On properties of the Ising model for complex energy/temperature and magnetic field

Abstract

We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable u = e−4K, where K = J/(kBT), 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/(kBT), these results provide new insights into features of our earlier study of this case. We also consider complex h = H/(kBT) and μ = 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., μ = eiθ, 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 contains a real line segment extending through part of the physical ferromagnetic interval 0 ⩽ u ⩽ 1, with a right-hand endpoint urhe at the temperature for which the Yang–Lee edge singularity occurs at μ = e±iθ. Conformal field theory arguments are used to relate the singularities at urhe and the Yang–Lee edge.