Equivalence between non-bilinear spin-$S$ Ising model and Wajnflasz model

TL;DR

This paper establishes a mapping between non-bilinear spin-$S$ Ising models and spin-crossover models, enabling analysis of complex spin systems through transformations to solvable Ising models, revealing residual entropy and frustration effects.

Contribution

It introduces a novel mapping of polynomial spin-$S$ states onto spin-crossover states, connecting complex non-bilinear models to solvable Ising models, and explores their thermodynamic properties.

Findings

Mapping of spin-$S$ polynomial to spin-crossover states

Transformation of non-bilinear spin-$S$ Ising model to effective spin-1/2 Ising model

Identification of residual entropy and frustration in the transformed models

Abstract

We propose the mapping of polynomial of degree 2S constructed as a linear combination of powers of spin-$S$ (for simplicity, we called as spin-$S$ polynomial) onto spin-crossover state. The spin-$S$ polynomial in general can be projected onto non-symmetric degenerated spin up (high-spin) and spin down (low-spin) momenta. The total number of mapping for each general spin-$S$ is given by $2(2^{2S}-1)$. As an application of this mapping, we consider a general non-bilinear spin-$S$ Ising model which can be transformed onto spin-crossover described by Wajnflasz model. Using a further transformation we obtain the partition function of the effective spin-1/2 Ising model, making a suitable mapping the non-symmetric contribution leads us to a spin-1/2 Ising model with a fixed external magnetic field, which in general cannot be solved exactly. However, for a particular case of non-bilinear spin-$S$ Ising model could become equivalent to an exactly solvable Ising model. The transformed Ising model exhibits a residual entropy, then it should be understood also as a frustrated spin model, due to competing parameters coupling of the non-bilinear spin-$S$ Ising model.