Complex-Temperature Properties of the 2D Ising Model with $\beta H = \pm i \pi/2$

We study the complex-temperature properties of a rare example of a statistical mechanical model which is exactly solvable in an external symmetry-breaking field, namely, the Ising model on the square lattice with $\beta H = \pm i \pi/2$. This model was solved by Lee and Yang \cite{ly}. We first determine the complex-temperature phases and their boundaries. From a low-temperature, high-field series expansion of the partition function, we extract the low-temperature series for the susceptibility $\chi$ to $O(u^{23})$, where $u=e^{-4K}$. Analysing this series, we conclude that $\chi$ has divergent singularities (i) at $u=u_e=-(3-2^{3/2})$ with exponent $\gamma_e'=5/4$, (ii) at $u=1$, with exponent $\gamma_1'=5/2$, and (iii) at $u=u_s=-1$, with exponent $\gamma_s'=1$. We also extract a shorter series for the staggered susceptibility and investigate its singularities. Using the exact result of Lee and Yang for the free energy, we calculate the specific heat and determine its complex-temperature singularities. We also carry this out for the uniform and staggered magnetisation.