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Y. Okabe, M. Kikuchi

To cite this version:



Colloque C8, Suppl6ment au no 12, Tome 49, dkembre 1988




Y. Okabe (I) and M. Kikuchi ( 2 )

(I) Department of Physics, Tohoku University, Sendai 980, Japan (2) Department of Physics, Osaka University, Toyonaka 560, Japan

Abstract. - The quantum XXZ model of spin 112 on the square lattice is numerically investigated. We use both the conventional quantum Monte Carlo method and the projector Monte Carlo method. The ground-state energy and the existence of the long-range order are examined

1. Introduction calculate the energy, the specific heat, the order pa- Two-dimensional quantum spin systems have re-

ceived a considerable attention quite recently+. The relevance t o high


superconductors is a hot topic from both theoretical [I] and experimental 121 sides.

In this paper we study the spin 112 quantum XXZ

model on the square lattice. The Hamiltonian is given by

Here we take J


0, and A = 0, 1 and -1 respectively correspond to the XY, the ferromagnetic (F) Heisen-

berg and the antiferromagnetic (AF) Heisenberg mod- els with an appropriate unitary transformation. We treat the intermediate models as well.

2. Methods of simulation

We use two methods of quantum simulation based on the Suzuki-Trotter formula [3]:

The first method (Method 1) is the conventional quantum Monte Carlo method [4, 51 to study the finite-temperature properties, and the other method (Method 2) is the projector Monte Carlo method [6] t o study the ground-state properties. In both sim- ulations we employ the cell-type decomposition into sub-Hamiltonian [7].

We should be careful with the ergodicity in the con- ventional quantum Monte Carlo simulation. To up- date the spin configuration we employ not only the local process but also the global processes which break the particle number and the winding number conserva- tions [8]. The calculation is fully vectorized by dividing the lattice into appropriate sublattices [9]. The sizes of system (L x L) are 8 x 8 and 16 x 16. Treating the Trotter size rn up t o 24, we follow the l/m2 extrapo- lation procedure t o get the m -+ oo data. The typical number of Monte Carlo Steps per spin is 2 x



rameter, an4 so on.

In the method 2, the operator e-PH is used as a pro- jection operator to the ground state. Multiplications of matrices are performed by a Monte Carlo (random walk) method. The advantages of this method are that we can calculate the ground-state expectation values of various quantities directly, and that the off-diagonal operators, such as the in-plane magnetization, can eas- ily be treated. We deal with systems up to 10 x 10

sites. The convergence of the data is checked for sev- eral values of ,O and rn. The typical number of the random-walker is 1

x lo6.

3. Results

In figure 1 we show the temperature dependence of the energy per spin for A = 1, 0.5, 0, -0.5 and -1.

The system size is 16


16. We plot the data in units of J. We see the systematic variation of the curves with respect t o A.

The size dependence of the ground-state energy for the X Y model and the AF Heisenberg model is

Fig. 1. - Temperature dependence of energy per spin for the quantum X X Z model with A = 1, 0.5, 0, -0.5 and -1.



Fig. 2. - Size dependence of the ground-state energy per spin for the XY model (A = 0) and the AF Heisenberg (AFH) model (A = -1).

16 8 6 L

0.2 4

shown in figure 2. We plot both results obtained by the method 1 and by extrapolating the data of the method 2. We consider the L - ~ dependence following Barnes and Swanson [lo]. We give the exact data for 4


4 by Oitmaa and Betts [ll]. The AF Heisenberg data by Barnes and Swanson


are systematically lower than ours. The linear dependence in this figure is clear, and our estimates of the ground-state energy of the infinite system are



E / J = -0.5490


0.0005 (XY model)

= -0.670 f 0.001 (AF Heisenberg model). 0 Method 1

0 Method 2


Ref. 11

Next consider the size dependence of the order pa- rameter in the ground state for the X Y model and the AF Heisenberg model. Following Reger and Young [12], we plot (M:) versus 1/L. Here, M, is the in-

plane magnetization per spin, and it corresponds to the staggered magnetization for the AF model in the usual representation. Our result of the AF Heiseng- berg model is consistent with those by other authors [12, 131. For both models the linear dependence sug- gests the existence of long-range order in the ground state. The estimates of the long-range order are




0.090 f 0.006 (XY model)

= 0.029 f 0.008 (AF Heisenberg model). Fujuki and Betts [14], and also Nishimori and Nakan- ishi [15] discussed the disappearance of long-range or- der in the study of quantum spin systems on the trian- gular lattice, assuming a nonanalytic size dependence. Miyashita [13] also suggested a sirhilar behavior in the AF Heisenberg model on the square lattice. We cannot exclude such a possibility because the present sizes are still small, although the linear 1/L dependence seems more straightforward.

I 1 nrn


Fig. 3.


Size dependence of the order-parameter in the ground state for the XY model (A = 0) and the AF Heisen- berg (AFH) model (A = -1).


We would like to thank H. Shiba, M. Takahashi, K. Nomura and S. Fujiki for valuable discussions.

[I] Anderson, P. W., Science 235 (1987) 1196. [2] Shirane, G., Endoh, Y., Birgeneau, R. J., Kast-

ner, M. A., Hidaka, Y., Oda, M., Suzuki, M. and Murakami, T., Phys. Rev. Lett. 59 (1987) 1613. [3] Suzuki, M., Prog. Thwr. Phys. 56 (1976) 1454. [4] Suzuki, M., Miyashita, S. and Ihroda, A., Prog.

Theor. Phys. 58 (1977) 1377.

[5] Hirsch, J. E., Sugar, R. L., Scalapino, D. J. and Blankenbecler, R., Phys. Rev. B 26 (1982) 5033. [6] Blankenbecler, R. and Sugar, 11. L., Phys. Rev.

D 27 (1983) 1304.

[7] Loh, E., Jr., Scalapino, D. J. and Grant, P. M.,

Phys. Rev. B 31 (1985) 4712.

[8] Marcu, M., Quantum Monte Carlo Methods in Equilibrium and Non-Equilibrium Systems, Ed. M. Suzuki (Springer, Berlin) 1987, p. 64.

[9] Okabe, Y. and Kikuchi, M., Phys. Rev. B 34 (1986) 7896.

[lo] Barnes, T. and Swanson, E., F'hys. Rev. B 37 (1988) 9405.

[ l l ] Oitmaa, J. and Betts, D. D., Can. J. Phys. 56 (1976) 897.

[12] Fteger, J. D. and Young, A. P., Phys. Rev. B 37 (1988) 5978.

[13] Miyashita, S., J. Phys. Soc. J p r ~ 57 (1988) 1934. [14] Fujiki, S. and Betts, D. D.,

C G ~ .

J. Phys. 64

(1986) 876; 65 (1987) 489.