Critical behavior of the classical spin-1 Ising model: a combined low-temperature series expansion and Metropolis Monte Carlo analysis

TL;DR

This study combines low-temperature series expansion and Monte Carlo simulations to analyze the critical behavior of the classical spin-1 Ising model across different dimensions and interaction types, highlighting the limitations of mean-field theory.

Contribution

It introduces a combined analytical and numerical approach to study spin-1 Ising models, including a transformation linking it to spin-1/2 models and explores long-range interaction effects.

Findings

Monte Carlo results highlight mean-field theory limitations.

Partition function reduces to spin-1/2 Ising model under specific conditions.

Critical temperature dependence on long-range interactions analyzed.

Abstract

In this paper, we theoretically study the critical properties of the classical spin-1 Ising model using two approaches: 1) the analytical low-temperature series expansion and 2) the numerical Metropolis Monte Carlo technique. Within this analysis, we discuss the critical behavior of one-, two- and three-dimensional systems modeled by the first-neighbor spin-1 Ising model for different types of exchange interactions. The comparison of the results obtained according the Metropolis Monte Carlo simulations allows us to highlight the limits of the widely used mean-field theory approach. We also show, via a simple transformation, that for the special case where the bilinear and bicubic terms are set equal to zero in the Hamiltonian the partition function of the spin-1 Ising model can be reduced to that of the spin-1/2 Ising model with temperature dependent external field and temperature independent exchange interaction times an exponential factor depending on the other terms of the Hamiltonian and confirm this result numerically by using the Metropolis Monte Carlo simulation. Finally, we investigate the dependence of the critical temperature on the strength of long-range interactions included in the Ising Hamiltonian comparing it with that of the first-neighbor spin-1/2 Ising model.