On Properties of the Ising Model for Complex Energy/Temperature and Magnetic Field

TL;DR

This paper analyzes the properties of the Ising model in the complex energy/temperature plane with a magnetic field, providing exact phase diagrams, zeros of the partition function, and phase boundary insights using analytical and conformal field theory methods.

Contribution

It offers new exact results for the Ising model's phase diagram in complex variables, extending previous real-field studies and exploring complex magnetic fields and their effects.

Findings

Phase diagram in the complex $u$ plane for 1D and quasi-1D strips.

Partition function zeros on the square lattice in complex $u$ plane.

Phase boundary contains a real line segment related to the Yang-Lee edge singularity.

Abstract

We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable $u=e^{-4K}$, where $K=J/(k_BT)$, for nonzero external magnetic field, $H$. Exact results are given for the phase diagram in the $u$ plane for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real $h=H/(k_BT)$, these results provide new insights into features of our earlier study of this case. We also consider complex $h=H/(k_BT)$ and $\mu=e^{-2h}$. Calculations of complex-$u$ zeros of the partition function on sections of the square lattice are presented. For the case of imaginary $h$, i.e., $\mu=e^{i\theta}$, we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the $u$ plane. We find that in this case, the phase boundary ${\cal B}_u$ contains a real line segment extending through part of the physical ferromagnetic interval $0 \le u \le 1$, with a right-hand endpoint $u_{rhe}$ at the temperature for which the Yang-Lee edge singularity occurs at $\mu=e^{\pm i\theta}$. Conformal field theory arguments are used to relate the singularities at $u_{rhe}$ and the Yang-Lee edge.