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### Submitted on 1 Jan 1988

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**MONTE CARLO STUDY OF QUANTUM SPIN**

**SYSTEMS ON THE SQUARE LATTICE**

### Y. Okabe, M. Kikuchi

**To cite this version:**

JOURNAL DE PHYSIQUE

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

**MONTE CARL0 STUDY OF QUANTUM SPIN SYSTEMS ON **

**THE SQUARE **

**LATTICE **

**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

**T, **

superconductors is a hot topic
from both theoretical [I] and experimental **T,**

**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**

### lo6.

Werameter, 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

### x

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.

JOURNAL DE PHYSIQUE

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

### x

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

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**

**h **

N

* E / J *= -0.5490

### f

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

### d

*(M:) *

### =

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 **

## I

Fig. 3.

### -

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

**XY****(AFH)**model (A = -1).

**Acknowledgments **

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.