Hard Squares for Z = −1

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 (πm/N). Here we tabulate the results for lattices of up to 12 columns with cyclic or free boundary conditions and the two obvious orien-tations. 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.