Ising ferromagnets in Ising-percolation square lattices, an example of Ising-Ising coupling

Abstract

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 ${T}_{q}$.