Hard squares with negative activity

TL;DR

This paper investigates the properties of the hard-square lattice gas at negative activity, revealing unique spectral and number-theoretic features, including eigenvalues as roots of unity and special partition function behaviors.

Contribution

It introduces conjectures about the eigenvalues and partition functions of the hard-square model at activity -1, supported by analytical and numerical evidence.

Findings

Eigenvalues of the transfer matrix are roots of unity.

Partition function equals 1 when lattice dimensions are coprime.

Eigenvalues form evenly spaced groups ('strings') around the unit circle.

Abstract

We show that the hard-square lattice gas with activity z= -1 has a number of remarkable properties. We conjecture that all the eigenvalues of the transfer matrix are roots of unity. They fall into groups (``strings'') evenly spaced around the unit circle, which have interesting number-theoretic properties. For example, the partition function on an M by N lattice with periodic boundary condition is identically 1 when M and N are coprime. We provide evidence for these conjectures from analytical and numerical arguments.