PHYSICAL REVIEW B VOLUME 19, NUMBER 11 1 JUNE 1979

High-temperature longitudinal susceptibility for the S = —, XY model:

Some numerical evidences

I. M. Kim and M. H. Lee'Department of Physics, University of Georgia, Athens, Georgia 30602

(Received 6 November 1978)

The longitudinal component of the XY-model susceptibility is nondivergent at 7 = T,. In an

earlier paper it was shown that as T T,+ the longitudinal susceptibility is determined essen-

tially by nearest-neighbor correlations and it behaves. as the derivative of the free energy with

respect to inverse termperature. We provide here numerical evidences in support of the earlier

conclusions.

The longitudinal component of the susceptibilityX"

for the 5 = 2. XY model, defined by the Hamiltonian

(ii) bcc lattice

1 —2K +K —6.105 SSK6

14 43948Ks+. . .

2K'+ K4 . 2 558 33K' 13 523 80Ks+. . .

(iii) sc lattice

X-=1-15K'+»5K'

is (unlike the transverse component) nondivergent atT = T,. For T & T„ it is convenient to write thelongitudinal susceptibility as a sum of nearest-neighbor correlations X and all other distant-neighbor correlations 5, i.e.,

X = I+X"+ 5~ .

As T ~, evidently X —1 X". In a recent paper, '

one of us showed using a somewhat involved asymp-totic argument that as T T,+, one still hasX —I X*'. In addition, X behaves as itF/BK,where F is the free energy and K = J/kT In this.brief note, we provide numerical evidences for theseconclusions.

Series expansions for X" may be obtained by amethod previously, given2 from which, by somemodifications, series expansions for X may also beobtained. 3 The nearest-neighbor correlation functionX" is an important quantity in spin dynamics. 4 %elist here our cubic lattice results through eight terms:

(i) fcc latticeI

1 3K2 4K3+0 5K4 OKs 32 408 333K

—124.416667K' —354,876 5K'+

X'=—3K2 —4K3+ 0.5K" +2.8Ks —21.690 27K6

97 38}78K' 254 98912Ks+. . .

3 337 5K' 10 697 4Ks+

X~=—1.5K +1.75K —2.568 75K

—7 63534K +-In Fig. 1 we show the behavior of X" and 1+X"

for the cubic lattices. The critical points K, =0.222,0.344, and 0.495, respectively, for the fcc, bcc, and scare from Betts et al. 6 As may be observed here, thetwo quantities are similar for the temperature range

T, & T & ~. Also observe that each of these quanti-

ties approaches the critical point with a finite slopeindicating nondivergent behavior at K = K, . In Fig. 2

we compare the susceptibility with BF/BK, with thefree energy obtained by Betts et a/. Although their

amplitudes are different, the two functions appear tobehave rather similarly as predicted.

W'e briefly discuss the general nature of our results

given here. The energy and susceptibility may beeach considered as a sum of certain short- and long-

range correlations. For nn models such as Eq. (2),the correlations which give rise to the energy are lim-

ited to short-range types, whereas the correlations forthe susceptibility are ordinarily dominated by long-

range types. In terms of high-temperature series ex-pansions one sees the difference readily. The expan-sion coefficients for the energy series are determined

by closed graphs, but those for the susceptibility

largely by open graphs. The expansion coefficientsfor X" for the XY model, however, are mhde up by

5815

I. M. KIM AND M. H. LEE 19

IO—

io'

IO

IO

I

O. I'

I

0.2 0.5I

OQ 0.5

3 l ~ X I t I I I t I I

O.i 0.2 ' 'O.I, 0.2 0.3 ' '

O.I 0.2 0.3 Q.4 Q,5K

FIG. 2. Susceptibility and the derivative of the free en-ergy shown as a function of K =J/kT for cubic lattices.Solid lines represent 1+X~ and dashed lines 8F/8K.A Ithough their amplitudes are different, both quantities ap-pear to behave similarly.

FIG. 1. Longitudinal susceptibility on cubic latticesshown as a function of K = J/kT. Dashed lines represent X~

and solid lines 1+X~. The critical points are 0.220, 0.344,and 0.495, respectively, for fcc, bcc, and sc lattices (seeRef. 6).

closed graphs, in fact, by precisely the same graphs asthose occurring in the expansion coefficients for theenergy series.

To show this further, it is convenient to define anoperator

r t 2

~ =Xs,*

The high-temperature expansion coefficents for X

are of the form TrA H", n =1,2, . . . , where Hisgiven by Eq. (2). Now one can show that for a givenll,

TrA H" = a„TrH",

where a„ is a c number. ' Hence the longitudinalcomponent of the susceptibility for the XYmodelbehaves as the energy. For the transverse com-ponents A and A~~ there are no such relationships.

ACKNOWLEDGMENT

This work is supported in part by DOE ERDAunder Contract No. EQ-77-S-09-1023.

To whom reprint requests may be addressed.~M. H. Lee, Phys. .Rev. 8 12, 276 (1975).2M. H. Lee, J. Math. Phys. 12, 61 (1971).3Essential modifications needed are given by certain identi-

ties derived in Appendix A of Ref. 1,For example, this quantity appears as a leading term in f-

sum rule. See M. H. Lee, Phys. Rev. 8 8, 3290 (1973).5A number of people have worked on. the longitudinal

component of the susceptibility: T. Obokata, I. Ono, andT. Oguchi, J. Phys. Soc. Jpn. 23, 516 {1967); K. Pirnie(unpublished); M. H. Lee, see Ref. 2; D. W. Wood andN. W. iDalton, J. Phys. C 5, 1675 (1972); J. Rogiers andR. Dekeyser, Phys. Lett. A 46, 206 (1973).

6D. D. Betts, C. J. Elliott, and M. H. Lee, Can. J. Phys. 48,1566 (1970).