THERMODYNAMIC PROPERTIES OF ANTIFERROMAGNETIC HEISENBERG MODEL (S = 1/2) ON THE SQUARE LATTICE

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THERMODYNAMIC PROPERTIES OF

ANTIFERROMAGNETIC HEISENBERG MODEL (S

= 1/2) ON THE SQUARE LATTICE

Seiji Miyashita

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C8, Supplkment au no 12, Tome 49, dCembre 1988

THERMODYNAMIC PROPERTIES OF ANTIFERROMAGNETIC HEISENBERG

MODEL

( S

=1/2)

ON THE SQUARE LATTICE

Seiji Miyashita

Departement of Physics, College of Liberal Arts and Sciences, Kyoto University, Sakyo-ku, Kyoto 606, Japan

Abstract.

-

Ordering process of the antiferromagnetic Heisenberg model on the square is investigated mainly by a

quantum Monte Car10 method. The effect of quantum fluctuation is studied by investigating the existence of long range order, the response function and the spin configurations. The magnetic susceptibility is also reported.

The effect of quantum fluctuation, in other words, the effect of non-commutativity of the order parameter and Hamiltonian, is one of the most inter- esting topic of quantum statistical mechanics. In par- ticular, this effect can be directly observed in magnetic systems as the spin reduction. In one dimension, var- ious exact or semi-exact result has been known and it turns out that the quantum fluctuation destroys the long range order (LRO). On the other hand in three dimensions, many models have been proved t o have LRO at low temperatures [I] and we believe the existence of LRO, where the spin wave theory gives a good estimation [2]. Between them in the two-dimensions,

we expect large quantum fluctuation and also relevant cooperative phenomena. Thus, the situation is difficult [3-61. Importance of quantum fluctuation is also enhanced for frustrated systems and the absence of LRO is discwsed as the picture of "spin liquid state" [7]. It has been thought t o be rather difficult t o study such marginal nature by approximation schemes.

Recently, more direct investigations have been in- troduced with the developments of computational fa- cilities. One of them is "finite lattice method" [5], where the energy spectrum of the Hamiltonian of finite lattice is studied as an eigenvalue problem of large matrix. Another one is "quantum Monte Carlo method (QMC)" [8-111 which can treat rather large systems at finite temperatures. In this paper we re- port the thermodynamic properties of antiferromagnetic Heisenberg model on the square lattice investigated by this QMC making use of the Suzuki-Trotter decomposition [8, 91. The Hamiltonian of this model is given by

H = J / 2 C a i a j .

{Gt

Fig. 1. - Temperature- and size-dependence of

(s2)

L - ~ .

Symboles (e,

=,

A, V and o) denote the data of L = 4, 6, 8, 10 and 16, respectively.

is the linear dimension of the system) is given. In this figure, we see a large enhancement of ordering fluctuation at low temperatures, which is a big difference from the frustrated cases [8]. The next interest is the existence of LRO in the ground state. From figure 1, the data for T = 0 is estimated as 5.8

f

0.1, 10.0 f 0.5, 1 5 . 5 f 1.0 and 21.5 f 2.0 for L = 4, 6,8 and 10, respectively. For L = 4, the exact value has been known by the finite lattice method as 5.90, [4] which guarantees the validity of the algorithm. The values of T = 0 are plotted in figure 2. Usually LRO, s, is estimated

The order parameter of this model is the staggered where the second term gives the intrinsic fluctuation. magnetization: Thus we expect the y-intercept gives s2 as far as the fluctuation is finite. The solid line gives an extrapola-

s

=

xo:

- C a t , (2) _{tion with the first three data, that is, data of systems}

i € A i € B _{with 8, 10 and 16 spins}_[4]._{This is called "Oitmaa and}

where A and B are sublattices of the square lattice. Betts (OB) line"

.

Here we see systematic deviation In figure 1, temperature dependence of

( s 2 ) /

L~ (L from the OB line. Thus we expect the value of s to

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C8 - 1392 JOURNAL DE PHYSIQUE

Fig. 2.

-

Size-dependence of L - ~ in the ground state. Solid, dashed, dot-dashed dotted lines denote

OB, equations (4), (5) and (6), respectively.

be smaller than that of OB. If we fit the data by a straight line,

s2 is estimated to be 0.1838 which is about 3/4 of OB's. On the other hand we can also fit the data in the form

which means the absence of LRO and algebraic decay of the spin correlation function. The data are well fitted by q =: 0.48, A = 0.63 and B = 1.4.

The spin wave theory predicts the fluctuation

( ( s 2 )

-

( s ) ~ )

/ L2 diverges proportionally to

L [12], and Reger and Young [ l l ] pointed out that the data are well fitted by

This gives s2 = 0.12, which agrees very well with the result of spin wave theory [2, 121.

The response function XSG is also estimated by the Monte Carlo method and it is shown that XSG is smaller than

(s2)

/ ~ B T and XSG.~BT becomes zero as T goes t o zero for finite lattices.

Fig. 3. - The magnetic susceptibility. Symbols denote the same as in figure 1.

The magnetic susceptibility

x

was also obtained. It

was found th'at

x

is finite as T goes lto zero and it has a broad peak around T

/

J = 2, see ;figure 3.

Finally we show two snap-shots of equilibrium spin configurations of L = 10 at T

/

J == 0.3 in figure 4. In figure 4a we see the staggered order but the order is disturbed in figure 4b. Microscopic investigation of quantum fluctuation is also interesti~ig problem in the future.

Fig. 4. - Spin configurations.

[I] Dyson, F., Lieb, E. H. and Spencer, T., J. Stat.

Phys. 50 (1976) 79.

[2] Anderson, P. W., Phys. Rev. 86 (1952) 692; Kubo, R., ibid. 87 (1952) 568.

[3] Neves, E. J. and Salvador, R., P'hys. Lett. A 114 (1986) 331.

[4] Oitmaa, J. and Betts, D. D., Chn. J. Phys. 56 (1978) 897.

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

Nishimori, H. and Nakanishi, H., J. Phys. Soc. Jpn 57 (1988).

[6] Iske, P. L. and Caspers, W. J., Physica A 142 (1987) 360 and ibid. A 146 (1987) 151.

[7] Anderson, P. W., Mater. Res. Btsll. 8 (1973) 153; Fazekas, P. and Anderson, P. W., Philos. Mag.

30 (1974) 423.

[8] Quantum Monte Carlo method in Equilibrium and Nonequilibrium systems, Ed. M. Suzuki (Springer-Verlag) 1987.

[9] Miyashita, S., J. Phys. Soc. Jpn 57 (1988) 1934. [lo] Handscomb, D. C., Proc. Cambridge Philos. Soc.

58 (1962) 594;

Manousakis, E. and Salvador, R., Phys. Rev. Lett. 60 (1988) 840;

Lee, D. H., Joannopoulas, D. and Negel, J. W., Phys. Rev. B 30 (1984) 1599.

[ll] Reger, J. D. and Young, A. P., IJhgs. Rev. B 37 (1988) 5978.