Hard Squares with Negative Activity on Cylinders with Odd Circumference

Abstract

Let $C_{m,n}$ be the graph on the vertex set $\{1, \ldots, m\} \times \{0, \ldots, n-1\}$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\pm 1)$ or $(a,b) = (c \pm 1,d)$, where the second index is computed modulo $n$. One may view $C_{m,n}$ as a unit square grid on a cylinder with circumference $n$ units. For odd $n$, we prove that the Euler characteristic of the simplicial complex $\Sigma_{m,n}$ of independent sets in $C_{m,n}$ is either $2$ or $-1$, depending on whether or not $\gcd(m-1,n)$ is divisble by $3$. The proof relies heavily on previous work due to Thapper, who reduced the problem of computing the Euler characteristic of $\Sigma_{m,n}$ to that of analyzing a certain subfamily of sets with attractive properties. The situation for even $n$ remains unclear. In the language of statistical mechanics, the reduced Euler characteristic of $\Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.