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

We study the complex-temperature phase diagram of the square-lattice Ising model for nonzero external magnetic field $H$, i.e. for $0 \le μ\le \infty$, where $μ=e^{-2βH}$. We also carry out a similar analysis for $-\infty \le μ\le 0$. The results for the interval $-1 \le μ\le 1$ provide a new way of continuously connecting the two known exact solutions of this model, viz., at $μ=1$ (Onsager, Yang) and $μ=-1$ (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 limaçon 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 $μ< 0$, 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 $μ=-1$ on the honeycomb and triangular lattices and discuss the relation between these and the corresponding zero-field phase diagrams.

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