Analytic expressions for estimation of the critical properties of inhomogeneous Ising models

TL;DR

The paper extends the analytic $m$-vicinity method, previously developed for homogeneous Ising lattices, to inhomogeneous ferromagnets and bicomponent lattices by Gaussianizing the density of states around $E_m$. It derives closed-form expressions for the critical temperature and its composition dependence, using parameters $E_0$, $q$, and $p$ that encode lattice inhomogeneity, and analyzes critical exponents, including a special regime where $q=4(1-p)$. Monte Carlo simulations on 3D bicomponent lattices validate the approach, showing sub-6% accuracy for $K_c$ and improved qualitative and quantitative agreement over mean-field theory. The results highlight distinct critical behavior at the boundary $q=4(1-p)$ and suggest avenues for extending the framework to systems with competing interactions, offering a fast, reliable tool for estimating critical properties in complex Ising models.

Abstract

In many applications of spin models, the fast estimation of their critical temperatures and other physical properties is of great importance. In this work, we present the analytical expressions estimating the critical properties of inhomogeneous Ising models with ferromagnetic interactions. The expressions were obtained within the framework of the m-vicinity method. The accuracy of the critical temperature estimations was evaluated through comparison with Monte Carlo simulations. Special attention was given to the case when the model consists of two interacting interpenetrating homogeneous sublattices, and relationships for the compositional dependence of the critical temperature were derived.

Figures (2)

• Figure 1: Temperature dependencies of the Binder cumulant for lattices of different sizes. The lattices have interaction parameters $W_3=1$, $W_4=1$ ($x = 1$). The intersection point of these curves yields the critical temperature for the infinite lattice ($N\to \infty$).
• Figure 2: Compositional dependencies of the critical temperature $K_c$. Comparison of MC simulation, $m$-vicinity method, and mean-field theory. The parameters of the sublattices are (a) $W_1=1$, $W_2=1$, $Y_1=0$, $Y_2=0$; (b) $W_1=1$, $W_2=0.5$, $Y_1=0.4$, $Y_2=0.2$; (c) $W_1=1$, $W_2=1$, $Y_1=0.5$, $Y_2=0.5$; (d) $W_1=2$, $W_2=0$, $Y_1=1$, $Y_2=0.5$.