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

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.