Dimer problem in statistical mechanics-an exact result

TLDR

The dimer problem models a lattice of rigid dimers occupying two adjacent sites, a statistical challenge solved exactly only in one dimension and previously approached by approximate methods. The study aims to provide an exact solution for the close‑packed dimer problem on the square lattice. An exact solution for the close‑packed dimer problem on the square lattice is obtained.

Abstract

Abstract An important, even though physically oversimplified model of a system (e.g. solution or gas) containing diatomic molecules is that of a lattice occupied by 'rigid dimers' (e.g. Fowler and Rushbrooke 1937). Each dimer fills two nearest-neighbour sites on the lattice and no site may be occupied by more than one dimer. This represents a statistical problem of some difficulty which has so far been solved exactly only in one dimension (Green and Leipnik 1960, Fisher and Temperley 1960). (It has been treated by various authors on the basis of approximate statistical methods, e.g. Rushbrooke et al. (1953). This note reports an exact solution for the dimer problem on the plane square lattice in the limiting case where the dimera completely fill the lattice (close-packed or high density limit).