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Physical review. B, Condensed matter

Finite-size calculations for the kinetic Ising model

Rahul Anil Pandit, Gabor Forgács, Pál Ruján

Published Aug 1, 1981• 9 citations

Abstract

Using a Hamiltonian formulation of the master equation, we carry out finite-size calculations for the dynamical critical exponent $z$ of the kinetic Ising model in one and two dimensions. In one dimension, different appropriately normalized transition probabilities give different values for $z$; this result is based on both exact and finite-size calculations. We show that this "nonuniversal" behavior is due to the fact that the critical temperature for the one-dimensional Ising model is zero. In two dimensions we combine our finite-size calculations with finite-size scaling theory to calculate $z$. Even in this case we find that $z$ depends on the form of the transition probability---a result that contradicts the universality hypothesis for dynamical critical behavior. We discuss the possible reasons for this contradiction.

DOI: 10.1103/physrevb.24.1576

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Physical review. B, Condensed matter•Jul 1, 1979•17 citations

Using a Hamiltonian formulation of the master equation, we carry out finite-size calculations for the dynamical critical exponent

z

of the kinetic Ising model in one and two dimensions. In one dimension, different appropriately normalized transition probabilities give different values for

z

; this result is based on both exact and finite-size calculations. We show that this "nonuniversal" behavior is due to the fact that the critical temperature for the one-dimensional Ising model is zero. In two dimensions we combine our finite-size calculations with finite-size scaling theory to calculate

z

. Even in this case we find that

z

depends on the form of the transition probability---a result that contradicts the universality hypothesis for dynamical critical behavior. We discuss the possible reasons for this contradiction.

The kinetic Ising model on a square lattice is studied by means of the finite-size scaling with the help of the Monte Carlo method. The results obtained are consistent with the universality hypothesis that the value of the dynamic critical expotent is universal for certain types of transition probabilities. The present results also suggest that the dynamic critical exponents for the relaxation times of the magnetization and energy are the same. Our estimation of the dynamic critical exponent z gives that z \cong2.2 ±0.1.

Critical finite-size scaling functions for the order-parameter distribution of the two- and three-dimensional Ising model are investigated. Within a recently introduced classification theory of phase transitions the universal part of critical finite-size scaling functions has been derived by employing a scaling limit which differs from the traditional finite-size scaling limit. In this paper the analytical predictions are compared with Monte Carlo simulation results. We find good agreement between the analytical expression and the simulation results. The agreement is consistent with the possibility that the functional form of the critical finite-size scaling function for the order-parameter distribution is determined uniquely by only a few universal parameters, most notably the equation of state exponent.

We determine accurate values of ordering temperatures and critical exponents for Ising-spin-glass transitions in four dimensions, using a combination of finite-size scaling and nonequilibrium scaling techniques. We find that the exponents $\ensuremath{\eta}$ and

z

vary with the form of the interaction distribution, indicating nonuniversality at Ising-spin-glass transitions. These results confirm conclusions drawn from numerical data for three dimensions.

We report a Monte Carlo study on the critical relaxation of the three-dimensional kinetic Ising model from a nonequilibrium initial state. It is shown that the details of the method, namely, a choice of the sampling procedure and a functional form of the transition probability do not affect an asymptotic power-law relaxation. The finite-size effects on the critical relaxation is also studied. Cross-over from the power-law relaxation to the exponential one is found, which is discussed in view of the dynamical finite-size scaling theory.

We make use of the calculation of Achiam and Kosterlitz and investigate the generalization of the Migdal-type renormalization-group calculation to critical dynamics, by looking at the one- and two-dimensional kinetic Ising model with no conserved magnetization. In the two-dimensional case the dynamical critical index

{Z}_{M}(\mathrm{magnetic}\mathrm{perturbation})=2.064

and

{Z}_{E}(\mathrm{energylike}\mathrm{perturbation})=1.819

for scale factor $\ensuremath{\lambda}=1$.

{Z}_{M}

involves the static $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ exponent, while

{Z}_{E}

involves $\frac{1}{\ensuremath{\nu}}$, and therefore, it is not surprising that

{Z}_{M}

is closer to the high-temperature expansion results, since $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ in the Migdal approximation for statics is much closer to the exact value than $\frac{1}{\ensuremath{\nu}}$. In one dimension we obtain

{Z}_{M}={Z}_{E}=2

, which is the exact result of Glauber.