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

Abstract

We solve the classical square-lattice dimer model with periodic boundaries and in the presence of a field $\mathbf{t}$ 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 $\mathbf{t}$-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.

arXiv comment16 pages, 4 figures, 1 table

References in corpus (15)

• Bosonization for Beginners --- Refermionization for Experts
• The "Coulomb phase" in frustrated systems
• Spin Ice, Fractionalization and Topological Order
• Coulomb and Liquid Dimer Models in Three Dimensions
• Classical dimers on the triangular lattice
• Interacting classical dimers on the square lattice
• Classical dimers with aligning interactions on the square lattice
• Properties of Resonating-Valence-Bond Spin Liquids and Critical Dimer Models
• Zero-Temperature Phase Transitions of Antiferromagnetic Ising Model of General Spin on a Triangular Lattice
• Dimer statistics on the Möbius strip and the Klein bottle
• Asymptotics of block Toeplitz determinants and the classical dimer model
• Integrability and conformal data of the dimer model
• Dimer representations of the Temperley-Lieb algebra
• Refined conformal spectra in the dimer model
• Loop statistics in the toroidal honeycomb dimer model

Cited by in corpus (4)

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• Derivation of field theory for the classical dimer model using bosonization
• Trivalent network model for d${}^3$ transition metal dichalcogenides in the 1T structure: Holography from local constraints
• Entropy of fully-packed rigid rods on generalized Husimi trees: a route to the square lattice limit