1, the two terms have equal magnitude when K= 3x. This shows that the sample size limiting our asymptotic expansion is the localization length, Lloc = Al N Tl= yxl N $x1. This is the maximal length for which we could have hoped. We note that the value 3x is independent of the strength of the coupling to the channels; this is also a satisfactory feature. 7.2. Calculation

of var( g)

Contributions to 2 arise from the last three terms given explicitly in Eq. (5.46). Proceeding as in Section 7.1, we supplement the list of contributions given there by the expansion of expression (5.57) for trg x2, which may be cast in the form 1 trg(;(!)‘=i 1.0

1 [T’, trg tJ12(e-l.k) tj2(ei.k) ,=1,x’ - (2T’, - T;) trg t{,(& . k) ti2 t& t;,($ . k) T’,) trg t{2t~,(d.k)

-(T’,--

t{2t{l(&ek)]

+ . ...

(7.13)

The first term is of order /1’, while the remaining terms given explicitly are of order /i-l. In calculating var( g), we have to evaluate the difference 2 - ( g)2. This cumbersome procedure can be avoided by observing that (g)’ subtracts from 2 precisely the disconnected terms, 2 in which case one need only extract the connected con’ We use a terminology

borrowed

from

diagrammatic

perturbation

theory.

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AND

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tributions to 2. The latter can easily be identified when Wick’s theorem is used, as in Sect. 7.1. In this way, one finds straightforwardly that terms of order /i and LI’ cancel, so that the leading-order contribution to var(g) is of order ,4O. This establishes, in the present framework, the universality of the fluctuations, It also implies that, while working to order no, we may drop the correction to 17, i.e., fi+ Z7. The actual evaluation of the O(n”) contribution is a rather cumbersome task. To enable the reader to check our procedure, we have listed, in Table 7.1, all contributions to 2 which are of order A ‘. To obtain var(g), the disconnected components of the terms should be omitted at the outset. The table contains three groups of contributions. Group (i) arises entirely from the expansion of the source terms: that is to say, the terms stemming from the expansion of the logarithm, Yi, of the “kinetic term,” go, and of the invariant measure, L&,, contribute only in zeroth order. Group (ii) comprises terms which contain at least one non-trivial contribution from the logarithm, but are of zeroth order in terms stemming from the expansion of the “kinetic term” and the invariant measure. Group (iii) contains terms which have at least one non-trivial contribution from either the “kinetic term” or the invariant measure. The physical reason for the decomposition should be obvious: Terms in groups (i) and (ii) are due to the coupling with the channels. They yield contributions of the form nf, s(n), where 9 is a polynomial in Z7,, , l7,, and nlK. Therefore, these terms fall off at least as Kp2 for large K. Terms in Group (iii), on the other hand, contain the “bulk contribution” to var(g) and yield, for K + CO, the weak localization contribution to var( g). The explicit evaluation of the terms given in Table 7.1 uses the contraction rules explained in Appendix A. After a lengthy calculation, we find +otA-‘)’

(7.14)

where 83 = 15y:, p4 = -2~;~(8y:

+ 15 Y:-~~Y:+~OY~Y,+~OY,Y~-~~Y:),

Ps= -4~;~(2~;-2Oy;‘+3Oy:y2+ /&=20~;*(4yf-8y;l+

12y;y,+4+

18y:-45y,y,-

(7.15) 15y,y,+45y:),

12y,y,+!+‘;).

In the special case of T; = 1 for all a, c, (whence yn = y for all n) the coefficients Bn reduce to /I3 = 15y3, /I4 = -2y3(8y + 15), /Is = -4(2y4 + 10y2 + 3), /I6 = 20(4y4 + 4y2 + 1). 7.3. Discussion

Equation (7.14) shows that for K-, CO, var(g) tends to the weak localization limit of 3. This value is consistent with the results of transfer matrix theory [7], and of impurity perturbation theory [20]. For K+ 1, var(g) attains the value i,

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Y

x

IIDA,

n

WEIDENMtiLLER,

,h

d

x

AND

ZUK

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0.6

20

0

40

60

80

100

K-i FIG. 7.1, The variance, var(g), (L - 1)/l as explained in the text.

for

Tz = 1 as a function

of the dimensionless

length

scale K-

1=

provided we assume Tz = 1. In this case, the solid curve in Fig. 7.1 shows that var( g) is nearly independent of K for 1 < Kc co. This result may seem somewhat disappointing. However, upon closer inspection, var( g) exhibits interesting features. The dashed curve in Fig. 7.1 is the contribution, (7.16) (UP to WOh f rom all terms listed under groups (i) and (ii) in Table 7.1, i.e., the contribution directly attributable to the coupling with the channels; while the dashand-dotted curve is the contribution,

var,(g)=

-l(jdEl

dEf;t, B

B

aBK(e-‘Y,““‘“+91)(e--(~o+~l.,)F

F

l)>cIc=o

(7.17)

(up to O(A”)), from all terms listed in group (iii). The “surface terms,” var,(g), decrease strongly with increasing K, as expected; but this fall-off is compensated nearly exactly by the rise of the “bulk contribution,” var,(g). The dotted curve in Fig. 7.2, calculated for the case of Tz = 0.1 for all a, c, shows that near compensation is a fortuitous consequence of our choice Tz = 1 yielding the solid curve. We note that the dotted curve is enhanced at K= 1 by almost a factor of two over the solid one, and that it approaches the asymptotic value of & rather more slowly than the dashed one, which gives the behaviour of var( g) for the case of TT, = 0.3. The reason for this last feature is related to the fact that if TT, = tx, 0 < c(5 1, for all a, c, then we have T= ~4 and yn = cPyO with y. = &. Thus for CL= 0.1, we have y = 8

257

UNIVERSAL CONDUCTANCEFLUCTUATIONS 1.0

.... . ‘.

n=0.3 '\

=

'\ ‘_

-.

0.8

5 kl >

CT=0 1

*.

‘.

a=l.O

-.

:\

-... \

---------\

-....__.____

0.6 --

--..__

-...

-----

0.4 lpr-.0

20

40

60

80

100

K-l FIG. 7.2. The variance, 1 =(L-I)/[.

var(g).

for Tb=cc

as shown,

as a function

of the dimensionless

length

scale

K-

and Lo = 1601, rather than y = & and L, = 53 .31 as obtained with c(= 0.3. We mention in passing that both g and var (g) depend on K only through the combination y(K- 1). This result holds generally, i.e., to arbitrary order in n-r and for any choice of sticking probabilities. In summary, we have seen that for L 5 L,, the length-dependence of both 2 and var(g) is strongly influenced by the coupling to the channels. It is only for L $ Lo that these functions attain values characteristic of weak localization, and independent of the coupling to the channels. For the model with site-diagonal disorder, we also have u2/12 - (k,l))’ < 1 since the argument of Section 6.1 continues to be valid. Thus, we should again expand in powers of u2/2’ in the generating functional, (4.15). Retaining terms in the exponent only up to order u’/J.‘, we lind that the generating functional becomes identical with that of the model with gauge-invariant disorder in the same approximation. Some details are briefly discussed in Appendix E. Therefore, our results apply equally well in the case of site-diagonal disorder.

8. SUMMARY AND OUTLOOK In this paper, we have developed a novel approach to the conductance problem for a two-lead measurement. Using the many-channel approximation to Landauer’s formula and statistical scattering theory, we have been able to take full account of the coupling to the open channels. We have calculated both g and var(g) in the framework of an asymptotic expansion which is valid (strictly for 2, but hopefully

258

IIDA,

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AND

ZUK

also for var(g)) for all length scales up to the localization length. For length scales by other authors who have used different approaches. This shows that the assumption of GOE statistics for the Hamiltonian is sufficient to yield universal conductance fluctuations. The results presented in this paper strictly apply to systems with transverse dimensions given by the elastic mean free path. It is possible, however, to extend the model to describe samples with larger transverse dimensions. The analytical calculations associated with this case are feasible, but are very involved and have not yet been completed. We have simultaneously used two asymptotic expansions-one in powers of N-l, the other in powers of /i-l. Since N/A is of order k,l, one may wonder whether terms of higher order in /i - ’ (such as the weak-localization correction to 2) can be meaningfully calculated without considering terms of order N-‘. We believe that the answer must be given in the affirmative for the following reasons. First, n is precisely defined as the number of transverse modes at the Fermi energy, E,, while N should also include states above E,, so that N/A N kF1 is an order of magnitude estimate only. Second, and perhaps more importantly, the terms in the two expansions have very different physical origins. Terms of order N-’ appear in exactly the same form also in the expansion of the one-point function, and therefore amount to a correction of the mean level density. Terms of order A-‘, on the other hand, relate to the long-range modes of the system which, in the absence of coupling to the external leads, would be Goldstone modes. It is these latter modes which determine the transport properties of the system. The model formulated in Section 2 hinges on the use of the elastic mean free path, and this may cause concern. We have convinced ourselves, however, that after scaling I + 21 and K- 1 + i (K- l), our generating functional, 2, remains invariant. We also recall that our results are the same for both the site-diagonal and the gauge-invariant models for disorder, thereby giving additional support to our belief that our results are of general validity, and not merely spurious consequences of some arbitrary choice of model. One strength of our approach-not yet utilized in this paper-is the ease with which it can be extended to other geometries of the sample. For instance, a twolead ring experiment is described by exactly the same formalism, save for the following modifications of the matrix Z7-’ in Eq. (5.67): The (1, K) and (K, 1) elements of n-’ are x rather than zero, the entries 1 + yL,R must be changed to 2 + yLsR and 2+yR has to be moved from the (K, K) position to the ((K+ 2)/2, (K+ 2)/2) position (assuming that K is even). A four-lead experiment requires another, almost obvious, modification of 17-l. We are confident that the relevant formulae for 2 and var(g) can also be worked out analytically in these cases. We feel that this would be particularly interesting to do for the four-lead experiments, where we expect the voltage fluctuations [l, 201 to depend significantly on the coupling to the open channels. Further extensions of the model are also called for. We have in mind, among other possibilities, the calculation of Aharonov-Bohm oscillations in the ring Lb Z, our results coincide with those obtained

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geometry [l], and of the coherence energy of the “magnetofingerprints.” Both of these require us to take account of the presence of a magnetic field, as would a comparison of var(g) with experimental data [21].

APPENDIX A

For the GOE, the matrices tlz, t,, satisfy the conditions (A.1)

t,t = k&, t 21 =rrtr

where k = k, - k,, except to say that

12

r

(A.2)

)

and r is a 4 x 4 graded matrix Pr=

1,

which need not be specified,

T2=k,

(A.3)

=0

(A.4)

and Ck, ri for c1= B, F. Let A(t), B(t) notation

be graded-matrix trgA tf2Bt{,

functions

of the t,,,

t,,

We introduce

= (trgatf2btl,)I,=,,,,.,=.,,,,

where the angle brackets ( . . . ) denote the Gaussian integration t,,, t,, given by Eq. (5.66). Similarly, r

trgA t&trg

I

(trgat12trgbt:,)I,=,(,,,,=,,,,.

Bt{,-

Then we can write the following contraction

the (A.5 1

over the matrices (‘4.6)

rules:

-

trgA t;,Bt$,=Z7,trgAtrgB, trg A ii,, trg Bii, = 17, trg AB, I # = trg A tf, trg Bt ;2 = LTli trg ATTBTT, r 1 trg A th, Bt;, = trg A tk, trg Bt& = Ll,j trg ATTBTT.

(A.71

1

trg A ti2Bt{,

(A.8)

In conjunction with Wick’s theorem, Eqs. (A.7) and (A.8) suffice to calculate any Gaussian integration over the t-matrices. It is useful to note such relations as

trg k, fi2 =trgk,t;,,

(A.91

260

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AND

ZUK

and trg k,t;,k,t

f2 = trg k, t& k, t&,

(A.lO)

etc., for CC,/? = 0, B, F, where k,, = 1. The discussion of the GSE is very similar [22]. The only difference is that the matrix f now satisfies r* = -k, and the left-most contractions in Eq. (A.8) acquire a minus sign. In the GUE, t i2, t2, are 4 x 4 graded matrices, and they satisfy only condition (A.l), i.e., there is no analogue of Eq. (A.2). Thus, the elements of t,,, t,, together constitute independent holomorphic integration variables. The contractions in Eq. (A.7) remain the same, but those in Eq. (A.8) must obviously vanish. Here, one should note that the overall factor of one-half appearing in the exponent in Eq. (5.66) is absent for the GUE. Relations such as Eqs. (A.9) and (A.lO) no longer hold.

APPENDIX

B

In this appendix, we derive Eq. (5.45), which expresses the sticking probabilities in terms of the diagonal elements of the average S-matrix. Let us begin by defining

(B.1) Then from Eq. (3.4), we have

Purely for the sake of convenience, we shall consider the sum over channels of the diagonal S-matrix elements, for which we have the simple relation 1 S;;= a For the purpose of performing source matrix, given by

-2i tr Q(c) D-l.

03.3)

the ensemble average, it is necessary to adopt a new

J[&]=(&“~k)Q(L)+(ER’k)Q(R).

(B.4)

Here, R,=

Ok,

k2=(;

;)@k,

(B.5)

where k is given by k = k, - k,, so that trg E=. R = 4(.s5+ ES)

WJ)

UNIVERSAL

for c = L, R. Plugging

CONDUCTANCE

261

FLUCTUATIONS

this source matrix into Eq. (4.1) yields the formulae

(J3.7)

The relevant part of the source term is given by exp{--Y&-,,,,,}=

-f

1 J=I.K

C trgx:+@E2)y

(B.8)

u

with x{ as shown in Eq. (5.17). Using the methods of Sections 5.2 and 5.3, and with the same notation, we obtain trgx=

-itrg

1

Ta

1

(~-~)cosS-~~(P~~~~~-~~COS~)

(B.9)

where, as usual, we have suppressed the indices a and j = 1, K. To lowest order in t,,, t,,, we have trgX=4i(l -/I)cos&~‘“e,, (B.lO) having set E?= 0. Combining

Eqs. (B.3) (B.7) (B.8) and (B.10) then yields

CE= u

-x2(1--fl)cosBe-‘“, u

(B.ll)

from which we infer I@p=

l-48(1

-p)cos’&

(B.12)

Thus we see that T; = 1 - ISc’,I’

(B.13)

holds in lowest order. After setting F>= 0, the higher order contribution (exp{-~]6trg~)=~ir:,e-iQ(e-‘“-2/?cosf?)

to Eq. (B. 11) is generated by

t

exp(-.Y]trgk1+Tt,,t2, Tt,zfz, I

, :, (B.14)

where exp( - 9 > denotes the contribution of terms in Y&, quartic or higher order In t,,, t,,, together with the contribution from the invariant measure, and the angle brackets signify the Gaussian integration of Eq. (5.66). But

i

exp(-9)

trgk

Tf,*f*, 1 + Tt,,tz, >

= 0.

(B.15)

262

IIDA,

WEIDENMtiLLER,

AND

ZUK

Therefore, Eq. (B.13) holds exactly. We finally proceed to outline the derivation of Eq. (B. 15). A typical term in the expanion of the left hand side of Eq. (B.15) in powers of the tlz, tZ, mvolves ’ the structure (

trg(kt{; t$ ... ) n trg(t;‘qt$ I

. . . )).

(B.16)

We note that the strings of t-matrices within each graded trace are alternating, in the sense that every occurence of t12 is followed and preceded by a matrix, t,,, of the opposite type, and vice versa. According to Wick’s theorem, each term of the type (B.16) is given by the sum over all possible pairwise contractions of the t12, tZ1. Let us consider a typical such contribution. We specify its evaluation by performing the Wick contractions in a prescribed order. First, all contractions between like t-matrices (i.e., those where both subscripts are either 12 or 21) should be carried out. It is clear from Eq. (A.2) and the contraction rules of Eq. (A.8) that this leaves an expression still involving the product of graded traces of alternating strings of t-matrices, as depicted in Eq. (B.16), but where only contractions between unlike t-matrices now appear. Next, we choose any remaining contraction which links two separate strings. Performing such a contraction will fuse the two strings together. After repeating this step as many times as possible, we arrive at a single string, trg(kt$ t& . . . ),

(B.17)

where internal contractions between unlike t-matrices remain. Due to the alternating nature of this string, an even number of t-matrices always stands between any contracted pair. Consequently, the rules of Eq. (A.7) imply that the final contraction must be of the form (trg kt& tj2, ), which vanishes.

C

APPENDIX

We want to study the eigenvalue problem 17corresponds to taking the continuum limit. We have

'f = iLf as K + co, which (C.1)

-~-l+vpfi+l=~Jj

for j= 1, 2, .... K- 1, and (1 +r)f1

-f2=Af1,

(l+Y)f,-fK-,=;lf,.

(C.2)

(C.3)

Now suppose that we have found the general solution of the difference equation (C. 1). Then we have expressions for fi, f2, fK, fK+ i, and these satisfy

-fo+2firfi=ilfi? -fK- 1+ 2f,-fK, 1= AfK.

(C.4)

(C.5)

UNIVERSAL

CONDUCTANCE

FLUCTUATIONS

‘63

But for this general solution to be the particular solution relevant to n - ‘, the pair f,, f2 and the pair fKp,, fK must also satisfy Eqs. (C.2) and (C.3), respectively. We can use Eqs. (C.4) and (C.5) to eliminate ;I from Eqs. (C.2) and (C.3). We thus obtain

(~-r)(fi-,h)-Yfo=0~

(C.6)

(l-Y)(fK+I-fK)-YfK+l=o.

Equations (C.6) constitute the boundary conditions for the eigenvalue problem pertaining to the [ - 1,2, - llK- z sub-matrix of I7 - ‘. Since the site label j varies between 1 and K, if we set T = j/K, then for K $ 1. T becomes a quasi-continuous variable which varies over the interval 0 I T < 1. Let us define f(z) by

f(j/K)=f,.

(C.7)

Then in the limit of large K, the system of difference equations (C.1 ), (C.6) reduces to the differential equation

subject to the Sturm-Liouville

boundary conditions

+&$f.=O $1

-y)$+yf=O

at

z=O,

at

T=l.

(C.9)

(C.10)

It is useful to note that the boundary value problem of Eqs. (C.8) (C.9), and (C.10) is invariant under reflection about the point r = 4. Thus, the eigenfunctions can be classified according to their parity. Consequently, the boundary condition at 7 = 1 can be replaced by the following simpler boundary conditions at T = i:

dfo

=o

for

parity even,

f(~)l,=1,2=O

for

parity odd.

dz

T= l/2

(C.11)

Let us introduce the new eigenvalue parameter w by +. Then the even parity eigenfunctions

(C.12)

have the form

f(z)=Acoso(s-4)

(C.13)

264

IIDA,

WEIDENMtiLLER,

AND

subject to Eq. (C.9), which serves to determine solutions of the equation wtan$o=-

ZUK

the admissible

KY

values of o as

(C.14)

1 --y’

We are interested in the region where KY + 1. Thus we write o = 2nm + &.I, m = 0, 1, 2, ...) so that 60 -P 0 as KY + 0. This leads to (2nm + 60) tan 4 60 = Ky + O(Ky2).

(C.15)

After making a Taylor expansion of tan 4 6w, we find the leading contribution &II to be given by

h= IJ- +O((Ky)3’2), 2 +O(Ky2), 2Ky

to

m = 0, (C.16) m2 1.

Hence, we obtain the series of eigenvalues

1, =

2Y z + W2L

n = 0,

n2n2 4y F + 2 + O(YZ)>

n = 2, 4, 6, ....

The odd parity eigenfunctions

(C.17)

have the form (C.18)

f(z)=Asino(r-$) subject to Eq. (C.9), which determines the admissible of the equation

values of o as the solutions

wcotgo= -- l-y’KY Again considering the region where Ky + 1, we write m = 0, 1, 2, .... so that 6w -+ 0 as Ky --* 0. This leads to ((2m + 1)~ + 60) tan 4 6w = Ky + O(Ky*). The leading contribution

01= (2m + 1)~ + 60, (C.20)

to 60 is thus (C.21)

UNIVERSAL

CONDUCTANCE

265

FLUCTUATIONS

for m = 0, 1, 2, . Hence, we complete the eigenvalue series with (C.22)

where rz= 1, 3, 5, .. D

APPENDIX

Our first task in this appendix is to fully invert the matrix 17 ~ ‘. For this purpose, let us introduce the family of r x r tridiagonal matrices Mll,!( y) defined by M&tY) = Cl, Y, ll,, where the main diagonal is given by the vector y, = .Y+ r hi, + ?’ dir, i = 1, 2, .... r. A straightforward leads to [Mgy(y)],p

application

= (-l)j+’

(D.1 I

of Kramer’s

rule for matrix

.det M!/;“()I)det Mb:,-“(y) det M $)( y)

inversion (D.2)

’

provided i I j. Elements corresponding to i > j are obtained from those above by observing that the matrix M$’ is symmetric. Now detM~~(~)=(?l+rl)(y+9’)P,~,(y)-(2y+9+’l’)P,

7(.v)+~K

4(~s),

(D.3

provided K 2 4, where P,(y) = det Mz( y). In particular, P,(-2)=(-l)‘(r+l). ye’= 1 - yR, y = -2 which yields

To

obtain

(D.4 I7,,

n,, =‘(I +(i- l)yLN1 + (K-j)yR) v .Y y L + yR +(K-l)yLyR

we take

f((Ctw)2)

+(T1)’

(D.5)

’

for i I j. To order no, 2 is composed of three types of contributions,

q= 1 -;‘I-,

as follows:

(trgk,t~2trgk,t~2)-T,(T2-~T,)

x (trgkd:,

trgk,tr2tt,t;Kz)

-T,(T2-~T,)(trgk,t~2trgk,t:2t:,t:,) = 2( T, )2 Z?,, -2T,(Tz-~T,)Ij,,(W,,+Ij,,) =2(T,)2(n,,-(n2),,)-2T,(T,-~T,) x fluAn, 59S;2,X,i2-4

+ 17,,) + ...,

(D.6)

266

IIDA,

WEIDENMtiLLER,

-~(T,)‘Tz

1 I=

= WA*

AND

ZUK

(trgk,t:,trgk,t~trgt:,t:,t:,t:,)

l.K

T~~IK((&)

+

(~KK)*)

+

“‘7

+

‘..

(D.7)

x trg tf,(t’,, - ti,,) t~,(t’,, - tir)) =x(T,)*

i

di+l,j[(17,,-2nq+Ujj)

i,j=

1

X cnlinjK +

+

211,(n,

nljlTiK)

-n,i)(UjK

-niK)]

.

(D.8)

We have derived the foregoing results without recourse to the actual form of the matrix 17. However, in order to simplify the final expression for 2, it is advantageous to make use of the properties that, for i 6 j, we have 17,=------

nljniK

(D.9)

nlK

’

and, with j = i + 1, nlj

-

nli

=

-YnlK,

(D.lO)

nj, - l7,, = YI;IIK. Collecting

all contributions

thus yields

+x(T,)*nlK

+

f i,j=l

niJzjK

+

bi+l,jCn,(nIj

l7$7,,]

+

0(/i

+niKm2Y2n~K)

- ‘).

(Dell)

On substituting Eq. (D.5) for the matrix ZT and performing the summation, we arrive at Eq. (7.11). Finally, we list the contributions to var(g) (i.e., with disconnected parts subtracted) from each of the items appearing in Table 7.1 in terms of the matrix 17:

(2) + -4T1

T&=2

-

(3) + -4(T,)’

(2T,

(4) + -I*

(T,

TI)(~IK)~ -

-

T,)(~,K)* T~)(~IK)~

(fl,,

+

(n,, (n,,

+ +

nKK) nKK) nKK)

UNIVERSAL

CONDUCTANCE

(6)~(T,)‘(2T,-T,)‘(n,,)’ (7) -+4(TlJ3 (8)

+

8(T,

C10(n,,)2+4~7,,17,,+5(n,,)2+5(n,,)21

VT,-4T2T2)’

(9) + -am

Tl)(n,K)2

U-M2

(W’,,)*

+

T&T2

- WVW2

x C6(n,1)3 + 6(fl,,)3 (10)

-+

-

16(T,14

267

FLUCTUATIONS

T3V7,d2

((fl,,)‘+

(n,,)*)

(n,,12)

+ n,,n,,)(n,,

+ (5VM2 ((n,,)’

+

+ n&l

WQ3)

(11)-4(T,)4(T2)2(J11K)2 x

C6(n,, j4 + W7,,)4

+

+ (171,n,, + (n,,)2)21

8n7,, flm(17,,)2

(12)+0 (13)+8x(T,)‘T,

6.rf

f

l.jn*KC(nii

+

n,y)(nlKnQ

+

n,injK)-4J7,,n;,nj,]

i,j=l

(14) + -DYE

(2T2-

T,)(n,,12

i

6i+l,j(n,,)2

[5fl,,(n,,flij

+ n,,n,,y)

i,j=l

(lb)+

AXE

; i,j=

6.I+ l,j C2nlK(2nlKn~

- ~,w,K4j)2

-BYTE

+

nlinjK)((n;i)2

1

(nlK)*

+ vL~j,)‘)l

5

i,i’,j,j’=l i
di+

I,j hi,+

1.1

+

(nj,)2)

268

IIDA,

x { -4(z7,,)2

WEIDENMtiLLER,

AND

ZUK

(zIii*17~* + Iiy7j,~)

+5C17,,(17v-niK)+nj~K(171,-17,K)1 x p7,iQ71j~ - z7,,,) + n,.(n,if - z7,,,)] -

2(nlinlj

x

(&z7,,

+

+

niKnjK)

z7,yJ7j,,)).

The relations (D.9) and (D.lO) have been used in deriving the expressions arising from items (13)-( 17). E

APPENDIX

From Eq. (4.15), we see that for the model with site-diagonal saddle-point equation reads 1

the

Id

El-V+iQ-Z,

1 = El + iQ( j) - C,(j) for j=

disorder,

+ o(u2p2)

(E.1)

1, 2, .... K, where

Expanding

G),” = QJCP (E.2) term in the exponent of Eq. (4.15) in powers of V yields

the logarithmic

-i’&trg(jlln(El-V+iQ-Z,)Jj) J

= -kc

trg(j(

ln(El+iQ-C,)

+t C trg(jl

1 V El+iQ-L’,

lj)

J

J

V

1

lj) + O(u4/A4)

El+iQ-XC,

(E.3)

since terms odd in V clearly vanish. The first term on the rhs of Eq. (E.3) gives rise to the source and logarithmic terms of the model with gauge-invariant disorder. Restricting our attention to the interior points j = 3, 4, .... K - 2, we have 1 V El+iQ-C,

trg(jl

V

1 El+iQ-C,

Id

K-l =

C k=2

trgvjkE1-L

(k)vkjElp~ G

(j) G

= + trg( jl vzG vzc, 1j) + o(u4/A4) * = $ trg( jl vcG voG lj) + o(04/,%4),

(E.4)

UNIVERSAL

where, in deriving the Z, = T ~ ‘ZO T and the The remaining terms does not apply directly

CONDUCTANCE

269

FLUCTUATIONS

second-last line, we have made use of the representation saddle-point equation to lowest order in V, viz. Eq. (E.1). of the summation in Eq. (E.3) to which the argument above are given by

1 trg El + J(j) + iQ(j) - C,(j) 1 = ,Z trg

1 El - C,(j+

1

El + J(j) + iQ(A - I,

1) C,(ji

1) + 0(2~2/A”)

for j= 1, K, with jk 1 chosen as appropriate. By virtue of the orthogonality (5.3). we can follow the reasoning of Section 5.1 to obtain tw

1

El + J(j) + iQ(A - C,(j) = (N- A) trg E, -lc

G

(E.5) relation

LG(j+_ 1)

(j) a,(j+

1 + C trg E+x~&‘.k+iXJJ-a~(j) CJ

1) OG(j + 11,

t-1

where it is understood that the “trg” on the rhs no longer involves a summation over the level index, p. Retaining only the contribution dominant for large N, we thus have 1 tr g

El + J(j) + iQ(j) -c,(j)

ZG(j+_ 1)

‘v $ trg frG(j) aG(j f 1) + O( r2/3.‘)

(E.7)

for j= 1, K. Consequently, we see that Eq. (E.4) holds to the desired accuracy for all j = 1, 2, .... K. Finally, we observe that --

4:2 I$ trg /

CG(~)

OG(.~ + $

c trg(jl

V~G Vo,

lj) (E.8)

On comparing with Eqs. (5.60) and (4.12), we conclude that the effective Lagrangians (i.e., the exponents of the generating functionals) for the models with site-diagonal and gauge-invariant disorder are equivalent up to terms of order &jW’.

270

IIDA, WEIDENMULLER,

AND ZUK

ACKNOWLEDGMENTS HAW thanks Y. Imry and we all thank K. Efetov and M. Zimbauer for helpful discussions, and A. D. Stone for communications. Note added in proof After this work was submitted for publication, the following paper was brought to our attention: R. A. Serota, S. Feng, C. Kane and P. A. Lee, Phys. Rev. B 36 (1987), 5031. They discuss the case where the thickness of the lead is much smaller than that of the sample; this situation may have some relation to our model with small sticking probabilities. We thank A. D. Stone for having informed us of this article.

REFERENCES 1. 2. 3. 4. 5.

S. WASHBURN AND R. A. WEBB, Adv. Phys. 35 (1986), 375. P. A. LEE, A. D. STONE, AND H. FUKUYAMA, Phys. Rev. B 35 (1987), 1039. B. L. ALTSHLJLER AND B. I. SHKLOVSKII, Sov. Phys. JETP (Engl. Transl.) 64 (1986), 127. K. A. MLJTTALIB, J.-L. PICHARD, AND A. D. STONE, Phys. Rev. Lert. 59 (1987), 2475. B. L. ALTSHULER, V. E. KRAVTSOV, AND I. V. LERNER, Sov. Phys. JETP (Engl. Transl.) 64

(1986),

1352. 6. Y. IMRY, Europhys. Left. I (1986), 249. 7. P. A. MELM, Phys. Rev. Left. 60 (1988), 1089. 8. F. J. WAGNER, Phys. Rev. B 19 (1979), 783. 9. D. C. LANGRETH AND E. ABRAHAMS, Phys. Rev. B 24 (1981), 2978. 10. E. N. ECONOMOU AND C. M. S~UKOULIS, Phys. Rev. Lat. 46 (1981), 618. 11. D. S. FISHER AND P. A. LEE, Phys. Rev. B 23 (1981), 6851. 12. P. A. LEE AND D. S. FISHER, Phys. Rev. Left. 47 (1981), 882. 13. A. D. STONE AND A. SZAFER, IBM .Z. Res. Dev. 32 (1988) 384. 14. C. MAHAUX AND H. A. WEIDENM~~LLER, “Shell-Model Approach to Nuclear

15. 16. 17. 18. 19. 20.

21. 22.

Reactions,” NorthHolland, Amsterdam, 1969. K. B. EFETOV, Adv. Phys. 32 (1983), 53. J. J. M. VERBAARSCHOT AND M. R. ZIRNBALJER, J. Phys. A 18 (1985), 1093. J. J. M. VERBAARSCHOT, H. A. WEIDENM~LLER, AND M. R. ZIRNBALJER, Phys. Rep. 129 (1985), 367. H. NISHIOKA, J. J. M. VERBAARSCHOT, H. A. WEIDENM~~LLER, AND S. YOSHIDA, Ann. Phys. (N. Y.) 172 (1986), 67. J. BLATT AND V. F. WEISSKOPF, “Theoretical Nuclear Physics,” pp. 386389, Wiley, New York, 1952. C.-L. KANE, P. A. LEE, AND D. P. DIVINCENZO, Phys. Rev. B 38 (1988), 2995. C. P. UMBACH, P. SANTHANAM, C. VAN HAESENDONCK, AND R. A. WEBB, Appl. Phys. Mr. 50 (1987), 1289. A. ALTLAND, private communication, and to be published.