Evaluation of ensemble averages for simple Hamiltonians perturbed by a GOE interaction

ANNALS

OF PHYSICS

153, 367-388 (1984)

Evaluation of Ensemble Hamiltonians Perturbed

Averages for Simple by a GOE Interaction

J. VERBAARSCHOT, H. A. WEIDENM~~LLER, M. ZIRNBAUER Max-Plan&Institut

fiir

Kerphysik,

Heidelberg,

Federal

AND

Republic

of Germanv

Received July 21, 1983

Using an expansion in powers of N- ‘, where N is the dimension of the Hamiltonian matrix, we evaluate ensemble averages of the resolvent, of products involving several resolvents, and of the moments of the Hamiltonian H, + 1V. Here, H, is arbitrary but fixed, and V is a GOE ensemble. The nature of the N-’ expansion is also discussed.

I. INTRODUCTION

Much insight into generic properties of nuclear spectra has been gained by the study of random Hamiltonians [I]. In the present paper, we extend such studies to Hamiltonians of the form H=H,+AV.

(1.1)

Here, V is an ensembleof symmetric N x N matrices with real elements V, which are uncorrelated, Gaussian distributed random variables with mean value zero. The second moments are given by Vij

V,,

=

(6i,

Sj,

+

6i, Sj,)

V *.

(1.2)

In other words, V is a GOE (Gaussian orthogonal ensemble). Because of the orthogonal invariance of V, the Hamiltonian H, can without loss of generality be assumedto be diagonal, and has the form H, = 5 ci (Qi)(QiI,

(1.3)

i=l

and the ]#i) are N-dimensional vectors. Both the si and ]#i) are (#i I 42 = Sik fixed (as opposed to being random variables). We are interested in a typical shellmodel situation where N % 1. Our interest in properties of the Hamiltonians (1.1) was prompted by our desire to understand how rapidly (in terms of the strength parameter A) the spectral properties of H, are washed out and are dominated by those of V. The results of this analysis 367 where

0003.4916184 $7.50 Copyright D 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.

368

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AND

ZIRNBAUER

are reported elsewhere [2]. In the course of our analysis, we were led to calculate ensemble-averages of various functions of H. This was possible by a generalization of techniques developed earlier for simpler cases [ 1, 31. We believe that both our technique and the results derived from it have an independent interest in the frame of statistical nuclear spectroscopy, and we communicate them in the present paper. The material is organised as follows. In Section II, we present the technique used to calculate ensemble-averages. It essentially consists in an expansion in powers of NP ‘. We use it to calculate the terms of leading order of the resolvent c(z) = (z - H) Pr, and of products of resolvents like N-’ Ci Gii(z,) N-’ cj Gjj(z2) etc. In Ref. [ 1 ] it was emphasized that the use of the N-’ expansion for the evaluation of averages of moments of V, i.e., of the quantities N-i xi I$., is limited to values of p 5 N since for p > N, the next-order corrections become as big as the terms of leading order in N-‘. This observation has prompted us to have a closer look at the N- ’ expansion. To prepare the discussion, we present in Section II also the evaluation of the correction terms of next order in N-’ for G(z) and for NP i Ci G,,(z,) N-’ cjGjj(z,). In Section III, we apply the NP ’ expansion to the explicit calculation of the averaged moments of H, defined by N- ’ Ci (HP) ii. We were motivated to do so by the need to calculate the branch points as explained in Ref. [2]. We introduce recursion relations for these moments which are used in Ref. [2], and employ such techniques to also discuss the corrections of order NP’. All this material enables us in Section IV to discuss the nature of the N-’ expansion. Some technical details are presented in Appendices A and B.

II. THE N-’

EXPANSION

FOR THE RESOLVENT

We demonstrate the method by calculating explicitly the average resolvent G(z) to leading and next order in N- ‘, and the correlation function of a product of resolvents to the sameorder (Sections 11.1and 11.2). In Section II.3 we present additional results without proof. In Appendix A we investigate the behaviour of terms of very high order in N-‘. 11.1. The Average Resolvent The resolvent G(z) is defined by G(z)=(z-H)-‘.

(2.1)

To evaluate the ensemble-averageG(z) of G(z), we expand G(z) in a power series with respect to LV and evaluate the average by calculating the average of each term in the series. With G,(z) = (z - HJ’ we have

G(z) = f

Go(ilVGo)m.

(2.2)

ttl=O

The average of (LVGo)m is obtained from the fact that a product of m Gaussian

EVALUATION

OF

ENSEMBLE

369

AVERAGES

random variables with zero mean values has zero average if m is odd, and that for even m the average is given by summing over all possible ways of taking the average of pairs of Vs. For m = 4, this rule implies that (vG,)4 = VG,G,

h$G,

+ LG, Ij.G,G,

?G, (2.3)

where the average of a pair of V’s is indicated by a contraction line connecting the pair. We emphasize that this rule is exact and entails no approximation. An approximation is introduced, and the number of terms contributing to fixed m is very considerably reduced, by omitting terms which are relatively small of order N- ‘. Using the uncorrelatedness of Vs with different indices and Eq. (1.2) we find easily that the last term on the r.h.s. of Eq. (2.3) is of order N-’ compared to the other two. As a general rule, we find [ 1,3] that the terms of leading order in N-’ are those in which contraction lines do not intersect. Application of this rule to the expansion (2.2) leads for C?(Z)to the integral equation [ 1,4] I I (O),-(z) = G,(z) + l’G,(z) V”‘G(z) v’“@(z). (2.4) The index (‘) indicates that c(z) has been evaluated to lowest order in N-l. Equation (2.4) implies via Eq. (1.2) that (“G(z) is diagonal in the representation (1.3), with diagonal elements given by

‘“‘c,(z)= [z- E,- p(‘O’G(z))] - I,

(2.5)

where we have used the definitions ,u = A2v2N

(‘“‘c;
and

5 ‘O’G,(z). Cl=,

(2.6)

Equation (2.5) was used in Ref. [2] to evaluate (C?(Z)) and C?,(Z) in terms of a continued-fraction expansion. It was first obtained by Pastur [5]. The same technique can be used to calculate corrections to order N- ’ for G,(z). We have to keep all terms of relative order N-‘, and omit terms of higher order. By inspection of the terms arising in the Born-von Neumann series (2.2), we find for c(z) the series G(z) = G,(z) + G,(z) V&k(z) I +

+

G,(z)lk(z) k(z) k(z) k(z) I I I I 1 G,(z)P-c?(z) Vc(z) V-G(z) h??(z)

k(z)

V-c(z)

+ ... .

(2.7)

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AND

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Using Eq. (1.2), the sum can be evaluated and yields 6,(z)

= Go,(z) + ego,)

~a(~)

+ N-‘~Go,(z)(G,(4)*[1 -~u(@%))~)l~*. This equation is valid to order N-r

and thus can be solved by writing

G,(z) = ‘“‘c,(z)

+ N- ’ “‘G,(Z)

+ .. * .

Collecting terms of lowest and next order in N- ‘, we retrieve for “‘GJz) (2.5) while “‘G,(z) is given by “‘&(z)[

1 - ~G,,(z)(‘~‘G(z))] = ~G,,(z)(‘~‘G,(z))~

To demonstrate the do not carry unduly case, where E[ = 0 G(z) + 0 as z --) 00

(2.8)

the result

- ,uG0,(z)‘“‘c,(z)(“‘~(<2)) [ 1 - /@“‘G(z))2)]

- ‘.

(2.9)

simplicity of this result, and to show that the N- * terms for (G) large coefficients, we apply Eqs. (2.5) and (2.9) to the pure GOE for all i, G,(z) = z-‘, and 1= 1. This and the condition that give

(‘O’~(z)) = ‘O’G,(z)= & (z - pq)

(2.10)

and this, when used in Eq. (2.9), (“‘G(z))

= “‘G,(Z)

1 z-@=F/L = y z2 _ 4cl

.

(2.11)

Both “‘@z) and (‘)G(z) have a branch cut along the real z-axis extending from -24 to +2&. The average level density p(x) is given by the discontinuity across the cut

D(x)= dw27w

l+i N

l”

x*-4/1

+&w2d4t~(x-2m.

(2.12)

The last bracket arises from the singularity at z = *I2 4. It is required to keep the normalization of p (i.e., the condition that j” p(x) dx = 1) to terms of order NP’. We see that except for the ends of the spectrum the N- ’ correction carries coefficients which are similar in magnitude to those of the leading order. This is a characteristic feature also of (G(Z)), but not of the high moments considered in Section III. The latter are strongly influenced by the ends of the spectrum where the corrections are large.

EVALUATION

11.2. The Average Two-Point

OF ENSEMBLE

371

AVERAGES

Function

This quantity is defined as (G(x))(G(y)). two-point correlation function

It is convenient to evaluate instead the

(2.13)

F(x, Y) = (G(x)XG(y)) - (G(x) )(G(Y)>-

which differs from the two-point function by terms evaluated under 11.1. Expanding G(x) and G(y) each in powers of V, we see that contributions to F arise only from terms in which at least one V in the series for G(x) is contracted with a V in the series for G(y). We refer to such V’s as “cross-contracted.” Whenever we do not wish to write down the entire expression for F(x, y), but only terms originating via the power-series expansion of G(x) [ or of G(y), respectively], we denote the cross-contracted V’s by the symbol I? In the terms of leading order in N-’ contributing to F(x, y), only non-interser+ting contraction lines appear. In the power-series expansion of G(x) with respect to V, the factor appearing between every pair of adjacent I% is just G,(x). This follows easily from the defirning Eq. (2.2). There are also terms preceding the first or following the last of the V’s in G(x). Because of the cyclic invariance of the trace appearing in the definition of (G(x)) [cf. Eq. (2.6)], such terms can be collected and written in the form Go Go + Go Go V-G,

+ Go &&Go

Go + Go VG,G,

+ ... .

(2.14)

It is straightforward to show that the expression (2.14) has the value -(d/dz) ‘“‘G(z). The total contribution to “‘F (x , y) arising from (G(x)) therefore has the form

d ‘O’G(x) {F++%3(x)VT+

dx ZZ

tt = - $

d ‘O’G(x) y’f? (‘O’G(x) dx 1 PC=0 c

+ ((‘“‘qx)

;)rm).

vy

(2.15)

It-1

Once again we have used the cyclic property of the trace. Contributions arising from (G(y)) are given by an analogous expression. Clearly, the numbers of crosscontracted V’s in (G(x)) and (G(y)) must be equal. Given a term with m factors v’in both (G(x)) and (G(y)), with m > 1, there are 2 . m ways of connecting the V’s pairwise without creating intersecting contraction lines. The factor 2 arises from the reflection symmetry of the term (2.15), and the factor m from its cyclic invariance. For m = 1, there is obviously only one way. To lowest order in N-‘, F(x, y) is therefore given by

312

VERBAARSCHOT,

‘O’q&y) = 2 &

WEIDENMijLLER,

AND

ZIRNBAUER

+ (‘OG(x)S>(FO’G(Y)) 1

++ (‘O’G(x) VO’G(x) V@O’G(y) d’O’G(y)) +3 <‘“‘C(x) ?‘“‘G(x) lJ’O’C(x) v)(s’“‘G(y) b’“‘C(y) ?‘O’G(y)) + ...

i

.

(2.16a)

Using Eq. (1.2) and the definition (2.6), we then find ‘O’F(x,

The derivative of “‘G(x)

d dx

y) = -

7i

&

ln[ 1 -p(‘“‘G(X>‘“‘@Y>)l.

(2.16b)

is easily found to be

(o’qx) = - ‘O’G(x)[ 1 -/L((‘“‘G(x))*)]

-’ ‘“‘G(x).

(2.17)

This completes the evaluation of “‘l;(x, y). It is instructive to use Eqs. (2.16) and (2.17) in the special case of the pure GOE (Ho = 0, A = 1). Using Eq. (2.10) and taking separately the discontinuity across the cut in x and y (this requires x # y), we find after some algebra for the correlation function of the level density the result (x # y) [ 91

&MY>____-P(X) P(Y) P(X) P(Y)

4P

4 - XYIP

= - N2 (x - y)2(4 - X2/P)(4 - y2/,Dj

(2.18)

Choosing p = 1 we find that this result agrees with Eq. (4.21) of Ref. [l] if we there put j? = 1. This fmding supports our confidence in the general procedure. We can also use Eq. (2.16) to calculate the correlation function of the level motion 6(x) which is defined as in Section 1V.D of Ref. [ 11. Introducing the average level spacing d we find 2 4x1 d(Y) =’ lim In d2 ?I2 v-0 ii

1 - j@%?(x + i?#“G( y + iv)) 1 - ,u(“‘G(x - ir#O’G( y + iv)) I! *

(2.19)

This result combined with the continued-fraction expansion for g(x) described in Ref. [2] makes it possible to calculate the 1.h.s. of Eq. (2.19) for arbitrary Ho. We turn to the N-’ correction to “‘F(x,y) which we denote by (“lr;‘(x, y). Contributions to “‘F(x, y) arise: (a)

From the N- ’ corrections to “‘G(x)

evaluated in Section 11.1. [These are

EVALUATION

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373

AVERAGES

directly given by replacing ‘“‘G(x) and “‘G(y) on the r.h.s. of Eq. (2.16) by ‘“‘G(x) + ,-l(l)G(x) and “‘@y) + N-““G(y), respectively.]. (b) If one pair of cross-contracted V’s is diagonal in the level indices. Such a pair will be indicated by a dotted line. (c) If two or more cross-contraction lines intersect in a nested way. (d) If a contraction line connecting a pair of V’s both appearing in G(x) [or in G(y), respectively] intersects a cross-contraction line. Not both V’s connected by the contraction line should belong to the end-factors considered in expression (2.14). We consider jointly the contribution of types (b) and (c). We use expression (2.15) and depict each of the terms ((‘“‘G(x) fim) and ((‘“‘G(y) ;)r,) as a ring to display the cyclic property of the trace. Omitting terms that were included already in (2,16), we find that the contributions can be depicted by

+[?J+YJ+W+... ] + [~+pj+)jj+... ] +[g+E+jijJ+...] +*..

(2.20)

The factor 4 multiplying the first curly bracket takes account of the fact that the other half of this contribution appears in (“F (x , y). Note that the very first diagram in (2.20) is identical to

Evaluation of the patterns (2.20) is tedious but straightforward following contribution to “‘F(x, y): 1 d2 ~““((‘o’~(x))2(‘o’~(y))2) Fdxdy [l -~('"'G(x>co)G(~>>]"

and yields the

(2.2 1)

In calculating this expression, we have used the cyclic symmetry of both rings which yields a factor m2. This last factor differs from the factor m leading to Eq. (2.17) because of the exceptional role played by a single, or a pair of cross-contraction lines in the patterns (2.20). Except for terms in the first square bracket the reflection symmetry does not give new patterns.

374

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WEIDENMtiLLER,

AND

ZIRNBAUER

of type (d) can be visualized as

E+fjJJ + .... +fjJ+fjj+ ....+....

+g+jfJ+ .... +jfJ+jGjJ+ .... +**** X-Y

+ [

1

(2.22)

The corresponding algebraic expression with the endpoints represented by derivatives (see Eq. (2.15)) is given by

N-3

d2 a:0 ‘uy(‘w(X))* -&&

(o’G(~~))‘((‘“‘~(x))‘)“‘(‘O’G(x)’O’G(y))”’+”J~”‘+“~+“~

n2>0 n,>o +

(2.23)

Ix-yJ.

Each pattern obtained by using the cyclic symmetry of both factors is generated twice. The associated combinatorial factors are cancelled by the factor 2 due to the reflection symmetry and the combinatorial factors accompanying the derivatives (see Eq. (2.15)). The brackets [X my] denote the analogous contribution arising from interchanging x and y. The geometrical series (2.23) can be summed and the result is given by d* ‘u3((‘o’G(X))2 ‘o’G(y))2 N-3 dxdy (1 -,~#~‘G(x))*))(l -~(‘“‘G(x)‘o’G(y)))2

+ Ix++ “’

(2.24)

Combination of the terms of type (a), and of the results (2.21) and (2.24) yields “‘F(x, y). The result again has the same features as the N-’ correction to (G(x)): The coefficients of the N-’ terms are of order unity. 11.3. Further

Results

We have also evaluated the terms of lowest non-vanishing order in N-’ for the average three-point function (G(x))(G(y))(G(z)), and for the four-point function (G(x))(G(y))(G(z))(G(w)). For purposes of our later discussion, we give here the result for the three-point function. We confine ourselves to the totally connected contraction patterns as partially connected contributions are given in terms of the results of Sections II.1 and 11.2:

EVALUATION

(WXG(Y))(W)

OFENSEMBLE

315

AVERAGES

lconnecteci

=-4; &g-& ~P3G(mY)m2[l -mmYw. [l -P(~(Y)W)l-‘[l -L@(z) W)l-’ + [~(x,y,z)[l -P(~(mY))1-‘[1 -/my) + cyclic permutations

of (x, y, z)] }.

Gz))l-’ (2.25)

The function F(x, y, z) is given by w> Y, z) = P2mmYN2

w

+lu3(~(x>(~(Y)>‘~((~o)2G(z)~~1

-Pu((aY))2)I-‘*

(2.26)

In all these formulas, G stands for “‘G as given by Eq. (2.5). These results and an analysis of our technique show that via the N-’ expansion, ensemble-averages of products of the resolvent are given in terms of traces of powers of G and of its derivatives.

III.

MOMENTS

OF H

In this section we evaluate the moments of the Hamiltonian H = Ho + AV, using the N- ’ expansion introduced in Section II. In Section III. 1, the moments are calculated explicitly to lowest non-vanishing order in N-’ by means of a combinatorial analysis. In Section 111.2, we derive a recursion relation in leading order, using Pastur’s equation for the resolvent. With the help of the same technique, N-’ corrections are considered in Section 111.3. Finally, in Section III.4 we derive recursion formulae that express moments of order higher than N in terms of those of lower order. 111.1. An Explicit Expression for the Moments The ensemble-averaged nth moment of H is defined as M,=;

$ {(Ho+~V)“}ii. 1-I

(3.1)

We calculate the leading-order contribution to M, in the N-i expansion, using the averaging techniques described in Section II. Since H, and V do not commute, the evaluation of the nth power in Eq. (3.1) yields all possible orderings of H, and V. Each intermediate summation gives rise to an extra factor N. Since Ho is diagonal, it does not produce such factors. Only even powers of V contribute to M,. We consider a term in Eq. (3.1) with p factors Ho and 2q factors V so that p + 2q = n, and with a fixed and nested contraction pattern connecting the V’s 595/153/2-l*

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pairwise. For the leading-order terms in N-’ each such pattern contains (q + 1) independent summation indices. Let nk denote the number of different indices that occur 2k times in the pattern. (The first and last index are counted twice; this rule simplifies the counting since contributions like H, VV and VVH,, have to be counted separately.) For instance, the pattern I? V? VT has n, = 3, n4 = 1 and the I 1 pattern V V 37 P!’ V has n, = 1 and n, = 3, all other nk)s being zero. The coefficients nk obey the conditions q+l

F’

n,=q

+ 1,

kzl

(3.2)

F’ kn, = 2q + 1.

(3.3)

kzl

The first relation expresses the fact that there are (q + 1) independent summation indices. The second relation expresses the fact that there are 2q indices in all. For a given contraction pattern, the p factors H, can be arbitrarily distributed among the Vs. The position of each factor H, within the contraction pattern determines its index. Factors H, occurring in different places in the contraction I I pattern may carry the same index. This is the case, for instance, for V H, @H,, V. This diagram gives the same result as bHi P?’ I! This leads us to group the contributions to M, as follows. We consider a partition {p,} of p into r parts with p, > 0, 1= l,..., p and (3.4) The integer p, denotes the number of times the factor (Hh) appears. Given a partition (3.4), the contribution of the H, factors to M, is given by (H,)P~(Hi)PL 9.. (Hi)“p, and M, can be written as M, =

x p+29=n

(3.5)

r=o (piI

The dashes on the summation signs indicate that the conditions (3.2) to (3.4) must be met. The combinatorial factor Ayn,, is defined as the number of different contraction patterns of q pairs of V with fixed index distribution {Hi}, and the combinatorial factor C$.$r counts the number of ways of inserting Ho into the contraction pattern without violation of the conditions (3.4). We show in Appendix B that

A ;?I,,=

41 n,.In,!

..-n4+,!

’

(3.6)

EVALUATION

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and that C$y;),l has the value

%,@k* - &J(nkl

- h,k, - &,I

*‘. hr - 4,kr - b*k,-

..a - 4&r).

(3.7)

The (li} are the factors appearing in Eqs. (3.4). They are determined uniquely in terms of the numbers p, r, {pi}. The dashed summation over {nil can now be rewritten, yielding (see Appendix P)

cj

'

'

qt1

41

F‘ ’ A yn., c$;~., = PI!P2!

*'*Pp!

r

z r-1

k,...k,=l

1

n;! n;! . . . fl’Qil’ I .

i=l

(3.8)

Here, {nl} is defined by the conditions 01

1

9+

n;=q+1-r;

1

s

i=l

inf=2q+

l-

i-l

2 ki. i=l

The sum over (n, can be evaluated with the help of the combinatorial \‘, zj

k! k,!+k,!=

(3.9)

identity [6] (3.10)

where {ki} is defined by the conditions xi ki = k and Ci ik, = n. After that, the summation over k , ,..., k, can be executed with the help of the identity (3.11) The final expression becomes astonishingly

simple. We find

(3.12) For p = 0 we retrieve the expression for the GOE moments [ 11. For r = 1, our expression coincides with the moments obtained [ 71 for the case where H, has only a single nonvanishing diagonal matrix element. Finally, we observe that Eq. (3.12) was first proposed by Pandey [ 1 ] with the remark that an unproven counting theorem had been used. The technique used above can also be applied to the calculation of the binary correlation coefficients of the moments, i.e., to expressions of the form ((H)p)((H)Q) and even to individual matrix elements like (a /HP(b). We have carried through this calculation, but the resulting expressions are lengthy and not given here.

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111.2. A Recursion Relation for the Moments Instead of calculating the moments directly in the manner described in Section 111.1, we can use the Pastur equation (2.5) to obtain a recursion relation for the moments. For this purpose, we write Eq. (2.5) in the form (we omit the upper index zero in this section)

(G(z))= + c [z- E,- p(G(z))]I. a

We expand the r.h.s. of Eq. (3.13) in a geometric series and use the binomial to obtain

(3.13) theorem

(G(z))= &x f z-k-’ 4 ( ; j &:-‘p’(G(Z))‘. a

On the other hand, the definition Eq. (3.1) shows that

k=O

1%

of (G(z)) in Eq. (2.6) and of the moments M, in

(G(z))= f

Z-P-M,.

(3.15)

p=o

Inserting this expression into Eq. (3.14) and comparing coefficients of equal powers of z-’ on both sides, we find

M,=

QT 6k+,+i,+...+i,.p~i(N~-i)(

(3.16)

kli,?i,=O k>l

This is a recursion relation for A4, as it is easy to see that the indices i, ,..., i, appearing on the r.h.s. are always smaller than p. It would have been interesting to use Eq. (3.16) for another derivation of Eq. (3.12). Combinatorial difficulties prevented us from doing so. 111.3. The N- ’ Corrections to the Moments While a direct calculation of these corrections along the lines of Section 111.1 is very cumbersome and was not attempted, the recursion relation of Section III.2 can be extended to include such corrections. For this purpose, we rewrite Eq. (2.9) and take its trace to obtain (3.17) In order to obtain a recursion relation for the moments we express ((“‘G(z))‘) and ((“‘G(z))~) in terms of the derivatives of (‘O’@(z)). This is accomplished by taking the first and second derivatives of Eq. (3.13). We obtain

EVALUATION

((‘“‘C(z)‘)

OF ENSEMBLE

379

AVERAGES

= - [ 1 -,U $ ((OQz))J

((‘O’G(z))3) = + [ 1 -p -$ ((W(z))]

(3.18)

-I $ (‘O’G(z)), -3 $

(‘“‘G(z)).

Inserting Eqs. (3.18) into Eq. (3.17) we obtain (“‘G(Z))

= +i

[ 1- p f

(W+)]

Multiplying both sides of this equation by the denominator moment expansion (W(z))

= q

(3.19)

- ’ -$ (‘“‘G(z)).

(i)J,$ z-P- I;

i=o,

on the r.h.s., using the

1,

(3.20)

p=0

and equating coeffkients multiplying relation

‘L’M,+p(p-

equal powers of z yields the following recursion

l)‘“‘Mp~,-p

y

?kf, - (k + ly”‘Mk.

(3.21)

ktl+2=p k./>O

By expanding the r.h.s. of Eq. (3.19) in powers of the derivative of (‘“‘G(z)) and again using Eq. (3.20), we can also obtain an explicit formula for the moments “‘M,. It reads ‘l’~pE+~

x kii ,...ik>O

(-p)k(Z + l)(Z + 2)‘O’M[ fi

82k+l+i,+...ik+2.p

(is + l)‘“‘Mit*

s=l

(3.22) For small values of ,u, p - N-‘, the leading contribution to Eq. (3.22) comes from the terms with k = 0, and for p > N we see that (‘)M, is of the order p . (“)Mp-2r so that N-’ “)M becomes comparable with (‘)A4 . The N-’ expansion for the moments ceases to ie meaningful for p 2 N. This wasPalready observed in Ref. [ 11. In the case of the GOE the zeroth order moments are given by the Catalan numbers

(O,M_ rup 2P-2p+1

t

2p + 1 p ’

1

(3.22a)

and the recursion relation (3.21) can be solved explicitly. (3.22b)

380

VERBAARSCHOT,

For large value ofp Stirling’s ficients. One obtains (O)M,, + L N

WEIDENMijLLER,

AND

ZIRNBALJER

formula can be used to approximate 22p

(I’M

the binomial coef-

II

.

2pNpP I fip&

(3.22~)

The conclusion is that for GOE matrices the N-’ corrections and the leading order terms are of equal magnitude for p - N2’3. The technique of Sections II.2 and 11.3, combined with the results of Section 11.2, can also be used to obtain a recursion relation for the leading-order and next-order terms in the N-’ expansion of the binary correlation coefficients of the moments of H. Similarly, the results of Section II.3 yield a recursion relation for tertiary correlation coefficients. III.4

A Recursion

Relation for High-Order

Moments of a Hermitean

Matrix

Given the first N moments of a Hermitean N X N matrix, all higher moments can be calculated via a recursion relation. We now derive this relation. The moments A4, of the Hermitean matrix are homogeneous polynomials of order p in the N real eigenvalues E, ,..., E,. We consider the generating function g(t)=

;

log(l -Ejt).

(3.23)

i=l

Expanding g(t) in powers of t, we find k’(t) = - pt, ;

M, tP,

with (3.25) On the other hand, exponentiating

Eq. (3.23) we find for g(t) the equivalent formula

g(t)=log

i: 1 I=0

,

(3.26)

Ei,Eil ... Ei,.

(3.27)

(-l)[a,t’

1

where a, =

T d

i,
Equating the r.h.s.‘s of Eqs. (3.24) resulting equation, we obtain

and (3.26)

and taking

the derivative

of the

(3.28)

EVALUATION

OF ENSEMBLE AVERAGES

381

Multiplying Eq. (3.28) with the denominator of the r.h.s., and equating coefficients multiplying equal powers of t, we obtain for p < N

m-P

;

m = l...., N.

(3.29)

Given the first N moments M,,, p
p = 1, 2,....

PtN=-

(3.30)

k=l

This is a recursion formula for the moments M,,,.

IV. THE NATURE OF THEN-'

EXPANSION

In Section III we have seenthat the NP ’ expansion breaks down for the moments M, of H with p ,> N: the N-’ corrections are as large as, or bigger than, the terms of leading order. On the other hand, application of the N-’ expansion to the calculation of (G(x)) gives perfectly reasonableresults, as shown in Section II. 1 by specialization to the GOE case, and in Ref. [2] by explicit numerical calculation. Finally, evaluation of the two-point function with the help of the N-’ expansion yields results which aie singular (and therefore unreasonable) in the limit x=x0 + i . E, y = x0 - i , E with E-+ 0; similar statements apply also to the NP1 correction term, and to the three- and four-point functions. Can we understand these results, and the underlying properties of the N-’ expansion? We base our reasoning on the observation that for p 2 N, the N- ’ expansion breaks down for the moments M,. The reason is plainly combinatorial: The number of contraction patterns neglected to lowest order in N -’ increasesfaster than or at least as fast as N, once p 2 N. As we shall seepresently, nature and properties of the N- ’ expansion are different for (c(x)) and for the n-point function (n 2 2), and we consider them separately. For reasons of simplicity, we consider the GOE case with H, = 0 and comment only occasionally on the changes of the argument required to cover the general situation. The power-series expansion G(z) = C,Yj’=, z -m ‘(2 V)” converges absolutely for IzI > z,, where z0 is bigger than the largest eigenvalue E,,, of the matrix AK For the GOE and L fixed, the spectrum of LV lies within the interval -2 & 26. Therefore, the ensemble-averagecan be calculated by interchanging the summation and the averaging procedure. The moments (llv)” are then evaluated using the N-’ expansion. Although for m 2 N the N-’ corrections may be as big as the “leading” term, the procedure is expected to yield perfectly reasonable results

382

VERBAARSCHOT,

WEIDENMOLLER,

AND

ZIRNBAUER

because for z > z,,, the weight of these contributions, and the ensuing N- ’ correction to the analytic function “‘c ( z), are small for N4 1. The expansion for “‘G(z) is then summed to yield the identity (2.5). This identity is by way of analytic continuation used to calculate “‘G ( z) outside the radius of convergence of the original powerseries expansion in z-i. This procedure yields the result (2.10). [In the general case of Ho + 1V, the identity (2.5) cannot be solved algebraically. Instead, it is used [2] to generate a continued-fraction or Padt approximant for “‘G(z). Standard features [ 8 1 of these expansions make us expect that this procedure provides an excellent approximation scheme for Iz ( < z. but z outside the spectrum of H, as is indeed found by numerical calculations [2].] For the N-’ correction, the power series in z -’ is again resummed, and an analytic expression for ‘“‘G(z) like (2.9) or (3.17) is found. This expression is again used outside the domain of convergence of the original power series, and in this way a perfectly sensible correction as given in Eqs. (2.1 l), (2.12) is obtained. We see that for G(z) the N-r expansion is perfectly well-behaved, because it combines the N-’ expansion in a z-domain where it works with analytic continuation in z. The situation is quite different for the n-point functions with n > 2 as is demonstrated by Eq. (2.18), which has a non-integrable singularity for x --t y. The result (2.18) for the density correlation function was obtained by evaluation of [9] lj$ (Im(G(x

- is)) Im(G(y

- ie)) - Im(G(x

- i&)) Im(G(y

- is))).

(4.3)

Taking the limit E + 0 and letting x and y approach each other, we find the singularity -(x - y))’ as given by Eq. (2.18). In Ref. [ 1] it was argued that the singularity in the two-point function was caused by the large error made in the evaluation of moments of order p 2 N and a regularization or cutoff procedure was used to regularize results like Eq. (2.18). This result yielded a well-behaved expression to lowest order in N- ‘. It is not difficult to see that the regularization procedure of Ref. [l] can also be implemented in our calculational scheme. We thereby retrieve Eq. (4.23) of Ref. [l] to lowest order in N-l. To next order we find, however, that the procedure yields diverging results for x + y. This suggests that the regularization procedure of Ref. [ 1 ] cannot claim general validity, and therefore is not useful. This conclusion is supported also by the preceding discussion on the properties of G(z) where no regularization ever enters. Closer inspection of the expansion used in Section II.2 reveals the origin of the divergence. The expansion (2.16) generates a geometrical series in ,~(‘~‘G(x)‘~‘d(y)). For the GOE, taking x = x0 + ic, y =x0 - ic with x, in the spectrum of AL’, and letting e--t 0, we find that ,~(‘“‘~(x)‘“‘~(y)) -+ +l, so that the series diverges at this point. The same argument holds mutatis mutandis for other n-point functions (n > 2), see Section 111.3. As mentioned above, the N-’ expansion as defined in Section II.2 cannot be regularized uniformly to all orders in N-’ for these functions. The conclusion is that the N-’ expansion can only be used in a domain which excludes a vicinity of the point x =y (for n = 2). The size of this domain follows from the

EVALUATION

OF ENSEMBLE

383

AVERAGES

requirement that the correction terms in the N-’ expansion (which also diverge x =y, and do so even more strongly) should be negligible. As shown explicitly for N-’ corrections in Section II.2 and in general terms for the NPk corrections Appendix A, the expansion proceeds eventually in inverse powers of N”‘(x Writing for the difference (x - y) approximately N-i kd where k is the number levels in the interval ]x - y ( and d the local average level spacing, we find from condition N”* (x - y ( 2 C for convergence of the N- ’ expansion kXC&.

for the in y). of the

(4.4)

However, the leading contribution to the correlation function is of order N-*(x - y))‘, and comparison with the exact result of Dyson and Mehta [lo] shows that it correctly describes those correlation properties of the spectrum which encompass, say, k = 5 or more adjacent levels. (Finer details involving properties of kth nearest neighbours with k 5 5 are not accessible in this way.) This suggests that the N-’ expansion in the way implemented in Section II.2 is an asymptotic expansion as the domain of validity is largest for the lowest order of the expansion. In a further analysis of the nature of the singularity in Eq. (2.18), we evaluate the correlation function for a finite distance s/2 from the real axis. The divergent part is then given by &* - (x-y)’ (4.5) N-2 (&* + (x-y)‘)” We see from this equation that for any nonvanishing value of E, the correlation function passes through zero at ]x - y ] = E and becomes positive for (x - y ( < E. In the limit E -+ 0 the positive term contracts to a delta function of infinite strength 2

N-2(c* + (e;-y)‘)’

N1~6(x-Y).

+ -7-

(4.6)

Since the correlation function given in Eq. (2.18) is negative everywhere, this shortrange behaviour is in fact needed to ensure that

li ‘,m@(xlP(Y)

____

- P(X) P(Y)) dxdy = 0.

(4.7)

The exact two-point correlation function for the GOE has been given by Dyson and Mehta [lo]. Their result is negative and finite for all values of x =y with a positive S-function of order N- ’ at x = y. The a-function term arises from autocorrelations of the eigenvalues. Comparing now with expressions (2.18) and (4.6) we see that our result corresponds in form to the exact one. (The long-range part agrees, in fact, with the result of Dyson and Mehta.) However, the d-function behaviour for small separations of x and y is grossly enhanced and has infinite strength. We therefore suggest the following interpretation of the singularity. The two-point function as defined in Eq. (2.13) contains information about the autocorrelations of

384

VERBAARSCHOT,WEIDENMijLLER,AND

ZIRNBAUER

the eigenvalues and thus “knows” about the &term at x =y. The N- ’ expansion, which has leading order N-*, tries to reproduce this d-function of order N- ‘, but of course it fails and, instead, generates the horrid singularity in Eq. (4.5). We contend that the singularity does not arise from improper evaluation of the moments of high order, but from an incorrect treatment of the autocorrelation terms. The present analysis suggests the following improved method for calculating the correlation function. As a first step, it is crucial to separate the autocorrelation terms from the two-point function (G(x))(G(y)). Next, the moments of this “regularized” two-point function are to be calculated. This may be possible via an expansion in inverse powers of N, as before. Finally, the moment expansion of the two-point function has to be summed and analytically continued to the spectrum of H, in the same way as was done for the average Green function. If the moment expansion cannot be summed in analytic form, then a Padi approximant may be used instead. We expect the procedure just described to yield finite and meaningful results also for x -+ y, and work along these lines is in progress. Nofe added in proof. After completion of the manuscript, further investigations have shown that (i) the two-point function has a genuine singularity at x =y; (ii) the N-’ expansion constitutes an asymptotic expansion; (iii) the expansion parameter is given by [N(x -JZ)]-‘. All other terms in the expansion (i.e., those which are not multiples of [N(x -J)] mkwith k integer) have vanishing coefficients.

APPENDIX

A: HIGH-ORDER

N-'TERMS

We consider terms of fixed order in N-‘, say N-k, and maximal order in (x - y)-‘. As explained in Section IV, the singular terms in (x - y) are generated by the geometrical series C,,,un(G(x) G(y))“. We therefore look for contributions where each line-crossing (i.e., each factor N-‘), is accompaned by as many factors (c(x) G(v)) ,U as possible. We distinguish two types of contributions to terms of order Ndk : (i) Contributions in which cross-contraction lines intersect. (ii) Contributions in which contraction lines interset. We consider these in turn and then the general case, which is a mixture of possibilities (i) and (ii). Contributions of type (i) are schematically depicted in Fig. A.l. It is seen that each set of crossed lines adds a factor N-’ and a factor (x -11))~; we see that contributions of order N-k carry a factor (x - Y))~~‘*.

A.1

FIG. A.

A.2

Leading contributions in (x -u)-’

A.3

to terms of order N- Ir.

EVALUATION

OFENSEMBLE

AVERAGES

385

Contributions of type (ii) are schematically depicted in Fig. A.2. Along the lines leading to Eqs. (2.22) and (2.23), it is seen that each set of crossed lines adds a factor N-’ and a factor (x -,v)-‘, which leads to the same conclusion as in the previous paragraph. Finally, mixed contributions as schematically shown in Fig. A.3 have the same behaviour. This shows that terms of order Nek carry (x -Y)-*~+* as the largest negative power in (X - y).

APPENDIX

B: DERIVATION

OF EQS. (3.6), (3.7), (3.8)

We begin with Eq. (3.6), A:“,, =

4! ‘n,! .a* nq+,! ’ n,.

(B.1)

where the integers n, > 0 obey q+l \’

fl,=q+

1,

kyl

(B.2) i-

kn, = 2q + 1.

k=l

We recall that APn,, is the number of different contraction patterns of q pairs of V with fixed index frequency {ni}. To count the patterns, we use a one-to-one mapping of patterns and rooted trees. We give first a few examples.

inr-

++-

(B.4)

ll-nil-

t3

(B-5) 034

(B.7)

inmr

t-3

(B-8)

i, Each vertex in the tree (denoted by a dot) index, and vice versa. Each contraction vertices. The open contraction lines at the leg (the root) of the tree. (We recall that

corresponds to an independent summation line corresponds to a leg connecting two end of the patterns correspond to the open the end-indices are doubly counted.) Each

386

VERBAARSCHOT,

WEIDENMijLLER,

AND

ZIRNBAUER

tree has (q + 1) legs. A vertex has k legs if the associated index occurs 2k times. We note that the trees (B.7) and (B.8) denote different patterns. The sequences {n,} with i= l,..., q + 1 belonging to the patterns (B.4) to (B.8) are given by { 1, 11, {2, 1, 1, O}, (3,0,0, 1 }, (2, 1, 1, O} and (2, 1, 1, O}, respectively. We see that several trees may belong to the same index sequence {ni}, as shown by the trees (BS), (B.7), (B.8): The numbers nk only give the number of times a k-vertex occurs in a rooted tree, without specifying where. Condition (B.2) states that each rooted tree has (q + 1) vertices. Condition (B.3) expresses the rootedness of the tree: There exists one and only one open leg. (Each tree is connected.) To count the trees, we associate with each tree a unique sequence of numbers (a “vertex sequence”) in the following way. If the tree starts with a k-vertex, we write k(A ,),..., (Ak-,) where the Ai are sequences of numbers defined analogously. For the examples (B.4) to (B.8), the vertex sequences are given by 2(l); 2(3(1, 1)); 4(1, 1, 1); 3(2(l), 1); and 3(1,2(l)), respectively. The sequences so defined also uniquely characterise the tree from which they are derived. Moreover, each such vertex sequence yields immediately the numbers (nil by counting. In general, this procedure maps each tree uniquely onto a sequence of (q + 1) positive integers {i, ,..., i, + , ) with Cik= (2q+ I). Next we wish to establish the condition that a sequence (i, ,..., i,, , } with Ci, = (2q + 1) defines a tree. With this aim in mind, it is convenient to associate a further sequence with the vertex sequence {ii} just constructed. This is done by counting the number of legs following each vertex. For this purpose, the vertices are arranged from left to right corresponding to the way the dial of a clock would pass them when it is rotated clockwise. For the examples (B.4) to (B.8) the arrangement looks as follows. -+I

1

(B.4’)

(190)

Cl,&

130)

(392, 130)

(B.5’)

(B.6’)

(B.7’)

(B.8’)

EVALUATION

OF ENSEMBLE

AVERAGES

387

The slashes indicate where the legs are counted. Each leg sequence /? = (I, ,..., 1,+ ,) has the properties that I,, , = 0, that li > 1 for i < q, and that Zi+ i > li - 1. Clearly, each tree uniquely defines the associated leg-sequence. The converse is also true: Any leg sequence with the properties just mentioned uniquely defines the associated tree. The vertex sequence {ik} and the leg sequence {I,} are related by I’ = 1 + (i, - 2), I, = l,-,

+ (i, - 2),

k = 2,..., q + 1.

(B.9)

Each tree defines a vertex sequence (ik}, i.e., a partition of (2q + 1) into (q + 1) positive integers. The converse is not true: A given sequence {s, ,..., s,, , ) with si 2 1, Csi = 2q + 1, does not define a tree unless all the integers 1, calculated from the sk via Eqs. (B.9) are positive for k < q and zero for k = q + 1. The sequence (1, 1, 3, 2) is an example for which I, = 0, I, = -1. However, for every partition {si} of (2q + 1) into (q + 1) positive integers there exists a uniquely defined cyclic permutation P such that P{si) defines via Eqs. (B.9) a sequence of positive integers li, i < k, with I,+ i = 0. To show this, we consider (k > i) the partial sums Cik = Ckzi (s, - 2). We define the index j by the conditions (B. 10) We permute the sequence cyclically so that s,~attains the last position, s,~+, the first. It then follows from the conditions (B.lO) and the identity Ci, = Zi, t C,, (i ( m < k) that the first q partial sums of the permuted sequence are all positive or zero, while the (q t 1)th partial sum equals (-1). This in turn is the condition that the integers 1, calculated via Eqs. (B.9) are all positive except for I,, , = 0. We now group the partitions {s, ,..., sq+ i } of (2q + 1) into classes specified by the sequence {ni} such that n, elements sj have value 1, n2 elements sj have value 2 etc. The number of elements in each class is N = (q t l)! [ns+: (n,)!]-‘. Since partitions which differ by a cyclic permutation define the same tree, the number of different trees in each class is obtained by dividing N with (q t l), the number of cyclic permutations. This is Eq. (B.l). We turn to Eq. (3.7). Given q and the index distribution {n,), the coefficient C$$l,, is equal to the number of ways of distributing the r factors Hh [each factor occurring p, times, with Z;p, = r] over r different indices. All different possibilities of choosing the r indices are included in C. When a factor HI, is inserted into an index that occurs 2k times (i.e., there are k index slots), there are (“:-‘) = (‘i!;‘) different possibilities. For a fixed choice of r indices occurring 2k, ,..., 2k, times, the result is (‘ii,!;’ ). This must be summed over all possible choices of (ki} [PI! ...p,!]P’n~=l consistent with the given index distribution {ni}: pInil PePi

-

(B.11)

388

VERBAARSCHOT,

WEIDENMijLLER,

AND ZIRNBAUER

The summation extends over all subsets of a set which contains the number k exactly nk times. The summation can therefore be written as

(B. 12) This is Eq. The {ni} summation Eq. (B.12) denominator

(3.7). summation in Eq. (3.8) can be executed and transcribed into the (n;} with the reStrictions (3.9) if it is observed that the n,-dependent factors in cancel some of the leading factors in the factorials appearing in the of A, see Eq. (B.1). This cancellation gives rise to the conditions (3.9).

REFERENCES 1. T. A. BRODY, J. FLORES. J. B. FRENCH, P. A. MELLO. A. PANDEY. AND S. S. M. WONG. Reo. Mod. Phys. 53 (1981), 385. 2. M. ZIRNBAUER. J. VERBAARSCHOT,AND H. A. WEIDENM~~LLER,Nucl. Phys. A 411 (1983). 161. 3. D. AGASSI. H. A. WEIDENMULLER, AND G. MANTZOURANIS, Phys. Rer. C 22 (1975). 145. 4. D. AGASSI. C. M. Ko, AND H. A. WEIDENM~~LLER,Ann. Phys. (N.Y.) 117 (1979). 407. 5. L. A. PASTUR. Tear. Mat. Fiz. 10 (1972). 102. (Theor. Mafh. Phvs. 10 (1972). 67). 6. J. RIORDAN, “Combinatorial Identities,” Wiley, New York, 1968. 7. R. C. JONES,J. M. KOSTERLITZ, AND D. J. THOULESS, J. Phys. A 11 (1978), L45. 8. G. A. BAKER AND J. L. GAMMEL, “The Padi Approximant in Theoretical Physics,” Academic Press, New York, 1970. 9. A. PANDEY, Ann. Phys. (N.Y.) 134 (1981), 110. 10. M. L. MEHTA, “Random Matrices and the Statistical Theory of Energy Levels,” Academic Press, New York, 1967.