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

TL;DR

This paper derives exact complex-temperature phase diagrams for the 2D Ising model on three heteropolygonal lattices, revealing novel non-analyticities in free energy that differ from traditional lattice cases.

Contribution

It provides the first exact analysis of complex-temperature properties of the Ising model on these specific heteropolygonal lattices, expanding understanding beyond standard lattices.

Findings

Exact complex-temperature phase diagrams for three heteropolygonal lattices.

Identification of nontrivial non-analyticities in free energy in two-dimensional algebraic varieties.

Comparison showing unique properties compared to square, triangular, and hexagonal lattices.

Abstract

Using exact results, we determine the complex-temperature phase diagrams of the 2D Ising model on three regular heteropolygonal lattices, $(3 \cdot 6 \cdot 3 \cdot 6)$ (kagom\'{e}), $(3 \cdot 12^2)$, 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.