Exact solution of the $2d$ dimer model: Corner free energy, correlation functions and combinatorics

Abstract

In this work, some classical results of the pfaffian theory of the dimer model based on the work of Kasteleyn, Fisher and Temperley are introduced in a fermionic framework. Then we shall detail the bosonic formulation of the model {\it via} the so-called height mapping and the nature of boundary conditions is unravelled. The complete and detailed fermionic solution of the dimer model on the square lattice with an arbitrary number of monomers is presented, and finite size effect analysis is performed to study surface and corner effects, leading to the extrapolation of the central charge of the model. The solution allows for exact calculations of monomer and dimer correlation functions in the discrete level and the scaling behavior can be inferred in order to find the set of scaling dimensions and compare to the bosonic theory which predict particular features concerning corner behaviors. Finally, some combinatorial and numerical properties of partition functions with boundary monomers are discussed, proved and checked with enumeration algorithms.

arXiv commentFinal version to be published in Nuclear Physics B (53 pages and a lot of figures)

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• Inhomogeneous field theory inside the arctic circle
• Integrability and conformal data of the dimer model
• Finite-size corrections and scaling for the dimer model on the checkerboard lattice
• Yang-Baxter Solution of Dimers as a Free-Fermion Six-Vertex Model
• Exact finite-size corrections in the dimer model on a planar square lattice
• Asymptotic Freedom at the Berezinskii-Kosterlitz-Thouless Transition without Fine-Tuning Using a Qubit Regularization
• Yang-Baxter Integrable Dimers on a Strip
• Pfaffian Correlation Functions of Planar Dimer Covers
• Derivation of field theory for the classical dimer model using bosonization
• Recurrence solution of monomer-polymer models on two-dimensional rectangular lattices
• Virasoro and Kac-Moody algebra in generic tensor network representations of 2d critical lattice partition functions
• Finite-size scaling of free energy in the dimer model on a hexagonal domain
• The expectation value of the number of loops and the left-passage probability in the double-dimer model
• Group Convolutional Neural Network for the Low-Energy Spectrum in the Quantum Dimer Model
• Exact finite-size corrections in the dimer model on a cylinder