Hard squares with negative activity

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 × 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.