Low temperature series expansions for the square lattice Ising model with spin S > 1

TL;DR

This paper develops low-temperature series expansions for the square lattice spin-$S$ Ising model with higher spins ($S > 1$), analyzing critical points, singularities, and critical exponents.

Contribution

It extends low-temperature series analysis to higher spin values and investigates the behavior of physical and non-physical singularities in the model.

Findings

Critical exponents are independent of spin $S$.

Number of non-physical singularities increases rapidly with $S$.

Estimates for critical points become less accurate for higher $S$ due to non-physical singularities.

Abstract

We derive low-temperature series (in the variable $u = \exp[-\beta J/S^2]$) for the spontaneous magnetisation, susceptibility and specific heat of the spin-$S$ Ising model on the square lattice for $S=\frac32$, 2, $\frac52$, and 3. We determine the location of the physical critical point and non-physical singularities. The number of non-physical singularities closer to the origin than the physical critical point grows quite rapidly with $S$. The critical exponents at the singularities which are closest to the origin and for which we have reasonably accurate estimates are independent of $S$. Due to the many non-physical singularities, the estimates for the physical critical point and exponents are poor for higher values of $S$, though consistent with universality.