Hard Squares for z = -1

TL;DR

This paper investigates the hard square model at activity z = -1, revealing surprisingly simple eigenvalue structures of the transfer matrix for small lattices and suggesting potential generalizations for larger sizes.

Contribution

It uncovers the simple eigenvalue structure of the transfer matrix for the hard square model at z = -1 and provides comprehensive tabulations for small lattice sizes with various boundary conditions.

Findings

Eigenvalues are zero, roots of unity, or solutions to x^3 = 4 cos^2 (pi*m/N)

Results are simple and extendable to larger lattices

Tabulated data for lattices up to 12 columns with different boundary conditions

Abstract

The hard square model in statistical mechanics has been investigated for the case when the activity z is -1. For cyclic boundary conditions, the characteristic polynomial of the transfer matrix has an intriguingly simple structure, all the eigenvalues $x$ being zero, roots of unity, or solutions of x^3 = 4 cos^2 (pi*m/N). Here we tabulate the results for lattices of up to 12 columns with cyclic or free boundary conditions and the two obvious orientations. We remark that they are all unexpectedly simple and that for the rotated lattice with free or fixed boundary conditions there are obvious likely generalizations to any lattice size.