Hard squares with negative activity on cylinders with odd circumference
Summary: Let \(C_{m,n}\) be the graph on the vertex set \(\{1, \dots, m\} \times \{0, \dots, n-1\}\) 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 \(-1\).
scientific article; zbMATH DE number 5576452