STATIC AND DYNAMIC PROPERTIES OF ISING SPIN ON A TRIANGULAR LATTICE WITH COMPETING INTERACTIONS

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STATIC AND DYNAMIC PROPERTIES OF ISING

SPIN ON A TRIANGULAR LATTICE WITH

COMPETING INTERACTIONS

Tatsuya Uezu, Kazuko Kawasaki

To cite this version:

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JOURNAL DE PHYSIQUE

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

STATIC AND DYNAMIC PROPERTIES OF ISING SPIN ON

A

TRIANGULAR

LATTICE WITH COMPETING INTERACTIONS

Tatsuya Uezu and Kazuko Kawasaki

Department of Physics, Nara Wornens's University,. Nara 630, Japan

Abstract.

-

Ising spin on a triangular lattice with antiferromagnetic nearest neighbor and ferromagnetic second nearest neighbor interactions is investigated by treating the master equation. Classification of solutions and explanation of phase transition in terms of bifurcation theory are presented. Relaxation processes are investigated numerically. Finally, a fluctuation-dissipation relation is obtained.

We have interest in statics and dynamics of systems s j are nearest or second nearest neighbor spins, we ig- which have many (meta stable) states [I]. Spin sys- nore the cuinulant. We call this the cluster decoupling tems on a triangular lattice are such systems and these approximation (CDA). Further, we assume that (si) have been studied by many researchers [2]. In this pa- and (sisj) depend only on which sublattices si and s j per, we treat Ising spins on a triangular lattice with belong t o and denote them by (s6) and (s6s7) when nearest neighbor antiferromagnetic and second near- si and s j are on the 6 and 7 sublattices, respectively est neighbor ferromagnetic interactions Jl and J2, re- (6, y =

I,

m, n). Thus, we obtain the simultaneous or- spectively [3, 41. It is known that the system exhibits

various types of equilibrium states depending on the value of a

=

252

/

J1 [5]. To describe the state, we have to divide lattice points into q sublattices, where q is the number of spins in the unit cell. Hereafter, we put q = 3, because the case of negative a which we

investigate corresponds to the configuration shown in figure 1.

The Hamiltonian of the system of N spins on a triangular lattice is written as

2 f f

H = - J I s ~ c ' s ~ s ~ - J ~ S C s i s j - ~ H C s i , (1)

where S is the magnitude of spin, C'

(c")

is the sum- mation over nearest neighbor (second nearest neighbor) spin pairs. Taking the transition probability from sj to -Sj per unit time as

w [sj] = 1

/

(27) (1

-

~ j t a n h {-PE ( ~ j ) ) )

,

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dinary differential equation (3),

d/dt X = F ( x ) , x = (xi

,...,

xg), (3) where XI = (s')

,

$2 = (sm)

,

x3 = (sn) ,x4 = (s'srn)

,

2.5 = ( s ~ s ~ ) , 2 6 =

(s~s')

2 7 = (s's')

,

2 8 = (smsm), xg = (snsn)

.

Let us consider the symmetry properties of the equation (3) in the general situation in which there are q sublattices. The equation is invariant under the permutation of lables of sublattices. There are q! permutations including the identity. Each permutation is generated by (q

-

1) operators, S I , ~ ,

...,

S,-I,,. Si,j exchanges i and jsublattices. let us assume H = 0. Then equation (3) is invariant under the operation S,, which transforms any spin si to -si. Assuming S t o

be any product of S1,2,

...,

Sq-l,,, and S,, we obtain the relation F (Sx) = SF (x)

.

Thus, if S x (0) = x (0)

,

then Sx (t) = x ( t )

,

i.e., S-symmetry space is invariant under flow F (x)

.

Although there are 2 (q!)

-

1 oper- ations, the number of symmetry spaces is less than

= ( k ~ ) - 1 , E =

-,p

( ~ ~ x l ~ ~

+

-

_{p ~ ,}2 (q!)

-

1. Now, let us return to the case of q = 3 and investigate symmetric steady solutions under these o p we obtain the equations of motion for averages of mo- erations. There are 11 operators and 8 symmetry sub- ments from the master equation. As these equations spaces. Table I.

have hierarchical structure, we have t o truncate the

hierarchy t o get closed equations. One method is _{Table I.}

_-

_{Symmetry operators and symmetry spaces.} the mean field approximation (MFA), which replaces _{Operators in parentheses satisfy different relations.}

E (sj) by E ((sj))

.

In [3], we treated this case. In this paper, we calculate correlation functions to consider fluctuation effects. Let us expand moments in terms of cumulants and retain up t o two body cumulants. Fur- ther, for a two body cumulant (s;si),

,

unless si and

Fig. 1.

-

Ground state configuration for a

<

0. I, m and n are labels for the three sublattices.

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C8 - 1384 JOURNAL DE PHYSIQUE

(1) Si,j (i

#

j) symmetric solution.

As an example, we take S I , ~ . From S I , ~ X = x, XI = x2, 2 5 = $6 and 2 7 = xs follow. If there is no other symmetry, x3 is not equal to XI, i.e., two of three sublattice magnetizations coincide with each other. Thus, the solution represents a 2 ferrimagnetic

(2FR) phase.

(2) SrSi,j (i

#

j) symmetric solution.

S r S 1 , 2 ~ = x means x i = -x2, 2 3 = 0, x5 = xe and xr = xs. This represents a partially disordered(PD) phase.

(3) S1,2S2,3 or S1,2S3,1 symmetric solution. S1,2S2,3~ = x means XI = x2 = $3, 2 4 = 2 5 = xe and 2 7 = xg = $9. If this does not have S, symmetry, X I = 2 2 = 23

#

0, i.e., this represents a ferromagnetic(F) phase.

(4) SrS~,2S2,3 or SrS1,2Ss,1 symmetric solution. SrS1,2S2,3x = x means XI = 2 2 = x3 = 0, 2 4 = x5 = XG and 2 7 = xs = xg. This represents the para- magnetic(P) phase.

All these solutions appear in our numerical results. In figure 2, a

-

T phase diagram is shown. Here, the equilibrium state is considered t o be the stable solution of equation (3). Phase transition is explained in terms of bifurcation theory as follows: For high T, P phase is stable and no other solution exists. At T = T2, symmetry breaking bifurcation takes place, P becomes unstable and 2FR and PD appear, one of which is st* ble. For -0.9

<

a

<

-0.05, at T = TI, the PD state loses its stability and the 2FR state becomes stable. By the continuity of bifurcation, it is plausible to consider that there is a stable 3FR state between the stable 2FR and the stable PD state as in the MFA case [3]. The 3FR state is defined as the steady solution in which all sublattice magnetizations are different from each other.

Next, we investigate the behavior of the system which is initially in a static magnetic field. In fig-

ure 3, we show a typical time sequence which exhibit

"turn round" behavior. All sublattice magnetizations firstly approach to P solution together, then split into

(+,

0, -) values, i.e., PD solution, and relax finally t o 2FR sb~ution, which is expected to be the equilibrium state. This behavior is quite similar to that for the MFA case [3]. In that case, in UF space, P is sta-

ble. Further, there are direct paths from P state to PD state. We conclude that the global flow structure is well approximated by that of the MFA case.

Finally, we give the expression for the frequency dependent susceptibility Xs (w) and the correlation function ( h m Y (0) h m 6 (t)),

,

where h m s E

j~

x

(si

-

(s;),) and (.), denotes an average at the

iE.6

au~librium.

Fig. 2. - CY

-

T phase diagram. T is scaled by the N&l

temperature TN in MFA. Tl and T2 denote critical temper- atures.

Fig. 3;

-

Typical time sequence of XI, 2 2 and x3 for T =

0.1 TN and CY = 0.2.

where 7j;s are eigenvalues of a 3 x 3 constant matrix and L (6, y, j) is a constant. The expression (5) shows the existence of three relaxation times. Equations (4) and (5) are derived in the framework of the linear response theory. See [4] for derivation. Comparing equations (4) and (5), we obtain the fluctuation-dissipation relation,

kT

/

(wP2) Imx (w) =

1/2

/-

dt (Am (0) ~m (Itl))T ~ X P but1

.

(6)

-m

where

x

(w) =

x

(w) and Am ( t ) = h m 6 (t)

.

6 6

In paper 161, a fluctuation-dissipatio~i relation was derived from the master equation in .the case of ferro- magnetism. Our treatment here is the extension of the paper [6] to the present antiferrclmagnetic system.

[l] For example, Bak, P. and von Boehm, J., Phys.

Rev.

B

2 1 (1980) 5279;

Nakanishi, K., J . Phys. Soc. J p n 52 (1983) 2449. [2] Miyashita, S. and Kawamura, H., J. Phys. Soc.

J p n 54 (1985) 3385;

Miyashita, S., J . Phys. Soc. J p n 55 (1986) 227. [3] Uezu, T. and Kawasaki, K., J. IJhys. Soc. J p n 56

(1981) 918.

[4] Kawasaki, K. and Uezu, T., J . IJhys. Soc. J p n 57 (1988) 3532.

[5] Kaburagi, M. and Kanamori, .I., Proc. 2nd Int. Conf. Solid Surfaces, J p n J . Appl. Phys. Suppl.

2. Pt. 2 (1974) 145.