Complex-Temperature Properties of the Ising Model on 2D Heteropolygonal Lattices

Abstract

Using exact results, we determine the complex-temperature phase diagrams of the 2D Ising model on three regular heteropolygonal lattices, $(3\cdot 6\cdot 3\cdot 6)$ (kagomé), $(3\cdot 12^2)$, and $(4\cdot 8^2)$ (bathroom tile), where the notation denotes the regular $n$-sided polygons adjacent to each vertex. We also work out the exact complex-temperature singularities of the spontaneous magnetisation. A comparison with the properties on the square, triangular, and hexagonal lattices is given. In particular, we find the first case where, even for isotropic spin-spin exchange couplings, the nontrivial non-analyticities of the free energy of the Ising model lie in a two-dimensional, rather than one-dimensional, algebraic variety in the $z=e^{-2K}$ plane.

Comments: 31 pages, latex, postscript figures