Topological sectors, dimer correlations and monomers from the transfer-matrix solution of the dimer model

TL;DR

This paper provides an exact solution to the classical square-lattice dimer model with a flux field, revealing topological sectors, explicit correlation functions, and monomer distributions, confirming previous Pfaffian results.

Contribution

We diagonalize a modified transfer matrix to solve the dimer model, explicitly derive topological sectors, and compute correlation functions and monomer distributions.

Findings

Partition function divided into topological sectors by flux.

Explicit formulas for dimer correlations and monomer distributions.

Asymptotic behavior of monomer correlations analyzed using Fisher-Hartwig conjecture.

Abstract

We solve the classical square-lattice dimer model with periodic boundaries and in the presence of a field $\boldsymbol{t}$ that couples to the (vector) flux, by diagonalizing a modified version of Lieb's transfer matrix. After deriving the torus partition function in the thermodynamic limit, we show how the configuration space divides into 'topological sectors' corresponding to distinct values of the flux. Additionally, we demonstrate in general that expectation values are $\boldsymbol{t}$-independent at leading order, and obtain explicit expressions for dimer occupation numbers, dimer-dimer correlation functions and the monomer distribution function. The last of these is expressed as a Toeplitz determinant, whose asymptotic behavior for large monomer separation is tractable using the Fisher-Hartwig conjecture. Our results reproduce those previously obtained using Pfaffian techniques.