Complex-Temperature Properties of the 2D Ising Model with $\beta H = \pm i \pi/2$

TL;DR

This paper explores the complex-temperature phase diagram and singularities of the exactly solvable 2D Ising model with a purely imaginary external field, providing detailed series expansions and analyzing critical behavior.

Contribution

It presents the first detailed analysis of the complex-temperature properties and singularities of the Ising model with $eta H = ext{i}rac{\pi}{2}$, including series expansions and phase boundary determination.

Findings

Identified complex-temperature phase boundaries and their properties.

Determined divergence points and critical exponents for susceptibility.

Analyzed singularities of specific heat and magnetization in the complex plane.

Abstract

We study the complex-temperature properties of a rare example of a statistical mechanical model which is exactly solvable in an external symmetry-breaking field, namely, the Ising model on the square lattice with $\beta H = \pm i \pi/2$. This model was solved by Lee and Yang \cite{ly}. We first determine the complex-temperature phases and their boundaries. From a low-temperature, high-field series expansion of the partition function, we extract the low-temperature series for the susceptibility $\chi$ to $O(u^{23})$, where $u=e^{-4K}$. Analysing this series, we conclude that $\chi$ has divergent singularities (i) at $u=u_e=-(3-2^{3/2})$ with exponent $\gamma_e'=5/4$, (ii) at $u=1$, with exponent $\gamma_1'=5/2$, and (iii) at $u=u_s=-1$, with exponent $\gamma_s'=1$. We also extract a shorter series for the staggered susceptibility and investigate its singularities. Using the exact result of Lee and Yang for the free energy, we calculate the specific heat and determine its complex-temperature singularities. We also carry this out for the uniform and staggered magnetisation.