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

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.