MONTE CARLO STUDY OF LOW-TEMPERATURE PROPERTIES OF QUANTUM FERROMAGNETIC XY MODEL ON THE TRIANGULAR LATTICE

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MONTE CARLO STUDY OF LOW-TEMPERATURE

PROPERTIES OF QUANTUM FERROMAGNETIC

XY MODEL ON THE TRIANGULAR LATTICE

Masako Takasu, Seiji Miyashita, Masuo Suzuki, Yasumasa Kanada

To cite this version:

JOURNAL DE PHYSIQUE

Colloque C8, Supplbment au no 12, Tome 49, dkembre 1988

MONTE CARL0 STUDY OF LOW-TEMPERATURE PROPERTIES OF QUANTUM

FERROMAGNETIC XY MODEL ON THE TRIANGULAR LATTICE

Masako Takasu, Seiji Miyashita

',

Masuo Suzuki ', Yasumasa Kanada

Department of Physics, Faculty of Science, Kanazawa University, 1-1 Marunouchi, Kanazawa Ishihawa 920 Japan

Abstract.

-

The S = 112 ferromagnetic XY model on the triangular lattice is investigated by quantum Monte Carlo methods based on the Suzuki-Trotter formula. The squared magnetization (M:)

/

N~ is calculated for finite temperatures and extrapolated t o T = 0. The existence of the long-range order is discussed. Kubo's susceptibility xxx is also calculated and compared with the squared magnetization.

In this paper, we study the following quantum ferromagnetic XY model;

where the sum is taken over the nearest neighbor spins on the triangular lattice. The classical version of this model has been studied by several authors [I, 21 and Kosterlitz and Thouless [l] found that the model un- dergoes a transition characterized by vortex-antivorex pairs. The nature of the phase transition of the quan- tum model on the square lattice has been studied by several groups [3-61 over the past years. The majority of the conclusions is that the phase transition of the quantum model is of Kosterlitz-Thouless (KT) type similarly t o the classical case.

Here, we study the low-temperature properties in order to know when and how the quantum model de- viates from the corresponding classical system. The method employed here is similar to what we used for Heisenberg models in the paper of Takasu et al. [7].

The squared magnetization (M:) shows a large n-

dependence of the data is show; in figure 1. At a fixed value of n, the temperaturedependence of the squared magnetization has a peak and this peak moves to lower temperature as the Trotter number n is increased. Therefore, when applying the 1

/

n2-expansion [8] at a certain temperature, we should be careful because the mdependence i s not monotonic. For the extrapo- lation, we have to use the data of large enough n where the 1

/

n2 dependence is monotonic.

The estimations of (M:) for n -+ co are given by the symbol (A) in figure 1 for N = 16.

In figure 1, we also plotted the susceptibility multi- plied by the temperature

xkT

/

N. Here the suscepti- bility

x

is calculated b y Kubo's formula. The quanti-

Fig. 1. - The temperature dependence of M:

/

N and xkT

/

N. The lattice size is 16.

ties (M:) and ~ l c T

/

N agree at the high temperature limit, but

xkT

/

N becomes small as the temperature is lowered, because the noncommutability plays an im- portant role. The data of (M:) also saturate near this temperature. Below this temperature, the spins are aligned in the xy-plane, but they tend to be uncor- related in the Trotter direction, which means that the off-diagonal matrix elements start t o give dominant ef-

l ~ e p a r t m e n t of Physics, College of Liberal Arts and Sciences, Kyoto University, Sakyo 606 Japan. '~epartment of Physics, Faculty of Science, University of Tokyo, Bunkyo, 113 Tokyo, Japan. 'computer Centre, University of Tokyo, Bunkyo, 113 Tokyo, Japan.

C8

-

1390 JOURNAL DE PHYSIQUE

fects. This behavior is characteristic of the quantum spin systems a t low temperatures. Similar behavior is found in the antiferromagnetic Heisenberg model on the square lattice 1121.

The ground state of the present model has been studied by many people [9]. First, Oitmaa and Betts

[91 studied by a finite lattice method for square and honeycomb lattices of spins N

<

18. Through extrap- olaton, they concluded that there remains a finite long- range order at thermodynamic limit;

( M : ) / N 2 - c + d / ~ (1)

where c = 0.116 for the square lattice and 0.104 for the honeycomb latice. Later Fujiki and Betts [lo] studied large lattices up to N = 21 and claimed that a new extrapolation formula is more plausible than that of Oitmaa and Betts. Namely,

Fig. 2.

-

The temperature dependence of M:

/

N for lat- tice size N = 16, 36, 64, 100 (symbols ( 0 , =, A, e) respec- tively.)

Recently, Nishimori and Nakanishi [ l l ] showed that the latter formula is more plusible 1~y studying larger systems N = 25,27, and estimated the value of q as

7) " 0.12.

For (M:)

,

these data are plotted in figure 2. The estimated values of (M:) at T = 0 are shown by the arrows on the vertical axis, for eqmations (1) and (2)

respectively. At these lattice sizes, we cannot conclude yet which formula is more plausible. Rather, the data for N = 100 are located in the middle of the points predicted by (1) and (2), which suggests that both extrapolations are not very good for large N. For an asymptotic form, there are many other possible fits

[12, 131. Investigations of larger lattices will be neces- sary for this problem.

The numerical calculations were ]performed on FA- COM M260D of the computer centre of University of Tokyo, and on FACOM M380 of Pllasma Institute of Nagoya University. Part of this work was carried out under the ISM Cooperative Research Program.

[I] Kosterlitz, J. M., Thouless, D. J., J. Phys. C 6

(1973) 1181.

[2] Miyashita, S., Nishimori, H., Kiiroda, A., Suzuki, M., Prog. Theor. Phys. 60 (1978) 1669;

Tobochnik, J., Chester, G. V., Phys. Rev. B 20 (1979) 3761.

[3] Suzuki, M., Miyashita, S., Kuroba, A., Pmg.

Theor. Phys. 58 (1977) 1377.

[4] De Read, H. et al., 2. Phys. B 57 (1984) 209. [5] Loh, E., Scalapino, D., J., Grant, P. M., Phys.

Rev. B 31 (1985) 4712.

[6] Onogi, T. et al., Springer Ser. Solid State Phys.

74 (1987) 75;

Marucu, M., ibid. 64.

[7] Takasu, M., Miyashita, S., Suzuki, M., Prog.

Theor. Phys. 75 (1986) 1254; $pringer Ser. Solid State Phys. 74 (1987) 114.

[8] Suzuki, M., Phys. Lett. A 113 (1985) 299. [9] Oitmaa, J., Betts, D. D., Can. J'. Phys. 56 (1978)

897;

Suzuki, M., Miyashita, S., Can. J. Phys. 56 (1978) 902.

[lo] Fujiki, S., Betts, D. D., Can. J. Phys. 64 (1986) 876.

[ l l ] Nishimori, H., Nakanishi, H., J Phys. Soc. Jpn 57 (1988) 626.

[12] Miyashita, S., J. Phys. Soc. Jprt 57 (1988) 6. [13] Reger, J. D., Young, A. P., Phys. Rev. B 37