Exact Series-Expansion Study of the Monomer-Dimer Problem

Abstract

The monomer-dimer problem is studied for several two- and three-dimensional lattices (coordination number $q$ by deriving between 8 and 16 coefficients of the exact series expansions in powers of the activity $z$ and density $\ensuremath{\rho}$ of dimers. Analysis of the series by the ratio and Pad\'e-approximant techniques shows that while there is no phase transition, close packing is a singular point so that as $\ensuremath{\rho}$ approaches $\frac{1}{q}$ (the close-packing density), $z\ensuremath{\approx}\frac{A}{{(1\ensuremath{-}q\ensuremath{\rho})}^{\ensuremath{\gamma}}}$. Our results enable us to conjecture that $\ensuremath{\gamma}=2$ for the close-packed lattices (irrespective of dimensionality), but for the loose-packed lattices $\ensuremath{\gamma}=1.75$ in two dimensions and 1.95 in three dimensions. Estimates of the amplitude $A$ are given.It is further shown how the various thermodynamic quantities may be accurately plotted over the whole physical region. The salient features of the plots are discussed and specific forms for the asymptotic behavior near close packing are given in terms of $\ensuremath{\gamma}$, $A$, and $\ensuremath{\varphi}$ (the molecular freedom per dimer). After relating our work to Nagle's activity-like expansions, we make accurate estimates of $\ensuremath{\varphi}$.