Spin-$S$ Ising models with multispin interactions on the one-dimensional chain and two-dimensional square lattice

TL;DR

This paper investigates spin-$S$ Ising models with multispin interactions on 1D and 2D lattices, revealing how interaction order and spin magnitude influence thermodynamic properties and phase transition characteristics.

Contribution

It provides a comprehensive analysis of multispin interactions in spin-$S$ Ising models, including transfer matrix formulation, numerical diagonalization, and multicanonical simulations, highlighting the effects on phase transitions.

Findings

For $S=1/2$, free energy is independent of $p$, and correlations increase with $p$.

For $S \\geq 1$, free energy varies with $p$, and correlations are enhanced at low temperatures.

First-order phase transitions occur at finite temperatures for all $S$ and $p \\geq 3$, with transition strength increasing with $p$.

Abstract

We study spin-$S$ Ising models with $p$-spin interactions on the one-dimensional chain and the two-dimensional square lattice. Here, $S$ denotes the magnitude of the spin and $p$ represents the number of spins involved in each interaction. The analysis is performed for $S=1/2,1,3/2,2$ and $p=3,4,5$. For the one-dimensional model, we formulate transfer matrices, and numerically diagonalize them to analyze the temperature dependence of the free energy and spin-spin correlations. In the case of $S=1/2$, the free energy does not depend on $p$, and the spin-spin correlations are uniformly enhanced across all temperature scales as $p$ increases. In contrast, for $S \geq 1$, the free energy varies with $p$, and the spin-spin correlations are significantly enhanced at lower temperatures as $p$ increases. For the two-dimensional model, by using multicanonical simulations, we analyze physical quantities such as an order parameter, internal energy, and specific heat. In addition, we define and examine an order parameter to distinguish ordered and disordered phases. It is found that a first-order phase transition occurs at finite temperatures for all $S$ and $p\geq3$, and increasing $p$ strengthens its nature. We present $S$ and $p$ dependence of the transition temperature and latent heat, and discuss effects of higher-order interactions on the nature of phase transitions.