Physica 108B (1981) 1067-1068 RA 2 North-Holland Publishing Company

ZERO TEMPERATURE SERIES EXPANSION FOR AN EASY-PLANE SPIN-0NE FERROMAGNET

Felix Lee and Yung-Li Wang

Department of Physics Florida State University

Tallahassee, Florida

The first six coefficients in the susceptibility series have been obtained for an easy-plane spin-one ferromagnet at zero temperature. The results are for a general lattice and for arbitrary range of interaction. We, however, present only the series for the fcc and the triangular lattices. Estimates of the critical points and the critical exponents are given.

I. INr RODUCT ION

Quantum critical phenomenon has been a topic of great interest in recent years. In some sys- tems, the critical temperature can be lowered to zero as an appropriate parameter is changed. While at finite temperature (T) it is generally believed that the quantum nature of the system plays no essential role in the critical behav- ior, the quantum fluc~ations becomes increas- ingly important as temperature is reduced to the vicinity of zero. At T=0, the critical behavior can be totally different from that at finite T and there is normally a crossover from the classical to the quantum critical behavior. I

Many models have been proposed for study. The Ising Model in a transverse field is perhaps the most extensively treated system. 2 In this paper we study by a series expansion method another magnetic system which displays a phase transi- tion at T=0. We consider an easy-plane spin- one ferromagnet. In contrast to the Ising model in a transverse field, the symmetry of our system is XY-like., The single-ion aniso- tropy which gives rise to the easy-plane plays a similiar (but different) role as the trans- verse field in the Isin~ system. Because of much greater complexity 0f the present system due to the presence of a single-ion anisotropy, a new series expansion technique has to be adopted. In the next section we outline a gen- eral series expansion method at T=0 for systems with complicated energy levels. We have com- puted the first six coefficients of the suscep- tibility series and obtained estimates of the critical point and the critical exponent. The results and discussions will be given in the third section.

II. SERIES EXPANSION METHOD FOR COMPLICATED LEVEL SYSTEMS

Series expansion methods for simple model sys- tems such as Ising and Heisenberg models have been proved powerful tools in finding the cri- tical temperatures and the critical exponents. Complications arise in the presence of (large) single-ion anisotropy. The major difficulty

lies in the evaluation of the canonical average of the spin-operator products. Generally, the time dependence of spin operators becomes very involved if a single-ion anisotropy term is included in the unperturbed Hamiltonian, and the standard-basis operators must be introduced to aid the calculations. Such series expansion technique has been recently developed by Wang and co-workers. 3 Following the same line of approach we have formulated a series expansion specifically at zero temperature. The simpli- fication in the zero temperature calculation has enabled us to obtain more terms in a series for an actually more complicated system than those studied.

We consider the Hamiltonian for an easy-plane spin-one ferromagnet.

H = DZ(siZ) 2 - Z. JijSi'Sj (i) i,3

where D>0 measures the easy-plane anisotropy, and the exchange interaction has been assumed isotropic. The mean-field theory predicts no magnetic ordering for D>4J (0) [J (0)=ZJij ] even

at T=0. It also shows that near the critical the susceptibility x~[D-Dc]-Y with y=l. point

At T=0, D = 4J(0). c

To incorporate the quantum fluctuations we treat the exchange interaction in a perturba- tion expansion and calculate the suscepti- bility X in two steps. Firstly X0 is com- puted by the formula:

f0[o f = 2i~ ~ dt d t l . . dt~ ×0 -,iN'n, ~ £1

<¢01T[S~(0)S~(t)Hl(tl)...Hl(tZ)]li0>c.(2)

Then X is related to X0 by

X = X0[1-2 J(0)X0 ]-1 (3)

Here ~0 = ~ 0 denotes the ground s t a t e of the

unperturbed Hamiltonian which consists of the anisotropy energy term only in Eq.(1). In

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fact, letting In> denote the eigenstate of S z with eigenvalue n, I~0> = I0> and the other two energy eigenstates can be chosen as [Ii>±I-i>]//2 which will be denoted by ~i and ~2" The opera-

tors in Eq.(2) are in the interaction represen- tation. The symbol T denotes the Dyson Time ordering and the subscript c instructs retaining only terms proportional to the number of sites. As mentioned above, the time-dependence of S~(t) is complicated. To facilitate the calculation, we introduce the standard-basis operators:

Lmn ~I,m><,nl.

We note that each such operator has a simple time-dependence.

Lmn(t) = Lmn(O) ei(em-en)t (4)

where Sm is the eigen energy of l~m>. The spin-

operators can be written as linear combinations of the standard basis operators. The ground state expectation values of the spin-operator products can then be evaluated as a sum of the ground state expectation values of the standard- basis operator products, each of which has a simple time dependent factor feasible for per- forming the multiple time integration. This is a crucial step in the series expansion calcula- tion.

III. RESULTS AND DISCUSSIONS

We have obtained the first six coefficients in the series expansion of ×0 defined in Eq.(2)

from which the same number of coeffients of the susceptibility × can be found. A computer has been used to handle the large number of terms. In the fifth order calculation a total of 58 diagrams have been evaluated. In the evaluation of the diagrams two to seven point cumulants of the standard-basis operators are needed. The number of cumulants used have exceeded four hundreds. Results have been obtained for a general lattice and for arbitrary range of exchange interactions. In table I we only show results for the fcc and the triangular lattices assuming nearest-neighbor-only interaction.

2 n X = ~ Z An[J(O)/D]

n 0 i 2 3 4 5

59 3821 326383 I039356i fcc i 4

4 72 1728 15552

27 769 316i 1168283 tr 1 4

2 18 24 2916

TABLE I Susceptibility Series for fcc and tr lattices

Employing the standard ratio test we estimate D a n d y b y :

c Dc(n)/j(O) = [nUn-(n-2)Un_2]/2 (5)

y(n)= l+[Un_2_Un]/[Un/(n_2)_Un_2/n ] (6)

where Un = An/An-l"

These values are presented in Table II.

Dc(n)/j (0) ¥ (n)

n fcc tr n fcc tr

3 3.397 2.747 3 1.178 1.456 4 3.431 2.791 4 1.150 1.419 5 3.449 2.858 5 1.129 1.322

TABLE II Estimates of D and y for fcc and tr c

lattices.

It is clear that the quantum fluctuations have substantially reduced the critical value of D from the mean field value 4J(0) to 3.45J(0) for fcc and 2.86J(0) for tr lattice. However, it is difficult to conclude at this point about the value of y because of the shortness of the series. On the other hand, according to a recent theory of Hertz, 1 we expect that the critical exponents for our system at T=O are the same as the corresponding ones for lattices of one higher dimensionality at finite T. Since our system has the XY symmetry, the value of y should approach i and 1.34 respectively for the fcc and tr lattices. While our results are consistent with this assertion, higher order terms in the series would be needed to reach a definitive conclusion.

This work is supported by NSF Grant DMR 7925369. One of us (FL) is on leave of absence from the National Tsinghua University, ROC, and is sup- ported in part by NSC of ROC.

REFERENCES:

[i] Hertz, John A., Phys. Rev. B 14, 1165(1976)

[2] Pfeuty, P., J. Phys. C 9, 3993 (1976). Elliott, R.J., and Wood, C., J. Phys. C 4, 2359 (1971). Yanase, A., Takeshige, Y., and Suzuki, M., J. Phys. Soc. Jpn. 41, 1108 (1976). Oitmaa, J. and Plishke, M., J. Phys. C 9, 2093 (1976).

[3] Wang, Y.L. and Lee, F., Phys. Rev. Lett. 38, 912 (1977). Johnson, J.W. and Wang, Y.L., J. Appl. Phys. 50, 7388 (1979).