Low-temperature series expansion of square lattice Ising model: A study based on Fisher zeros

TL;DR

This paper derives new low-temperature series coefficients for the square lattice Ising model, analyzing their asymptotic behavior and relation to Fisher zeros, revealing insights into physical and non-physical singularities.

Contribution

It provides explicit expressions for the series coefficients using Fisher zeros density and determines their asymptotic behavior for both zero and imaginary fields.

Findings

Convergence radius depends on Fisher zeros' accumulation points.

In zero field, the critical point is the dominant accumulation point.

In imaginary field, the dominant point is a non-physical singularity.

Abstract

Low-temperature expansion of Ising model has long been a topic of significant interest in condensed matter and statistical physics. In this paper we present new results of the coefficients in the low-temperature series of the Ising partition function on the square lattice, in the cases of a zero field and of an imaginary field $i(\pi/2)k_BT$. The coefficients in the low-temperature series of the free energy in the thermodynamic limit are represented using the explicit expression of the density function of the Fisher zeros. The asymptotic behaviour of the sequence of the coefficients when the order goes to infinity is determined exactly, for both the series of the free energy and of the partition function. Our analytic and numerical results demonstrate that, the convergence radius of the sequence is dependent on the accumulation points of the Fisher zeros which have the smallest modulus. In the zero field case this accumulation point is the physical critical point, while in the imaginary field case it corresponds to a non-physical singularity. We further discuss the relation between the series coefficients and the energy state degeneracies, using the combinatorial expression of the coefficients and the subgraph expansion.