Complex-Temperature Properties of the 2D Ising Model for Nonzero Magnetic Field

TL;DR

This paper explores the complex-temperature phase diagram of the 2D Ising model under nonzero magnetic field, connecting known solutions and analyzing singularities through zeros of the partition function and series expansions.

Contribution

It introduces a continuous connection between the exact solutions at zero and infinite magnetic field, using complex-temperature zeros and series analysis for the first time.

Findings

Identifies complex-temperature phase boundaries and singularities for various magnetic field regimes.

Shows the breaking of the inner branch of the limaçon in the ferromagnetic phase under nonzero field.

Provides exact phase diagrams for the model at specific magnetic field values on different lattices.

Abstract

We study the complex-temperature phase diagram of the square-lattice Ising model for nonzero external magnetic field $H$, i.e. for $0 \le \mu \le \infty$, where $\mu=e^{-2\beta H}$. We also carry out a similar analysis for $-\infty \le \mu \le 0$. The results for the interval $-1 \le \mu \le 1$ provide a new way of continuously connecting the two known exact solutions of this model, viz., at $\mu=1$ (Onsager, Yang) and $\mu=-1$ (Lee and Yang). Our methods include calculations of complex-temperature zeros of the partition function and analysis of low-temperature series expansions. For real nonzero $H$, the inner branch of a lima\c{c}on bounding the FM phase breaks and forms two complex-conjugate arcs. We study the singularities and associated exponents of thermodynamic functions at the endpoints of these arcs. For $\mu < 0$, there are two line segments of singularities on the negative and positive $u$ axis, and we carry out a similar study of the behavior at the inner endpoints of these arcs, which constitute the nearest singularities to the origin in this case. Finally, we also determine the exact complex-temperature phase diagrams at $\mu=-1$ on the honeycomb and triangular lattices and discuss the relation between these and the corresponding zero-field phase diagrams.