New Series-Expansion Method for the Dimer Problem

Abstract

A new series-expansion technique is presented for the grand-partition function for the dimer problem with no attractive interactions. The zeroth-order term in the expansion recovers the Bethe approximation. Higher order corrections involve the weighted summation of closed subgraphs (no vertices of degree one). The weight formula is given and is a simple function of the topological type of the subgraph and the number of edges. From this series expansion, the series in powers of the dimer activity valid at low density of dimers can be recovered. The series expansion is also applicable for high density of dimers. In particular, it provides an improved approximation technique for estimating the molecular freedom per dimer at close packing, as can be seen by comparing the approximate values obtained by other authors and those obtained using this technique with the exact values known for the two dimensional lattices. Finally, this series method is used to discuss the thermodynamic behavior.