Numerical method to evaluate the dynamical critical exponent
M. Silve
´
rio Soares,* J. Kamphorst Leal da Silva, and F. C. Sa
´
Barreto
Departamento de Fı ´sica, Instituto de Cie ˆncias Exatas, Universidade Federal de Minas Gerais, Caixa Postal 702,
30161-970 Belo Horizonte, Brazil
Received 24 September 1996
A finite-size scaling approach is used to show numerically that dynamical scaling occurs for short and long
times independently of the initial conditions. Its main idea is to construct particular quantities scaling as L
0
in
the thermodynamic limit L → , L being the linear size of the system. These are the quantities for which the
dynamic scaling occurs for short and long times. This approach is applied to obtain the critical dynamical
behavior of two- and three-dimensional ferromagnetic Ising models, subjected to Glauber dynamics.
S0163-18299700302-0
Although the static critical properties of classical spin sys-
tems are well described by renormalization group theories
theoretical framework and calculation procedures,
1
the dy-
namic critical properties
2
are not as well understood. In par-
ticular, the value of the dynamic critical exponent z is still an
open question even for the two-dimensional Ising model
3–6
when dynamics with local flips of spins are considered. The
determination of the exponent z for classical models in dif-
ferent lattice dimensions has been done by using several ap-
proaches: field-theoretical dynamical renormalization group
methods,
2,7
Monte Carlo simulations,
8–10
renormalization
group methods,
11–14
damage spreading,
6,15,16
non-
equilibrium relaxation,
17,18
and series expansion.
4,19
For the
Ising model the various methods obtain in two dimensions
2.10z 2.52 and in three dimensions 1.95z 2.35. Usu-
ally, some of these methods obtain the z exponent from long-
time behavior. This limit is hard to be reached because of the
critical slowing down that always appears, except for clusters
algorithms.
20
Besides, it is also very hard to obtain good
statistics in these procedures. Critical slowing down and poor
statistics are among the reasons why different calculations
produce so many different values for z .
Our objective in this work is to present a finite-size dy-
namical scaling approach which overcomes the usual diffi-
culties pointed out above. The method is then applied to two-
and three-dimensional kinetic Ising models with single spin
flips. The method can also be applied to systems with more
complex ordering behavior subjected to different dynamics.
Recently, a method has been proposed
21
to evaluate the
z exponent from short-time behavior. It is based on the scal-
ing relation for the dynamics at early times.
22
In this time
regime the magnetization initially grows, characterizing a
new universal stage of the relaxation of the magnetization,
the so called ‘‘critical initial slip.’’ However, it turns out that
the initial condition zero magnetization and very short cor-
relation length is essential to obtain the dynamical exponent.
This happens because the critical initial slip sets right in after
a microscopic time scale and eventually crosses over the
long-time regime. The characteristic time associated with the
critical initial slip is t
0
=m
0
-z / x
, where m
0
is the initial mag-
netization and x is a new exponent. If m
0
=0, we have that
t
0
→ and the early time scaling overlaps with the expected
long-time scaling.
The short-time behavior obtained by the approach pre-
sented in this paper does not depend on the initial condition.
It is based on the same ideas developed in a recently pro-
posed renormalization group calculation used to evaluate the
static properties of Ising systems.
23
We analyze the relax-
ation to the equilibrium of well-chosen variables which
scales as L
0
( L is the linear size of the system. In order to
show that the scaling occurs already for short times indepen-
dently of the initial conditions, we chose two different initial
conditions, both with zero correlation lengths. The first one
consists of up and down spins, in such a way that the initial
magnetization and a chosen quantity of our schema both
have the value zero. In this case the characteristic time scale
of the critical initial slip is t
0
→ . The second initial condi-
tion consists of all spins in the up direction. Now, m
0
and the
chosen quantity of our approach have their maxima values,
namely, 1. In this case the time scale t
0
1 is essentially
microscopic. The reasons for introducing this finite-size dy-
namical scaling approach are the following: i The weak
dependence of the procedure on L implies that the value of
z is very good even for the smallest lattices; ii the critical
temperature and the static exponents are easily obtained with
good accuracy; iii the chosen variable relaxes exponentially
after a transient behavior, thus allowing a good traditional
way of calculation of the relaxation time; and iv the dy-
namical scaling occurs for short times independently of the
initial condition, hence the evaluation of the z exponent from
short-time simulations.
The finite-size dynamical scaling approach FSDSA is
based only on the finite-size dynamical scaling hypothesis
2
which, for a thermodynamic quantity P and a finite system
of linear dimension L , can be expressed as
P
' , H ' , t ' , L ' =l
-
P
, H , t , L , 1
for arbitrary values of the scaling factor l . We are here
adopting the ferromagnetic language where H is the external
field, the reduced coupling constant is K , =K -K
c
, where
K
c
is the critical coupling, ' =l
1/
, H ' =l
y
H , t ' =l
-z
t ,
and L ' =l
-1
L . Here, , 1/y , and z are the correlation length
and magnetic and dynamical critical exponents, respectively.
Equation 1 has its general validity for L → and near the
PHYSICAL REVIEW B 1 JANUARY 1997-II VOLUME 55, NUMBER 2
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