Nuclear Physics B290 [FS20] (1987) 87-110 North-Holland, Amsterdam
SINGLE ELECTRON IN A RANDOM POTENTIAL AND A STRONG MAGNETIC FIELD Hans A. WEIDENMULLER* W. K Kellogg Radiation Laboratory, California Institute of Technology, Pasadena CA 91125, USA
Received 2 July 1986 (Revised 13 April 1987)
An electron in an infinitely extended plane is subject to a strong transverse magnetic field. The degenerate states pertaining to a single Landau level are coupled by a random potential. We develop a matrix algorithm that maps this problem onto a path-integral problem in two dimensions. For a class of random potentials, we calculate the average level density for an arbitrary Landau level, and we show that the problem differs from the localization problem without magnetic field only by the presence of the topological term. The coefficient multiplying this term is given explicitly.
I. Introduction T h e interpretation of the integer q u a n t u m Hall effect requires localized electronic states in the tails of each L a n d a u band, and a group of extended states (or at least a single extended state) near or at the centre of the b a n d [1, 2]. Does theory support this interpretation? The renormalization group analysis of the localization p r o b l e m in d = 2 dimensions in the absence of any magnetic field [3] indicates that all states are localized. Studying the problem in the context of a non-linear sigma model, Pruisken [4] and collaborators [5] have argued that the magnetic field causes the presence of an additional term in the effective lagrangian, and that this "topological t e r m " delocalizes the state(s) at the b a n d centre. O n the other hand, the explicit calculation of the average level density for the lowest L a n d a u level and a white-noise potential [6, 7] has (not surprisingly) failed to produce any singularity that might be indicative of a mobility edge. Such an edge has likewise not been found in various r e s u m m e d perturbation expansions for the conductivity [8, 9]. On the contrary, the m o s t extensive and recent of these studies [10] finds a finite value for the conductiv* On leave from Max-Planck-Institut ftir Kernphysik, Heidelberg, FR Germany. 0619-6823/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
88
H.A. Weidenrnfiller/ Randompotential
ity at and near the centre of the Landau band with a smooth behavior that does not suggest any singularity. It is the purpose of this paper to contribute towards the clarification of this situation. More specifically, we develop the theory of the localization problem for an electron constrained to a single Landau level and subject to a random potential. This case is opposite to the problem [3] without magnetic field. Constraining the electron to a fixed but arbitrary Landau level introduces into the theory the projection operator onto this level. This causes considerable technical difficulties. For the case of the lowest Landau level and the one-point function, the problem was solved in refs. [6] and [7] by expressing the projection operator in terms of holomorphic functions. We follow a different route, applicable to any Landau level. Using gauge transformations and their matrix representations, we construct for each Landau level a complete set of unitary matrices Utah(P). Here m and n label the degenerate states within each Landau level, and p-space is two dimensional. Every operator A with associated matrix representation Amn c a n thereby be mapped in a one-to-one fashion onto a function A(p). The functional dependence of A on p is determined both by the operator A and by the projector onto the L a n d a u level j under consideration. In this way we associate with the correlation function of the random potential a function wJo(p). This technique allows us to formulate the problem using methods of functional integration in p space without reference to the projection operator. This step can be made for any random-potential model. For a specific class of potential models involving an integer N we can push the theory one step further by considering the limit N ~ oe. This leads to a non-linear sigma model which can then be compared with the corresponding model for the localization problem without magnetic field [11], and with the model suggested by Pruisken et al. [4, 5]. The average level density can also be calculated. The various random-potential models considered in this paper are introduced in sect. 2. We argue, in particular, that models involving the integer N >> 1 are not purely formal extensions of the standard white-noise potential, and that the parameter N incorporates physical reality. The unitary matrices Urn,( p ) are constructed, and their completeness is shown, in sect. 3. These results are used in sect. 4 to map the matrix elements of the correlation function of the random potential onto a single function w 6 ( p ) which contains the entire information relevant for Landau level j. We calculate w~(p) explicitly for standard examples of correlation functions. In sect. 5, we give the path-integral representation of the generating function in p-space, valid for any of the potential models introduced in sect. 2. It is satisfying that we obtain a universal expression in which the form of the correlation function, the strength of the random potential, and the label j of the Landau level under consideration only affect a single parameter )t and the functional form of w~(p). For the random potentials involving the parameter N we consider in sect. 6 the limit N ~ oe, keeping the product N Ipl 2 fixed. This yields a non-linear sigma model
H.A. WeidenmFdler/ Random potential
89
which we compare with the corresponding model for the localization problem without magnetic field [11], and with the model proposed in refs. [4] and [5]. Our conclusions are drawn in sect. 7.
2, Random potentials The electron moves in an infinitely extended plane with cartesian coordinates (x, y). It is subject to a random potential V(r) which simulates disorder. We assume that V(r) has a gaussian probability distribution with mean value zero. The various models considered in this paper differ by their choice of second moments. We use a bar to denote the ensemble average. The standard model [4, 6, 7] is the "white noise" model, defined by
V(rl)V(r2)
= (Xl)
a(r,
-
r=),
(2.1)
with X a strength parameter of dimension energy, and l a length scale. This model is the special case of a wider class of models [12] for which =
7([rl - r2]=/d=).
(2.2)
The dimensionless correlation function f depends on a correlation length d which on physical grounds is expected to be of the order of atomic dimensions. Translational and rotational invariance require that f depends only on the distance
It1 -r21. A further generalization of the models (2.1) and (2.2) is obtained by considering the random potential as a matrix V~,(r) in some suitable space, the indices /L and v running from 1 to N. Two choices are
~2 (2.3) and [9] Vvv(rt)V#,(r2) = ~ f ( [ r
1 -- r212/d2)~,,,~r#.
(2.4)
The first of these is invariant under orthogonal transformations of the matrix V,~(r) and is based on the postulated symmetry V~,('r) = V,,(r) (time-reversal symmetry), while the second does not impose this symmetry and is invariant under unitary transformations. For N = 1, both cases reduce to the model (2.2). We may view both models (2.3), (2.4) as formal extensions of the model (2.2), the parameter N offering us the possibility of taking the limit N ~ o0 and thereby considerably reducing the complexity of the theory. This is the point of view
90
H.A. Weidenmiiller / Random potential
expressed in ref. [9]. We believe that both models also incorporate important features of the microscopic scattering processes which are simulated in terms of the r a n d o m potential. We devote the remainder of this section to a justification of this view. The electron is scattered by impurities, dislocations, etc. We label all these individual scatterers by an index i, with i = 1, 2 , . . . . In the scattering process, the scatterer will change its intrinsic quantum state [ia) and make a transition to lift). Let I#) be the product of all the actual quantum states of all the scatterers, Ii~) = F l i l i a i ) where a i labels the actual state of scatterer i. A microscopic description of the scattering of the electron in the medium requires us to introduce the matrix V~,(r), defined in terms of the matrix elements of the electron-scatterer interaction with respect to the states/~ and v. A full description of the dynamics of the problem would also require us to introduce the intrinsic hamiltonian of the scatterers. We neglect this quantity because the energy loss of the electron in the scattering process lia) ---' lift) with fl ~ a is negligible. (Full degeneracy of the states l i a ) for all values of a.) We are thus naturally led to consider a matrix model involving the potential matrix V~(r). It also stands to reason that the complexity of the problem is such that we consider V~v(r ) a gaussian-distributed random variable. But are the distribution laws (2.3) and (2.4) realistic? The answer is no, and the models (2.3) and (2.4) simplify a situation which is actually more complex. In a microscopic scattering event, we expect the quantum state of a single scatterer to change, a i --* r , with fli = a, for all index values but one. Both eqs, (2.3) and (2.4) do not impose this restriction but allow the indices ~t and to differ in the quantum states of an arbitrary number of scatterers. Likewise, when the scattering takes place on the scatterer labelled i, the spatial localization of this scatterer makes us expect that the function f be localized at the position of the ith scatterer. The translational invariance embodied in eqs. (2.3) and (2.4) does not observe this constraint. An attempt to incorporate the more realistic conditions just mentioned into a stochastic treatment of the potential leads to difficulties which seem insurmountable, and forces us to make do with the models of eqs. (2.3) and (2.4). We observe, however, that N is proportional to the number of quantum states of each scatterer, and that N ~ ~ is a reasonable limit to take. The difference between the models (2.3), (2.4) and a realistic description is similar to the difference between the gaussian ensembles [15] often used in nuclear physics, and the more realistic embedded ensembles which defy analytical treatment but do seem to have the same fluctuation properties as the gaussian ensemble [15]. (The gaussian ensembles do not, however, yield a realistic average level density.) Concerning the difference between eqs. (2.3) and (2.4), we observe that the quantum states of the scatterers are in general expected to be uncharged and therefore not affected by the magnetic field (unless they carry spin). Then, timereversal symmetry allows us to choose the matrix V~(r) symmetric which leads to
H.A. Weidenmiiller / Random potential
91
eq. (2.3). We are therefore inclined to view eq. (2.3) as the more realistic model. However, the model (2.4) is considerably simpler to handle. In the bulk of this paper, we therefore focus attention on this latter model. In the appendix, we show that for N ~ oe the model (2.3) yields the same effective lagrangian as the model (2.4). This justifies our procedure. 3. Landau levels In this section we focus attention on symmetry properties of the degenerate set of states lying within one Landau level. We consider the kinetic part H o of the hamiltonian, H o = (2m)-l[(h/i)V
- (e/c)A] 2
(3.1)
defined in the two-dimensional plane with cartesian coordinates (x, y), with a constant magnetic field B perpendicular to the plane, and a vector potential A given in symmetric gauge by ½B(-y, x). Using the length l o = ( 2 h c / ( e B ) ) 1/2 we define the dimensionless variables (2, y) = lol(X, y). We scale H o with the harmonic-oscillator frequency o~o = e B / ( m c ) [the distance between neighboring Landau levels is given by h~0] and have with
17o = Ho/( h,,,o)
0 ) + f f 2 + 5,2.
02 0z 0-)7-0--2 4/40 = - 0 y z - 05---5 + 2i ffff-f
(3.2)
Introducing the complex variable z = 2 +/)7 with z* = 2-/)7, we define the operators 0 I
_ _
a = ~z + Oz* '
0 a t
x ,
= ~z
0 b=½z*+
__
Oz'
_
Oz '
(3.3)
0 b t =1
5z
_
_ _
Oz*"
(3.4)
They obey the commutation relations [a, a t ] = 1 = [b, bt], [a, b] = 0 = [a, b ' l , [at, b ] = O = [ a t ,
bt].
(3.5)
When written in terms of these operators, /lo takes the form I t o = a t a + ½.
(3.6)
H.A. Weidenmfiller / Random potential
92
The vacuum state 100) with (00100) = 1 is defined by al00) = 0 = bl00 ) .
(3.7)
For m and j positive integers or zero, we define the states
Imj) = ( m ! j ! ) - l/2( bt )m( a't )Jlo0) "
(3.8)
We have
(mjJnk) = 8,,,Sjk ,
&l.,J> = (s +
(3.9)
)lmj>.
(3.10)
All states { [rnj), m = 0,1 .... } belonging to a Landau level j are degenerate. The states Irnj) can also be obtained from a generating function F ( u , / 3 ) = exp( aa t +/3b t }.
(3.11)
We have
Imj) = ( rn!j!)-l/2 Oajam+gO~3"F( a, /3 )100) ~=o=1~
=(rn!j!)l/2(21ri)-2 fda~d~a-J-1/3-m-1F(et,/3)[O0).
(3.12)
The contour integrals enclose the origin. We have used a particular gauge, with (Ax, Ay)-= ½B(-y, x). Under a gauge transformation, A ~ A + v S - ~ A ' with S = ½ B ( x o y - xyo) we shift the origin so that (A~, Ay) = ½ B ( - y -Yo, x + x0). The shift r ~ r + r0 is expressed in terms of a unitary transformation, !
A ' = exp{ iro • ( V / i ) } A
exp( - i v o • ( V / i ) } .
(3.13)
This shift is compensated if we multiply all the state vectors by the phase factor e x p { - i e S / ( h c ) } . Therefore, the gauge transformation is induced by the unitary matrix (note that ro • VS = 0) U = exp { iro ( V / i )
- ieS/(hc) }.
(3.14)
Converting U into the notation introduced in eqs. (3.3), (3.4) we find, with Zo = Xo + 070, U = exp{ zob - zgb* }.
(3.15)
H.A. Weidenmi~ller / Random potential
93
N o t surprisingly, the matrix elements U,,, of U play a fundamental role in the theory. To clearly exhibit the unitarity of U, we define ip = z o and have
Um,( p ) = (m[exp{ ipb + ip*bt }ln ) .
(3.16)
The matrices Urn.(p) have a number of properties which are sufficiently different from those of ordinary plane waves to be listed here. Trivially, we have
t:2o(P) = G , ( - P ) , Um,(O) = 6,,°, ~ U m ~ ( p ) U , ] ( p ) =8,,,.
(3.17)
s
The dagger denotes the adjoint. The product of two U's with different arguments Pl, P2 is calculated using completeness and the relation exp(A)exp(B) = exp(A + B 1 + ~[A, B] + -.. ). This yields
EU,.s(pl)Us.(pz)
= e x p { ~PlP2 1 *
1
~ P ,2 p l } U m , ( P l + P 2 ) •
(3.1~)
s
We see that under multiplication, the U's form a closed but non-commutative algebra. To interpret the phase factor on the r.h.s, of eq. (3.18), we consider the product U ( P l ) U ( p 2 ) U ( - p l - P z ) - It differs from unity by this very phase factor. The phase equals twice the area of the triangle spanned by the three vectors (Pl, P 2 , - P x - P 2 ) which is taken to be positive (negative) if, following the three vectors, one circumscribes the triangle in a counterclockwise (clockwise) direction. In the space of states {Ira), m = 0,1 .... }, the matrices Urn,(p) form a complete set. This is expressed by the two relations
(2 r) fd -1
2
pUm.(--P)G (+P)=rmsrnk, E Uk~(Pl)Usk(P2)=
(27r)32(Pa
(3.19)
+P2)"
(3.20)
k,s
The volume element d2p stands for d p d p * = 2 d R e p d I m p , and 8 2 is the twodimensional delta function, 32(p) = ½3(Re(p)) 3 (Ira(p)). To derive eq. (3.19), we use eqs. (3.12) and (3.16) to write Utah(P) in the form ~rn+n
U,,,,(p) = (m!n!) -1/2 -Oa~' - Oa~ exp{(al + iP)(a2 + ip*) + l p p , } [~=0=,2" (3.21) Using this form, we easily verify eq. (3.19).
H.A. Weidenmfiller / Random potential
94
Eq. (3.21) implies the important fact that for given p, the elements U,,n(p) have the same numerical value for all Landau levels j. (The r.h.s, of this equation does not depend on j.) We use this statement to derive eq. (3.20), confining ourselves to j = 0. Using eq. (3.18) we notice that eq. (3.20) is equivalent to
Y', Umm( p ) = (2~r)82(p).
(3.22)
t'n
With the help of eqs. (3.4), we write Umm(P )
as
U,,,,,,( p ) = (mr)-X(OOIb"exp{ipb + ip*b t }(bt)m[O0) = ( m ! ) - l(OOlbmexp( ipz* + ip*z }(b t) mexp{ -ipa* - ip*a }lO0) = (m!)-'(001exp( ipz* + ip*z } ( z ' z ) ml00)exp{ + ½pp*}. Summing over m and using the explicit representation
lO0) =
(2¢r)- 1/2exp ( - ½zz* }
(3.23)
we find
EU,,,,(p)=(2~r)-t fd2zexp{ipz*+ip*z}exp{+½pp*).
(3.24)
m
This yields the right-hand side of eq. (3.22). Multiplying eq. (3.19) with an arbitrary matrix Ask and summing over s and k, we obtain
A,,,, = (2qr)-i
f d2p U m , ( - p ) A ( p ) ,
(3.25)
where
A(p) = ~, Uks(p)Ask.
(3.26)
sk
This is a Fourier theorem for matrices defined in the space of a single Landau level. To each matrix A,,, we can uniquely associate a function A(p) via eq. (3.26), and conversely via eq. (3.25). The uniqueness of both operations is assured by eqs. (3.19) and (3.20). Under a gauge transformation, the states Im) are mapped onto the states Ira') =~,Um,(po)[n ). The matrix A,~, is transformed into (U(po)AU*(po)),~,,,. Using the representation (3.25) and eq. (3.18), we find that A(p) is mapped onto exp( PoP* - Po*P }A(p).
H.A. Weidenmiiller / Random potential
95
4. The correlation matrix We recall (sect. 2) that all random potential models introduced there have in c o m m o n a gaussian probability distribution with mean value zero, and a second moment the space part of which contains the correlation function
S([r, - r212/d2).
(4.1)
In this section, we use the results of sect. 3 to derive a representation of the matrix elements of f pertaining to a single Landau level j. The matrix elements of f with respect to the states Im), m = 0 , 1 , 2 . . . ) define the correlation matrix W. Wrnks n = ~ ( m i 2 ( k [ f ] s ) l [ n ) 2
(4.2)
The indices 1,2 indicate whether the state refers to the variable r 1 or r 2. Interchanging the names of the integration variables 1 ~ 2 a n d / o r using hermiticity, we find that W has the following symmetry properties. Wr,~s° = W~m,, = Ws*,mk.
(4.3)
If the correlation function f is proportional to a delta function, we have in addition mmks=Wkmsn.
(4.4)
It is convenient to express W in terms of the unitary matrices Um,(p) introduced in sect. 3. We introduce dimensionless variables as in sect. 3, write z~ = ~ + iy~ with 1 = 1, 2, and express f in terms of its Fourier transform
f([ rl - r212/d 2)
-= (27r)-1
f d2p exp( ip*(z 1 - z2) + ip(z r - z~ )}
×f(d2pp*/lg).
(4.5)
We use this in eq. (4.2) which yields
W,~k,. = (2~r)-i f d2pf'(d2])p*/12o)-i ( m l e x p ( ip*z 1 -P i102'1"}IS)I × 2(klexp { - ip*z2 - ipz~ }[n)2.
(4.6)
The two matrix elements appearing on the r.h.s, of eq. (4.6) can be expressed in terms of the matrices U,,,(p). Omitting the indices 1 and 2, using eqs. (3.3), (3.4)
H.A. Weidenmfiller / Random potential
96
and completeness, we have (m]exp{ ipz* + ip*z }Is)
= (j!)-l(m!s!)-i/Z(OOlaJbmexp{ip(at
+ b) + ip*(a + bt)}(bt)S(at)J[O0)
= Ums(p)Ujj(p*).
(4.7)
We define
wd(p)=fld2pp*/12)
• I ~j(p*)l 2
(4.8)
and find for W
W,,ks . = (2~r)-'
f d~pw~(p) Ums(p)Ukn(-p).
(4.9)
The relations (4,3) imply w
(p) =
=
(4.10)
Using eq. (3.20), we find
~, Wm,snU, k( po) = W:o( Po)Ums( Po) .
(4.11)
nk
This shows that eq. (4.9) provides the decomposition of the matrix W into a sum of orthogonal projectors onto the eigenvectors, the weights being given by the eigenvalues WJo(p). Using eq. (3.18), we find
E W,,,k,,,Usk(P) = ~d(p)U~,,(p),
(4.12)
k,s
where ff6(p) = (2~r)-1 f d 2 p l exp{ pp? - P I P * } w~(Pl) •
(4.13)
This shows that we can also write
winks.= (2,r)-l f dap~vd(p) U m n ( p ) G s ( - p ) .
(4.14)
Eqs. (4.9) and (4.14) provide two different decompositions of the correlation matrix into orthogonal projectors onto eigenvectors.
H.A. Weidenmfiller / Random potential
97
We pay special attention to the case of a delta-function correlation. In this particular case, defined by f = (2~r) -1, we denote the eigenvalues by v0J and g~, respectively. The additional symmetry (4.4) implies, together with eq. (4.8),
o~(p)=Of~(p)=(2w )_ iUyj(p
)[2.
(4.15)
We use the explicit representation
Ujj(p*)
= (2~')-1(j!) -1 fdz
f dz*(zz*)j exp{ -zz*
+
ip*z + ipz* + ½pp*} (4.16)
Therefore, (4.17) Ll=0 This result can also be derived by combining the symmetry properties (4.10) with the requirement that vd(p) is invariant under Fourier transformation, eq. (4.15). We note that vd(p) is positive except for isolated circles in the complex plane where it vanishes. For a correlation function f of finite range, the typical functional dependence is expected to be gaussian, yielding for f also a gaussian shape. Since
w6(p)
-- 2~rfv~(p)
(4.18)
we see that w~(p), too, is positive except for a finite number of circles. For functions f which depend only on the distance ]rl - r : [ , the Fourier transform f-depends only on the magnitude (pp*)l/2 of the variable p. Combining this with eqs. (4.17) and (4.18) we see that both w~ and w~ are invariant under rotations in the complex plane around the origin,
WJo(Pe'~)
=
w~(p),
~Jo(pei~) ~ ( p ) =
(4.19)
for any real a. We note that the effect of a finite correlation length on the formulas is easily worked out. Choosing f proportional to e x p { - ~[r ~ 1 - r2]=/d=) we find that f i s proportional to e x p { - 2pp*d 2 } with d = d/l o. The function (4.18) is proportional to exp{ -pp*(1 + 2,~2)). Rescaling p ~ p(1 + 2d2) 1/2, we leave the functional form of WJo(p) the same. The factor (1 + 2el2) appears explicitly in eq. (4.17) as well as in
H.A. Weidenmi~ller / Random potential
98
the normalization of the p-integrations as in eq. (4.9). In this way, scaling arguments can be used to derive the dependence of observables on the correlation length d. In view of recent work [12], this may be of interest. Using the gauge transformations introduced in the last paragraph of sect. 3 and the representations (4.9) and (4.14) we see that W/o(p) and # ~ ( p ) remain unchanged under these transformations. The gauge invariance of our formulation is thus explicitly demonstrated.
5. Generating function We construct this function for one of the three models introduced in sect. 2, i.e. for the model with time-reversal-symmetry breaking, eq. (2.4). The cases of the white-noise model, eq. (2.1), and of eq. (2.2) are obtained from this model by specializing to the case N -- 1, and by dropping the indices/~ and i,. The case of eq. (2.3) which keeps time-reversal symmetry gives rise to additional complications, although in the limit N ~ ~ it eventually yields the same result as the time-reversal violating case of eq. (2.4). This fact is briefly demonstrated in the appendix. The generating function is introduced as in ref. [13]. Following sect. 2, we define the dimensionless operators/4~ -- H~J(hwo), It o and 1 7 ( r ) = V~(r)/(hwo) with / 4 =/~08~ + 17 (r). We confine ourselves to the Landau level j, and choose the origin of the (dimensionless) energy scale E at ( j + ±) We introduce the complex 2 " commuting integration variables ~,,~(l), and the anticommuting integration variables Xm,(l ). Here,/~ runs from 1 to N while m = 0 , 1 , 2 , . . . . The argument l = 1,2 distinguishes the two blocks needed for calculating the two-point function. We combine these variables into four-dimensional graded vectors ~m~ with x/,x = (~pm~(1), ~,~(2), Xm~(1), Xm~(2)) and components denoted by x/, . The vector in an infinite-dimensional space with components (x/%~; a = 1. . . . . 4; /~ = 1. . . . , N; m = 0, 1, 2 . . . . ) is denoted by g'. Let L ~ be a graded diagonal 4 × 4 matrix with diagonal elements ( 1 , - 1, 1, - 1 ) . In the infinite-dimensional space, we define the matrices
e = (
+=
8 = { 8L~8~.Sm. }, D=E+iS-~-
V+J,
{ o.
(5.1)
The quantity 6 is positive infinitesimal and E and e are defined implicitly in eqs. (5.5) and (5.6) below. The matrix J in eqs. (5.1) has the form (S, f i ~ J ~ } with Jr~n given by (-Jmn(1), -Jmn(2), +Jmn(1), +Jm,(2)) for a - - 1 . . . . . 4 and Jinx(l) nonsymmetric.
H.A. Weidenmiiller/ Randompotential
99
With the volume element d[q'] defined as d[q'] = ]-I mp.l
d(Rego,,u,(l))d(Imeo,.~,(l))dX*~,(l)dX,,,~(l)
(5.2)
we introduce the generating function
Z( E,e; J ) = S d['I'lexp( ½i't'*L1/2DL'/2'I') .
(5.3)
Z(E, e;0) = 1.
(5.4)
It obeys
The quantities of physical interest are the average one- and two-point functions. They are, for instance, given by
E(mi[(e+ #
i8- P)-']..i.) -
1 0 2 oJ, m(1 ) Z ( E , 0 ; J )
(5.5) J~O,
E (mi[(E1 -p i~-- g)-lJli#iis)~kl[ (E 2- i~- V)-l]l,i,l#/) 1
0 2
= 4 Ojsm(1 ) Ojnk(2 ) Z(½(E1 + E2)' E2 - El; J)lJ=o"
(5.6)
To calculate the average of Z, we use eq. (2.4) and eq. (4.14) and find exp{
- ½iq'tLt/2VLt/2g " }
= exp
x2 E w,.ks,(q'2,oGo%a)(q';*,Gd'o~)}
8N mks,~ aByau
=exp{+(4~rN)-lfd2pff~dtrg[A(p)A(-p)]),
(5.7)
where A ( p ) is the graded 4 × 4 matrix
(A(p))~B=~iX
~
(Lt/Z)~(xI%,vU,~,(p)g'~8)(L1/2)~ ~.
(5.8)
mntl.y8
It has the symmetry property trg( At( p )) = - trg( A( - p )).
(5.9)
100
H.A. Weidenmiiller / Randompotential TABLE 1 Parametrization of o (p) al(p) a~'2( - p ) /11(-P) /112(--P)
alz(P) a2(p)
~12(--P) /12(--P)
/1~'(P) ~(P) ibl(p) ibm2(-P)
/1~'z(P) /1~"(p) iblz(P) ibz(p)
We perform the Hubbard-Stratonovich transformation in the way described in ref. [13]. For each value of p, we introduce a graded 4 x 4 matrix o(p) with elements given in table 1. The variables with Latin (Greek) labels are complex commuting (anticommuting), respectively. We also have a/(p) = at*(-p), b/(p)= b~'(-p), l = 1,2. To define the volume element, we write with l = 1,2 { v~Re(a,(p)), 0 ( P ) = V~-Im(at(p)), { v~-Re b,(p) sl(P) -
¢2Imb,(p)
for Rep/> 0 for Rep < 0, for Rep >/0 for Rep < 0.
(5.10)
Then, with d[ a I = 1-I [drl(P) dst(P) drz(P) ds2(P) da12(P) d a ~ ( p ) p
X db,2 (p) d b~'2(p) d ~ ' ( p ) d~l (p) d~/~(p) d~2 (p) d~t2 (p) d~h2(p)
(5.11)
x d~'~(p) d~'~2(p)], we have
f d[o]exp{--~N f d2Pff~d(p)trg[o(p)o( - p ) ] )
= 1.
(5.12)
All commuting variables range from -oo to + m. We shift the integration variables, o(p) ~ ~(p) + (l/N) A(p), and find
exp{+(47rN)-'Sd2p#d(p)trg[A(p)A(-p)]} N = f d[aJexp{- ~ f d2p.d(p)trg[o(p)o(-p)]
1
}
2~rf d'p~g/(p)trg[o(p)A(-p)] .
(5.13)
H.A. Weidenmiiller / Random potential
101
In the transformation formulas (5.12), (5.13) we exclude the singular points p for which ~ ( p ) = 0. The integration over the original variables of integration 'P can now be done. We obtain
Z(E,~; J-~= fd[olexp
N -Z-4~ f d 2 p ~ d ( p ) t r g [ ° ( p ) ° ( - p ) ]
-Ntmr trg ln[ E3mn3~fl - ~3,~,L,~e + i33m~L~B
+J.~.8~¢-~T f dZp~Vd(p)Um.(-p)o(p)
.
(5.14)
A simplification arises for a delta-function correlation. Then, the symmetry (4.4) causes terms like Era, ,X*(l)Um,(p)X,(l) with l = 1, 2 to disappear in the exponent on the r.h.s, of eq. (5.7). As a consequence, we may put b l = 0 for l --- 1, 2 in table 1 and in eqs. (5.12) and (5.14), with a corresponding change in the volume element (5.11) as well as the normalization condition on the remaining variables of integration. From this case, the case of the white noise potential, eq. (2.1), and the case of eq. (2.2) are obtained by putting N = 1 in eq. (5.14). 6. T h e limit N -->
As in ref. [13], we take the limit N ~ ~z by determining the saddle points of the exponent on the r.h.s, of eq. (5.14). We must decide which quantities in the exponent are small of order N-1 and therefore can be dropped in the saddle-point equation. As in ref. [13], the energy difference e is destined to probe fluctuations of the order of a level spacing, and therefore is formally considered to be of order N-1. The conductivity is determined by the behavior of the two-point function for small e and p2. This behavior is expected [9] to be of the form lie + Dp2] -1. Therefore, we consider likewise pZ to be formally of order N -1. The quantities J,~, and 3 are also suppressed. To derive the saddle-point equation with these omissions, it is convenient to cast eq. (5.14) into a different form. We write ~ 6 ( p ) = 1 - 2 . # . p p * + ... and keep only the unity. (This incidentally fixes the normalization of ~2 in eq. (4.1). This normalization is independent of the Landau level j under consideration, cf. eq, (4.17).) It is convenient to Fourier-transform o(p), o ( x ) - ( 2 ~ r ) - l f d2p exp{ ipx* + ip*x } a(p), o ( p ) - (2rr)-i f d2x exp{ - ipx* - ip*x } o ( x ) .
(6.1)
H.A. Weidenmfiller / Random potential
102
Then, the kinetic term in eq. (5.14) takes the form
N fdZxtrg[oZ(x) ]
(6.2)
4~r
We expand the logarithm in a power series and observe, using eqs. (3.18) and (3.20), that tr[U(-pl)U(-P2)-.. U(-p,)] =2~rexp(+½ Z ( p * p k - p : p ~ ) } 6 2 ( - p l - p 2
.....
p,).
(6.3)
/
We replace the exponent by unity and find, using eqs. (6.1), that the logarithmic term takes the form N 2~r
f dZx trgln(E - Xo(x)).
(6.4)
We add the expressions (6.2) and (6.4) and vary o(x) to determine the stationary point. This yields the saddle-point condition (with o(x) replaced by Q(x)) X
Q(x) - E - 2tQ(x) "
(6.5)
It is known [13,14] that the solutions of eq. (6.5) have the form
Q ( x ) - [To(x)]-1QoTo(x),
(6.6)
with (Q0),~¢ = (E/(2)~))B~ - iAL~I~, A = [1 - (E/(2~))2] 1/2, and To(x ) a matrix in the coset space of U(1/1; 1/1)/[U(1/1)® U(1/1)]. Explicitly, To(x ) is parametrized as*
r0(x)
= [ (1 + t12t21) 1/2
-itzl
it12
(1 + tzlt12
)1/2
(6.7)
The 2 x 2 graded matrices t12 and t21 are displayed in table 2. Their form is derived from the condition TotLKT = LK where K is a 4 × 4 diagonal matrix with elements ( + 1 , +1, +1, -1). We now expand about the saddle-point (6.6), writing
o(x) = Q(x) + [T0(x)] 1P(x)To(x). ~' The arrangement of rows and columns in eq. (6.7) differs from that of table 1.
(6.8)
H.A. Weidenmfiller / Random potential
103
TABLE2 The matrices qa ( x ) and t21( x ) tl2(X) a(x) az(X )
t21(X) ia~(x) iz( x)
a*(x) ial( X )
a~(x) iz*( x)
Here, P(x) has vanishing elements in the [1, 2] and [2,1] blocks, i.e., at the positions of the matrices itlz and - i t 2 1 in eq. (6.7). The symmetries of the matrix in table 1 impose further constraints on P(x) which we do not display. In the spirit of the saddle-point approximation, we define/~(x) = N1/ZP(x) and drop all terms which vanish as N--* oe. At the same time, we expand in powers of pp*, with Npp* considered to be of order unity, and again drop all terms which vanish as N ~ oe. (When written in terms of the o(x), this actually amounts to a gradient expansion.) We also use eq. (6.5) and the fact that terms linear in Q cancel by virtue of the saddle-point condition, and that trg(QZ(x)) = 0. We recall that Ne is of order unity, and we keep only terms of up to second order in J~n as only these terms are needed in eqs. (5.5) and (5.6). We display the result for a particular choice of J,~,. It consists in transforming J,~, according to eqs. (3.25) and (6.1) and putting J ~ ( x ) = 2Tr62(x)J ~ with ja given by ( - J ( 1 ) , +J(1), -J(2), J(2)) for a = 1,...,4. (Other choices are possible, of course, if other observables are needed.) All this yields for the exponent on the r.h.s, of eq. (5.14) the expression - (2~r) i f d2 x trg[/3Z(x)(1 - ( E / ( 2 X ) ) 2 + i A ( E / ( 2 ~ ) ) L ) ] N + e . ~ - ( 2 ~ r ) - l f d 2 x trg[LQ(x)]
fd2xtrg[(-~xQ(X))(-~Wx, Q(x))l
1 -NcS(2~r) -
N
O
O
N
- -~trg[O(O)J 1 + ~ t r g [ Q ( O ) J Q ( O ) J ] .
(6.9)
We can use eq. (6.9) to calculate the one-point function, eq. (5.5). For N ~ ~ , the level density within each Landau level has the shape of a semicircle with radius 2~. This is the generalization of a result derived by Streit [9] for the lowest Landau level and for a delta-function correlation.
104
H.A. Weidenmi~ller / Random potential
For E within the spectrum the term proportional to /~2(X) in expression (6.9) is negative definite. The integration over P(x) can thus be done and yields unity [13]. It remains to simplify the fourth term in expression (6.9). We expand the logarithm in powers of )~, use eq. (6.3), and expand the exponential on the r.h.s, in powers of the pj, keeping only terms of zeroth and second order. The zero-order terms add up to a vanishing contribution. We replace -pTpk+p~pj by [( O/Oxj)(O/Ox~) - (O/Ox*)(O/Oxk)], the latter operator after partial integration acting on the product of the Q's and being equal to ~iVj× Vk in cartesian notation. We write x=(xl, x2) and Qj instead of OQ/Oxi with i = 1 , 2 . After integrating over all momenta, we find
E8.,.8.¢-~
Ntr trgln
fd'pVm.(-p)O(p)
.=2
E-
j=l
k=2
k>j
× tr?([(Q(x));-1Qx(Q(x))k-'-lQz(Q(x))'-k]-[1~ 2]}. (6.10) We write n -1 foldpp"-1. Then, the summations over j, (k - j ) and (n - k) can be carried out independently. With 6 = (Xp/E)Q, the expression (6.10) takes the form ---
¼iN(2~r)-' fo'dpp-'fdZx tr? {(i- Q)-2QI(1
- Q)
162-(1
(--)2)}.
(6.11) From the saddle-point equation (6.5), we have
(
)~
1--EOQ
)(
We use this and the relation Q2 = form
~INfodPP |
,
×trg~
1
([(
_
1-
)
)k P- Q = 1 + . . 1+ ~1 P ~-21_-- O
g.
(6.12)
EQ/X - 1 to rewrite the expression (6.11) in the
l(2"Ir)-lfd2xg-3()-~) 2 E(~Z O)
+ E ~ i - ;;' ~ 91 1 + E ( 1 - p) (6.13)
H.A. Weidenmfiller / Random potential
105
Because of the cyclic invariance of the trace, only the third-order terms in Q give a non-vanishing contribution, and the result takes the form
LaiNh(E)(2v)-X f d2xtrg{Q(x)[Ql(x),QR(X)] } .
(6.14)
The real function h (E) is given by
[ 0 2 ~ t 2 p2 ] 2. "E; Jod0t-~p-p)e[ 1+ ( \E-] ~
h(E)=(2t\3rl
(6.15)
We obviously have
h(-E) To calculate the branch of and obtain
h(E)
h ( e ) = f]/xd,
= -h(E).
(6.16)
with E > 0, we make the substitution r =
1+~ 2-
,
e>0.
E/()tO) (6.17)
We have h(0 + ) = ~r 1 and h(2X)= 7. 1 For negative values of E, we use eq. (6.16). We see that h jumps by ½~r at E = 0. Collecting everything, and recalling that the integration over/3 for the decomposition (6.8) of o(x) introduces [13] the invariant measure d/.t[t12(x)], we obtain
Z(E,e;
J)=
eN fd/*[q2]exp{~-(20r)lfdRxtrg[LQ(x)]} × e x p { - N c J ( 2 0r)
× e,
'fd2xtrg(f-~Q(x)~-~Q(x)]) d2xtrg(O (x)[ e,(x), e2(x)])}
×exp - g trg[Q(0)J] + ~
trg[Q(O)JQ(O)J] .
(6.18)
To define the measure d#[t12 ] = Fix d/~[txz(X)] we observe that Q(x) itself can be written in the form LTo(x ) with To(x) given by eq. (6.7), and parametrised as in table 2. With this parametrization for Q(x), d/t[ta2(x)] is for each x given by [14] 1 (X i + X2) 2 d ~ t [ q 2 ( x ) ] - 4 (X]X~) '/2 dadc~*dzdz*dcqdc~'dc%dc~'.
Here, X1, X2 are the eigenvalues defined in ref. [14].
(6.19)
106
H.A. Weidenmgdler / Random potential
We note that the effective lagrangian occurring in the exponent on the r.h.s, of eq. (6.18) contains, aside from the usual interaction term and a term proportional to e, a term involving trg{Q(x)[Ql(x ), Qz(x)]}. This is the "topological term" first introduced by Pruisken [4]. We see that the random-matrix models of eqs. (2.3) and (2.4) allow us to give a stringent derivation of the lagrangian containing this term, together with a precise definition of the function h(E) in eqs. (6.16) and (6.17). Using partial integration, we can write [4, 5] the topological term as a surface term. Since partial integrations were also used to derive eqs. (6.9) and (6.10), it is important to ascertain that these latter integrations do not contribute additional surface terms. For the derivation of the topological term in eq. (6.10), it is easy to see that the surface terms arising from partial integration cancel pairwise. For the double gradient term in eq. (6.9), the vanishing of the surface terms follows from trg(Q2(x)) = 0 and from trg[Q(x)(O/Ox)Q(x)] = 0 . The last two relations are implied by the parametrization (6.6). We recall the definition (6.6) of Q(x) and the fact that Q0 = (E/2~)6~# - iAL~B. We note that the part of Q0 which is proportional to the unit matrix does not contribute to the effective lagrangian in eq. (6.18). This suggests the definition Q ( x ) = [T0(x)]
~LTo(x).
(6.20)
Replacing Q(x) by Q(x) everywhere on the r.h.s, of eq. (6.18), we are led to make the following substitutions:
e ~ -iAe, cJ ~
-- A2C j '
ih(E) ~ O(E) = - A 3 h ( E ) ,
(6.21)
and correspondingly for the source terms. Comparing this result with refs. [4,5], we see that the effective lagrangians obtained in either approach are Very similar. A central difference, caused by the use of the replica trick in refs. [4, 5] lies in the parametrization of To(x), and of Q(x). The authors of refs. [4, 5] use either a compact or a non-compact parametrization, depending on whether they employ replicas with anticommuting or with commuting variables. They eventually opt for the former choice as only here the topological invariant assumes non-trivial (i.e. non-zero) values. Such an ambiguity does not arise in the present context. Arguments of symmetry and convergence [13] stringently require the matrix To(x ) to contain both compact and non-compact symmetries. This combination of symmetries is contained in the parametrization of To(x ) given in eq. (6.7) and table 2. The topological invariant attains non-trivial values in the fermion-fermion block [14]. Our result goes beyond that of refs. [4, 5] in yet another respect: We obtain an explicit expression for what is called the "bare Hall conductance" O°y in refs. [4, 5].
H.A. WeidenmFdler / Randompotential
107
Indeed, our form of the topological term in the fermion-fermion block coincides with that of refs. [4, 5] if we put
o° = 4N@(E)/(2~r).
(6.22)
Combining eq. (6.22) with eqs. (6.21), (6.16), and (6.17), we find that N-lo°y vanishes at E = _+2X (the endpoints of the spectrum) and that it has vanishing derivatives there. In the interval - 2 X ~
~ since each of the eigenvalues of Q(x) has modulus unity, cf. eq. (6.6). The results (6.14) to (6.17) are therefore strictly valid in those parts of the spectrum which are defined by X < [E I < 2~. For IEI ~< X, the results (6.14) to (6.17) are obtained by analytic continuation. To compare our results with those of ref. [9] we use eqs. (6.18) to (6.21) and introduce ft2 = N1/2tt2, and fzt = N1/2t21 • We expand the effective lagrangian in powers of i12 and i21. Terms of second (fourth) order carry a factor N o ( N -1 respectively). This makes it possible to perform a N - ~ expansion. To leading order, we retrieve Streit's result [9]. To next order, we find a logarithmic singularity, also in keeping with ref. [9]. It is caused by the fact that the t12 modes are massless and therefore produce an infrared divergence. The fourth-order terms arising from the topological term vanish by partial integration in the sector where the topological quantum number is zero. This confirms the statement that this term is not accessible to perturbation theory [4, 5].
7. Summary and conclusions Within the space 5p spanned by the degenerate states of a single Landau level, we have used gauge transformations and the associated unitary matrices to construct a matrix algorithm. It allows us to map any operator A,~n in 5 p onto a function A ( p ) in two-dimensional p-space. The map is one-to-one and has a close formal similarity with the Fourier transformation. Through this map, we avoid using the projection operator onto a specific Landau level which tends to complicate the algebra.
108
H.A. Weidenrniiller / Random potential
For a random-potential model with correlation function f, the map associates with f the function w~(p). This function corresponds to the projection of f onto Landau level j. Using path-integral techniques, we have rewritten the random-potential problem in terms of an effective lagrangian involving graded 4 x 4 matrices in p-space. For the model of eq. (2.4), we have considered this lagrangian in the limit N ~ oo, with Npp*= constant (long wavelength approximation), using the saddle-point approximation. We have shown (i) that the spectrum of any Landau level is given by a semicircle law. The radius of the semicircle is the same for all Landau levels. We have also shown (ii) that the effective lagrangian reduces to the non-linear sigma model of eq. (6.18). This model contains a topological term of the form first introduced by Pruisken [4]. The term has its origin in the non-commutativity of gauge transformations within a given Landau band and, ultimately, in the presence of a strong magnetic field. We have explicitly calculated the factor h ( E ) multiplying the topological term. The function h ( E ) depends only on the variable Elk which characterises the spectrum and is independent of the correlation function f and of the label j of the Landau level under consideration. Reference to this label is only made via the coefficient c ~ = ½ ( j + 1 + 2 d 2) multiplying the interaction term. Identical results are obtained when one starts from the random-potential model (2.3) and considers the same limit N ~ oo with Npp* fixed. It is shown in the appendix that the orthogonal symmetry present in eq. (2.3) leads to the occurrence of additional massive modes, but that the non-linear sigma model obtained for N ~ ~ is again governed by the unitary symmetry caused by the magnetic field. In comparing our work to that of refs. [4, 5], we have pointed out that our parametrization of the Q-matrices makes use of the correct combination of compact and non-compact symmetries. For the function multiplying the topological term, it yields an energy-dependence which is in line with the general considerations of these papers, save for a discontinuity in the centre of the Landau band. It would be interesting to investigate whether the formulas of sects. 3 and 4 make it possible to calculate the average level density for the white-noise potential (2.1) for other Landau bands than the lowest. This would extend the work of refs. [6, 7]. It would likewise be interesting to extend our approach to a finite geometry of the type considered in refs. [1, 2]. We could then also compare with the work of ref. [16]. What do our results imply with regard to the situation described in the introduction? Our most important result in this respect, based on a comparison between our eq. (6.18) and the form of the non-linear sigma model derived [11] for the localization problem without magnetic field, lies in the statement that the tWO models differ only in the presence of the topological term. This lends further support to the claim [4, 5,17] that this term is responsible for delocalization. In view of the work of ref. [17], we expect a delocalisation transition at the band centre, E = 0. We find it surprising that the calculations reported in ref. [10] and earlier works show evidence for a domain of extended states of finite width in energy.
H.A. Weidenm~ller / Randompotential
109
Combining our result with the argument of universality we would not have expected this. In as much as delocalisation is due to the topological term (which is not accessible to perturbation theory) it would also be interesting to understand how the resummation of the perturbation series carried out in ref. [10] manages to produce extended states. We hope that a detailed comparison between our approach and the method used in ref. [10] may clarify this issue. Most of this work was done while the author was visiting the Division de Physique Th6orique of the Institute de Physique Nucl6aire at the Universit6 de Paris Sud at Orsay, France where the author received partial support. The work was completed at Caltech under the auspices of the Sherman Fairchild Distinguished Scholarship program and was supported in part by National Science Foundation grants PHY85-05682 and PHY82-07332. I am grateful to my colleagues at both institutions for their warm hospitality. I have benefitted from discussions with E. Br6zin and F. Wegner. Comments by D. Boos6 and H. Nishioka helped me in clarifying the contents of sects. 2 and 3. M. Zirnbauer suggested using the Fourier-transformed quantity o(x) in determining the saddle-point manifold, and deriving the topological term. These suggestions were instrumental in shaping the content of sect. 6.
Note added in proof. F. Wegner has kindly pointed out that aside from the discontinuity at E = 0, the function a°v (E) grows linearly with the number of states below E. While the results of ref. [10] did not rule out the possibility of a singular behaviour of the localization length at E - - 0 , positive numerical evidence for such behaviour has recently been presented by S. Hikami [19].
Appendix THE CASE OF ORTHOGONAL SYMMETRY
We use the model of eq. (2.3) and proceed as in sect. 5. The occurrence of the extra term on the r.h.s, of eq. (2.3) causes additional terms to appear in the exponent in the centre expression of eq. (5.7). All the terms which arise can be arranged in the form of 8 x 8 graded matrices. This is done in table 3. (For the model of eq. (2.4), this table would contain only the 4 X 4 matrix in the upper left-hand corner.) Graded 8 x 8 matrices appear also in the GOE problem of ref. [13]. The difference between that problem and the present one lies in the admissible transformations of the integration variables. In the present case, we consider arbitrary pseudounitary transformations of the graded four-vector qe,,~, while in ref. [13] one deals with pseudounitary orthosymplectic transformations of graded eight-vectors. It is not difficult to show that the former transformations form a subgroup of the
H.A. Weidenmiitler / Random potential
110
TABLE 3 The graded 8 x 8 matrix Amksn appearing in 2 for the mode/of eq. (2.3) IL1/2~
.I,
.I,*
i L l ~ 2~.
1/2
**
1/2
E ~ v ( L 1 / 2 ).~k,w.~lks~v (L 1 / 2)v~ * * * E~,13~(L ),',~Pk~,SlPswr( L )v~
~Bv~ J ~ ' ~ *" ~ * 12)v~ E~,Bv(L1/2 ),,t~k~'l~lk'm'v ( n / )v~
All entries are 4 × 4 graded matrices. The fourth-order term in the exponent of 2 is proportional to trg(A,,k.,,A k.......). Aside from factors, the function A(p) in eq. (5.8) is related to the entry in the upper left-hand corner by a transformation of the form (3.25).
latter. The restriction to this subgroup causes the models (2.3) and (2.4) to coincide for N ~ ~ . The Hubbard-Stratonovich transformation is now performed using a set of o-matrices which are likewise of dimension 8 and incorporate the symmetries of table 3. For these matrices (which still carry the labels m, n of the degenerate Landau levels) we obtain a saddle-point equation which has the general structure of eq. (6.5), with x replaced by the indices m , n , k , s and with Q an 8 × 8 graded matrix. The solutions to this equation consist of a multiple of the unit matrix, (with respect to the indices m, n), and matrices involving the Goldstone modes. The latter correspond to the breaking of graded symmetry by the level density and have the general form of eq. (6.6), with To(X ) a graded 4 × 4 matrix. This last fact is caused by the restriction to the subgroup of 4 × 4 pseudounitary matrices mentioned in the last paragraph. Therefore, the Goldstone modes of the model (2.3) have the same form as those of the model (2.4), and the two models give identical results in the limit N ~ oo. References [1] R.B. Laughlin, Phys. gev. B23 (1981) 5632 [2] B.I. Halperin, Phys. Rev. B25 (1982) 2185 [3] E. Abrahams, P.W. Anderson, D.C. Licciardello and T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673 [4] A.M.M. Pruisken, Nucl. Phys. B235 [FSll] (1984) 277 [5] H. Levine, S.B. Libby, and A.M.M. Pruisken, Nucl. Phys. B240 [FS12] (1984) 30, 40, 57 [6] F.J. Wegner, Z. Phys. B51 (1983) 279 [7] E. Brezin, D.J. Gross and C. Itzykson, Nuel. Phys. B235 [FSll] (1984) 24 [8] S. Hikami, Phys. Rev. B29 (1984) 3726; Prog. Theor. Phys. 72 (1984) 722 [9] T.S.J. Streit, J. Physique Lett. 45 (1984) L713 [10[ R.P.P. Singh and S. Chakravarty, Nucl. Phys. B265 [FS15] (1986) 265 [11] F. Wegner, Phys. Rev. B19 (1979) 783; L. Sch~ifer and F. Wegner, Z. Phys. B38 (1980) 113 [12] S. HikamJ and E. Br6zin, J. de Physique 46 (1985) 2021 [13] J.J.M. Verbaarschot, H.A. Weidenmtiller and M.R. Zirnbauer, Phys. Reports 129 (1985) 367 [14] J.J.M. Verbaarschot and M.R. Zirnbauer, J. Phys. A17 (1985) 1093 [15] T.A. Brody, F. Flores, F.B. French, P.A. Mello, A. Pandey and S.S.M. Wong, Rev. Mod. Phys. 53 (1981) 385 [16] H. Aoki, J. Phys. C15 (1982) L1227; 16 (1983) 1893 [17] T. Affleck, Nucl. Phys. B265 [FS15] (1986) 409 [18] D. Ebert and H. Reinhard, Nucl. Phys. B271 (1986) 188 [19] S. Hikami, Tokyo preprint (1986)