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Physical review. B, Solid state

Equilibrium States of the Ising Model in the Two-Phase Region

Giovanni Gallavotti, Salvador Miracle-Solé

Published Apr 1, 1972• 58 citations
Publisher

Abstract

We prove that at low enough temperature all translationally invariant equilibrium states for the Ising ferromagnet are a superposition of only two extremal states, i.e., the positively and negatively magnetized pure phases. In particular this proves, at low temperature and in two dimensions, the identity of the spontaneous magnetization and the Onsager's value ${M}_{0}={[1\ensuremath{-}{(sh\ensuremath{\beta})}^{\ensuremath{-}4}]}^{\frac{1}{8}}$.

DOI
10.1103/physrevb.5.2555

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Joint International Information Technology, Mechanical and Electronic Engineering Conference (JIMEC)•Jan 1, 2016•1 citation
We prove that at low enough temperature all translationally invariant equilibrium states for the Ising ferromagnet are a superposition of only two extremal states, i.e., the positively and negatively magnetized pure phases. In particular this proves, at low temperature and in two dimensions, the identity of the spontaneous magnetization and the Onsager's value M0=[1\ensuremath−(sh\ensuremathβ)\ensuremath−4]18{M}_{0}={[1\ensuremath{-}{(sh\ensuremath{\beta})}^{\ensuremath{-}4}]}^{\frac{1}{8}}M0​=[1\ensuremath−(sh\ensuremathβ)\ensuremath−4]81​.
The partition functions of ferromagnetic Ising models of square lattices in a finite magnetic field is deduced using topological considerations within a heuristic graph-theoretical approach. These equations are derived separately for low and high temperature regimes while the exact solution of Onsager is obtained therefrom when the magnetic field is zero. The derived partition function equations here are almost similar to those given by Onsager, thus indicating a straight-forward protocol, even when the magnetic field is present. The spontaneous magnetization derived here using the Helmholtz free energy is identical with that arising from the exact solution. The partition functions lead to the known series expansions of the magnetization and zero-field susceptibility.
The equilibrium magnetization distribution for an Ising-spin model in the mean-field approximation is used to present a simple study of the ferromagnetic phase transition in the absence of external magnetic field (spontaneous magnetization). For temperatures below a critical one (the Curie Temperature) this distribution exhibits two peaks, showing that two states of magnetization are possible. At the Curie temperature the distribution presents a plateau, showing large fluctuations in the magnetization. A scaling-like analysis of the distribution shows that the Curie point can be characterized as an unstable fixed point of a recursion relation.
We study a two-dimensional ferromagnetic Ising model in which spins are updated using modified versions of the Metropolis and Glauber algorithms. These update rules do not obey the detailed balance condition. The steady-state behavior of the model is studied using molecular field theory and Monte Carlo simulations. This model is found to exhibit a nonequilibrium phase transition from a "paramagnetic" state with zero magnetization to a "ferromagnetic" state with nonzero magnetization as the variable that plays the role of temperature in the spin updates is decreased. From detailed Monte Carlo simulations using the modified Metropolis algorithm, we demonstrate explicitly the nonequilibrium nature of the transition and show that it cannot be described as an equilibrium transition with an effective temperature different from the temperature used in the spin updates. The critical exponents that characterize the singular behavior near this continuous phase transition are calculated from finite size scaling of specific heat, magnetization, susceptibility, and correlation length. We find that the values of these exponents are the same (within error bars) as those of the equilibrium Ising model in two dimensions.
Using the Ising Model, the phase transition between ferromagnetic and paramagnetic phases of a 2-D square spin lattice are explored. A phase transition into the ferromagnetic phase occurs when the temperature of the system falls below some critical value. The ferromagnetic phase is characterized by a non-zero magnetization on the macroscopic level. Using the Mote Carlo method to obtain system's configurations (i.e., sets of microstates) in thermal equilibrium, we will explore system's macroscopic properties such as magnetization, magnetic susceptibility, and specific heat capacity as functions of temperature. We will evaluate the transition temperature, and discuss how the lattice size affects the results.