Abstract
We solve the classical square-lattice dimer model with periodic boundaries and in the presence of a field 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 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.
Highlights
The dimer model is a paradigmatic example of a stronglycorrelated system, in which dimers cover the edges of a lattice subject to a close-packing constraint, i.e., each vertex touches exactly one dimer
Fisher and Stephenson’s Pfaffian calculation of the monomer distribution function in 1963 [9] implies that, due to the entropy of the background dimer configuration, the monomers interact through an effective Coulomb potential, which is logarithmic in two dimensions
We provide a general framework for the calculation of expectation values and explicitly calculate dimer occupation numbers, dimer–dimer correlation functions and the monomer distribution function
Summary
The dimer model is a paradigmatic example of a stronglycorrelated system, in which dimers cover the edges of a lattice subject to a close-packing constraint, i.e., each vertex touches exactly one dimer. Fisher and Stephenson’s Pfaffian calculation of the monomer distribution function in 1963 [9] implies that, due to the entropy of the background dimer configuration, the monomers interact through an effective Coulomb potential, which is logarithmic in two dimensions They have shown that dimer–dimer correlations are long-range with algebraic, rather than exponential, dependence on separation. Perhaps a more elegant solution of the dimer model is Lieb’s transfer-matrix method [10], analogous to the well-known solution of the Ising model by Schultz et al [11], which maps the problem to free fermions In this approach, the partition function is expressed in terms of a transfer matrix, which, given a configuration on a row of vertical bonds, generates all dimer configurations compatible with the close-packing constraint on the subsequent row of horizontal and vertical bonds.
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