Low-Temperature Series Expansions for the Spin-1 Ising Model

TL;DR

This paper extends low-temperature series for the spin-1 Ising model on a square lattice using a new transfer matrix formalism, confirming critical exponents match the spin-1/2 case and estimating critical parameters.

Contribution

A novel transfer matrix approach enables series expansion up to 79th order, improving analysis of the spin-1 Ising model's critical behavior.

Findings

Critical exponents match those of the spin-1/2 Ising model.

Precise estimates of critical temperature and singularities.

Evidence for a non-analytic confluent correction with exponent ~1.1.

Abstract

The finite lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the spin-1 Ising model on the square lattice. A new formalism is described that uses two distinct transfer matrix approaches in order to significantly reduce computer memory requirements and which permits the derivation of the series to 79th order. Subsequent analysis of the series clearly confirms that the spin-1 model has the same dominant critical exponents as the spin-$\frac{1}{2}$ Ising model. Accurate estimates for both the critical temperature and non-physical singularities are obtained. In addition, evidence for a non-analytic confluent correction with exponent $\Delta_{1} = 1.1 \pm 0.1$ is found.