Critical dynamics of finite Ising model

Jafar Cheraghalizadeh, M. N. Najafi

In this paper we consider the Ising model (called the dynamical Ising model with dynamical temperature $T$) on the Ising-correlated percolation lattice. To construct this lattice, another Ising model (which is called the quenched Ising model with quench artificial temperature ${T}_{q}$) is employed to model the correlations between imperfections. We argue that for each quenched temperature, there is a particular dynamical temperature ${T}_{c}({T}_{q})$ for which the dynamical Ising model becomes critical. We present some finite-size arguments (based on the moments of the magnetism and the energy as well as the spanning cluster probability) to extract the critical points and also show that they are compatible with the finite-size scaling of the singular point of the magnetic susceptibility. The model is thoroughly characterized in and out of these points. We find that the critical behaviors of the model change significantly with respect to the regular Ising model as well as the Ising model on the uncorrelated percolation lattice. It is shown that for the critical lattice, ${T}_{q}={T}_{c}^{\text{square}\phantom{\rule{4.pt}{0ex}}\text{Ising}\phantom{\rule{4.pt}{0ex}}\text{model}}\ensuremath{\approx}2.269$, the critical temperature for the dynamical Ising model in the thermodynamic limit is ${T}_{c}({T}_{q}={T}_{c}^{\text{square}\phantom{\rule{4.pt}{0ex}}\text{Ising}\phantom{\rule{4.pt}{0ex}}\text{model}})=1.94\ifmmode\pm\else\textpm\fi{}0.005$, and the fractal dimension of the exterior perimeter of geometrical spin clusters is ${D}_{f}^{T={T}_{c}({T}_{q}=2.269)}=1.408\ifmmode\pm\else\textpm\fi{}0.002$. Many quantities, such as the dynamical critical temperatures, all local and global critical exponents, and the fractal dimension of loops ${D}_{f}$, scale with the quench temperature in a power-law fashion, with some critical exponents that are reported. Significantly we see that ${D}_{f}^{T={T}_{c}({T}_{q})}\ensuremath{-}{D}_{f}^{T={T}_{c}({T}_{q}=2.269)}\ensuremath{\sim}\frac{1}{\sqrt{\ensuremath{\zeta}({T}_{q})}}$ in which $\ensuremath{\zeta}({T}_{q})$ is the correlation length of the quenched Ising model at temperature ${T}_{q}$.

Masuo Suzuki, 英雄 猪狩

Michel Droz, J. Kamphorst Leal da Silva, Anna‐Sapfo Malaspinas

M. Cemal Yalabık, J. D. Gunton

The time constant which characterizes the critical slowing down of the kinetic Ising model is studied as a function of system size for n×n lattices with periodic boundary conditions, for n from two to ten. In particular, it is shown that the behavior of this time constant at the temperature at which the specific heat exhibits a maximum is consistent with Suzuki's finite size dynamical scaling theory and yields a dynamical critical exponent of z≃2.

Rahul Anil Pandit, Gabor Forgács, Pál Ruján

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.

Hans-Otto Heuer