Exact Solution of a Monomer-Dimer Problem: A Single Boundary Monomer on a Non-Bipartite Lattice

TL;DR

This paper provides an exact solution to the monomer-dimer problem on a non-bipartite lattice with a single boundary monomer, revealing a central charge of -2 in the associated conformal field theory.

Contribution

It introduces a novel mapping approach to solve the monomer-dimer problem on a non-bipartite lattice, overcoming limitations of previous methods.

Findings

Exact solution for the monomer-dimer problem on a non-bipartite lattice.

Finite-size analysis and asymptotic expansion of free energy.

Identification of the central charge c=-2 in the conformal field theory.

Abstract

We solve the monomer-dimer problem on a non-bipartite lattice, the simple quartic lattice with cylindrical boundary conditions, with a single monomer residing on the boundary. Due to the non-bipartite nature of the lattice, the well-known method of a Temperley bijection of solving single-monomer problems cannot be used. In this paper we derive the solution by mapping the problem onto one on close-packed dimers on a related lattice. Finite-size analysis of the solution is carried out. We find from asymptotic expansions of the free energy that the central charge in the logarithmic conformal field theory assumes the value $c=-2$.