Mixed-spin Ising model with one- and two-spin competing dynamics
M. Godoy and W. Figueiredo*
Departamento de Fı ´sica, Universidade Federal de Santa Catarina, 88040-900 Floriano ´polis, SC, Brazil
Received 6 July 1999
In this work we found the stationary states of a kinetic Ising model, with two different types of spins:
=1/2 and S =1 . We divided the spins into two interpenetrating sublattices, and found the time evolution for the
probability of the states of the system. We employed two transition rates which compete between themselves:
one, associated with the Glauber process, which describes the relaxation of the system through one-spin flips;
the other, related to the simultaneous flipping of pairs of neighboring spins, simulates an input of energy into
the system. Using the dynamical pair approximation, we determined the equations of motion for the sublattice
magnetizations, and also for the correlation function between first neighbors. We found the phase diagram for
the stationary states of the model, and we showed that it exhibits two continuous transition lines: one line
between the ferrimagnetic and paramagnetic phases, and the other between the paramagnetic and antiferrimag-
netic phases.
PACS numbers: 64.60.Ht
I. INTRODUCTION
In this work we studied the nonequilibrium states of a
two-sublattice ferromagnetic Ising model with mixed spins
=1/2 and S =1. The time evolution of the states of the
system is governed by two competing dynamical process:
one simulating the contact of the system with a heat bath at
a fixed temperature T, and the other mimicking an input of
energy into the system. If a system is subject to an external
flux of energy, it can exhibit the self-organization phenom-
enon. Self-organizing structures are well known in chemical
reactions and in fluid dynamics. The book by Nicolis and
Prigogine 1 and that by Haken 2 present interesting ex-
amples of these phenomena. In our open ferromagnetic spin
system, the contact with the heat bath is simulated by the
Glauber stochastic process 3, where both and S spins
relax through single-spin flips. In our model, the flux of en-
ergy into the system favors states with the highest energy,
generating a competition with the one-spin flip Glauber pro-
cess. The increase in the energy states is obtained when we
simultaneously flip a nearest neighbor pair of spins and S.
This is not a Kawasaki exchange process 4, as used, for
instance, in the work of Tome
´
and de Oliveira 5 to induce
a self-organizing phenomenon in the kinetic Ising model. In
their model, the stochastic Kawasaki dynamics conserves the
order parameter. Here our particular interest is to investigate
the competition between two dynamical processes when the
order parameter is not conserved. This is easily achieved
with the two-sublattice Ising mixed-spin system, after a si-
multaneous flipping of a pair of nearest neighbor spins.
We used the dynamical pair approximation 6 to de-
couple the hierarchy of equations of motion which follow
from the application of the master equation approach. We
attribute a weight p to the one-spin flip Glauber process, and
a weight (1 - p ) to the two-spin flip process, which increases
the energy of the system. We found the stationary states of
the model as a function of temperature and of the parameter
p, which accounts for the competition between the two dy-
namical processes. We determined the phase diagram of the
model in the plane of temperature T versus competition pa-
rameter ( Q =1 - p ), and we noticed the presence of three
different phases: for very small values of Q small flux of
energy, we obtained a ferrimagnetic phase. Increasing the
flux of energy, the ferrimagnetic phase becomes unstable,
and appears to be a paramagnetic phase. However, when the
flux becomes large, we observed a transition from the para-
magnetic phase to the ordered antiferrimagnetic phase. In
Sec. II, we describe the model and derive the equations of
motion for the sublattice magnetizations and the correlation
functions of interest. In Sec. III, we apply the pair approxi-
mation decoupling scheme to find a closed set of equations
of motion. In Sec. IV, we find the stationary states of the
system, and exhibit the phase diagram of the model. Finally,
in Sec. V, we present our conclusions.
II. MODEL AND EQUATIONS OF MOTION
We consider a ferromagnetic Ising model in a square lat-
tice with mixed spins =1/2 and S =1, in a bipartite lattice,
with the spins occupying the sites of one sublattice, and
the S spins occupying the sites of the other one, each sublat-
tice containing N sites. A state of the system is represented
by ( , S ) (
1
,...,
l
,...,
N
; S
1
,..., S
m
,..., S
N
),
where the spin variables
l
can assume the values 1 and
the spin variables S can assume the values 0,1. The energy
of the system in the state ( , S ) is given by
E
, S =-J
( i , j )
S
i
j
, 1
where the sum is over all nearest neighboring pairs of spins,
and J is taken to be positive. Let us call p ( , S ; t ) the prob-
ability of finding the system in the state ( , S ) at time t. The
equation of motion for the probability of the states of the
system is given by the gain and loss master equation 7 *Electronic address: wagner@fisica.ufsc.br
PHYSICAL REVIEW E JANUARY 2000 VOLUME 61, NUMBER 1
PRE 61 1063-651X/2000/611/2185/$15.00 218 ©2000 The American Physical Society