Exact solution of close-packed dimers on the kagome lattice

TL;DR

This paper derives an exact closed-form expression for the free energy of close-packed dimers on the kagome lattice using two methods, and finds that dimer correlations vanish beyond a certain distance.

Contribution

It provides the first complete analytical solution for close-packed dimers on the kagome lattice, using Kasteleyn and vertex model approaches.

Findings

Exact free energy expression $(1/3) \, \ln (4 x y z)$ derived.

Correlation functions vanish for dimers separated by two or more lattice spacings.

Both methods yield consistent results.

Abstract

It is well-known that exact enumerations of close-packed dimers can be carried out for two-dimensional lattices. While details of results are now known for most lattices, due to the unique nature of the lattice structure, there has been no complete analysis for the kagome lattice. Here we derive the close-form expression $(1/3) \ln (4 x y z)$ for the free energy of close-packed dimers on the kagome lattice, where $x,y,z$ are dimer weights. We use two different approaches, the Kasteleyn method of evaluating a Pfaffian and an alternative vertex model formulation. Both methods lead to the same final expression. The correlation function between two dimers at a distance equal or greater than two lattice spacings is found to vanish identically.