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RELAXATION FUNCTION OF 3D XY M O D E L t

M.H. LEE:~ and R. D E K E Y S E R Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, B-3030 Hevedee, Belgium

The Method of Mori has been extended in an effort to study the relaxation function. The difficulty posed by continued fractions has been in some special cases overcome to obtain exact solutions for the relaxation function. The physical significance of our solution is discussed with aid of the XY model.

Dynamical quantities of interacting many- body systems, such as scattering cross sections, conductivi ty, and others, can all be deduced f rom the relaxation funct ion ~,(t), which des- cribes how a given system relaxes in time t after a perturbat ion has been in some suitable way removed at t = 0. At present there is in spin dynamics, for example, considerable interest in calculating this quanti ty starting f rom model Hamiltonians. Of the various existing ap- proaches, the one by Mori [1] developed from the ideas of Brownian motion appears to be based on the soundest ground. But there are in this approach certain inherent difficulties which have hitherto prevented it f rom being fully ex- ploited. The approximations of ten employed to c i rcumvent these difficulties are of questionable nature. Here we shall describe some of our preliminary advances made in developing the approach of Mori in connect ion with our study of the dynamics of the X Y model.

Mori has shown that by writing

E( t ) = ~ - ~ dz eZ'~.(z), (1)

E(z) is expressible as a continued fract ion in terms of certain time independent correlat ion functions An

1 A1 A2 A3 ~(z) = - (2)

Z'q'- Z "[- Z q- Z -t-

These correlation functions are given by time derivatives as shown below: If H is the Hamil- tonian of the system and Co the order parameter of the system, e.g., co(K)= Y~ eiKe'si, one can

t This research has been supported in part by grants from NATO (No. 1024) and Research Corporation. Permanent address: Department of Physics, University of Georgia, Athens, Georgia 30602, USA.

define the n th time derivative of Co as: c. = [n, Cn_l] for n t> 1. The correlat ion functions are then scalar products of c . ' s [1, 2].

For nontrivial Hamiltonians, it is not a simple matter to obtain c~ even for n relatively small. Thus, one does not expect to have explicit forms of A. except for possibly the first few of them. It is, nevertheless, evident that all cor- relation functions A. exist as n ~ oo. This implies that the continued fract ion (2) does not in general terminate, i.e., it is an infinite series; and one unavoidably encounters the mathematics of continued fractions. A standard procedure, first considered by Mori on physical grounds, is to replace =.(z) by

1 A 1 A 2 ~ ' (z) = , (3)

z + z + Z + Z o

where z0 is a parameter . The above is a 3-pole approximation to a funct ion with an infinite number of poles. Although this approximation has on occasion given some interesting results, it is all the same an unsound approximation as is discussed later. Here we show that the con- tinued fract ion can be made tractable without truncation if certain assumptions may be made on the behavior of A. as n ~ oo. In some suitable limits these assumptions appear to be realizable in the X Y model and we find some interesting physics concerning the excitations, dynamical modes, and others.

The continued fract ion for ~(z) may be ex- pressed as [3]

-=(z) = lim det N._~(z)/det D.(z), (4) n - ~ o o

where Dn(z) is an n-dimensional tridiagonal matrix whose elements dq are: di~ = z and dq = - ajSij_+l, where a. = iA~./2; and N._l(Z) is the same matrix but without the first row and first

Physica 86-88B (1977) 1273-1274 © North-Holland

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column. With (1) we observe that .E(t) may be determined if the zeros of det D , ( z ) as n --* oo are first determined. We can rewrite D,(z) as (suppressing n)

D ( z ) = z 1 - 9 , (5)

where I is the unit matrix and 9 = - D ( z = 0).

The requirement , d e t D ( z ) = 0 , implies that there exist a set of orthogonal e igenvectors 2(, r ~ , satisfying the eigenvalue equations

9 X , = z~X,. (6)

The eigenvectors represent the dynamical modes of the system. The zeros of det D ( z ) are, thus, the eigenvalues of 9 and to obtain the eigenvalues Zr we need to diagonalize the infinite dimensional matrix 9 .

The infinite dimensional matrix 9 is di- agonalizable under suitable conditions on A and, hence, ~ ( t ) may be in these cases ex- plicitly determinable. Utilizing the propert ies of tridiagonal matr ices we obtain the following 3- term relation: For n t> 3

det D. = z det D,_I + An_ l det Dn-2. (7)

The above is in the fo rm of the general recur- rence relation of or thogonal polynomials [4]. Thus, depending on A,, we can obtain the eigenvalues z, and the eigenvectors X, directly f rom orthogonal polynomials with the n ~ limit appropr ia te ly taken af terwards.

In this way it is possible to determine ~ ( t ) for all {A,} which belong to some set of or thogonal polynomials (e.g. Hermi te , Gegenbauer) . Now, the physical informat ion is contained entirely in A,. Depending on these values of A,, we can obtain a whole new class of funct ions for ~( t ) , whose t ime-dependent behavior can be different f rom the usually assumed exponential form.

We shall discuss here two simplest examples which are realizable in the X Y model. (i) At high tempera tures , if the coordinat ion number of the sys tem is large, we find [2] that A, = A for all n, where A is a constant. Then (7) becomes

the recurrence relation for Tschebyshev poly- nomials of the second kind. Hence , the eigen- values are the zeros of the polynomials , i.e. z = 2 i a c o s 0 , where 0~<0~<~- and a 2=A. The eigenvalues form a cut on the imaginary axis of the complex-z plane f rom z = +2ia to z = -2 i a . The eigenvectors , which can be constructed f rom the Tschebyshev polynomials , are of plane wave form. The corresponding relaxation func- tion obtained by integrating along the cut is

=-(t) = 2J l (2a t ) /2a t , (8)

where Jl is the Bessel funct ion of order 1. The short and long time behaviors of ~ ( t ) are given by the asymptot ic propert ies of the Bessel function:

~ ( t ~ 0) = cos at (9)

and

~ ( t --~ ~ ) = (87ra3t3) - m COS (2at -- 37r/4). (10)

The short- t ime behavior agrees with a result earlier obtained via a mean field approximat ion valid for high frequencies [5]; and the long-time behavior is to our knowledge new. (ii) Also at high tempera tures , if the sys tem is one-dimen- sional, we find [2] that A1 = 2A, A, --- A for n >/2. Then (7) becomes the recurrence relation for Tschebyshev polynomials of the first kind and the corresponding relaxation funct ion is , ( t ) = Jo(2at). The short and long t ime behaviors fol- low, again, f rom the asymptot ic propert ies of the Bessel function. A fuller account of our result will appear in a future publication.

References

[1] H. Mori, Prog. Theor. Phys. (Kyoto) 33 (1965) 423, 34 (1965) 399.

[2] M.H. Lee, AlP Conf. Proc. 29 (1976) 472. [3] M.H. Lee, Letts. Appl. Eng. Scis. 4 (1976) 63. [4] G. Szeg6, Orthogonal Polynomials (Am. Math. Soc.,

Providence, R.I., 1937) p. 42. [5] M.H. Lee, Phys. Rev. B8 (1973) 3290.