Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

2d dimer model, correlation functions and combinatorics

Nicolas Allegra

Published Apr 5, 2016• 0 citations
OpenAlex

Abstract

In this present 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 ıt via the so-called height mapping which will be extensively used to interpret exact results. 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.

Related papers

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

Nicolas Allegra

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 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 predicts 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:1410.4131•Nuclear Physics B•Apr 5, 2015•20 citations

Yang–Baxter solution of dimers as a free-fermion six-vertex model

Paul A. Pearce, Alessandra Vittorini-Orgeas

Abstract It is shown that Dimers is Yang–Baxter integrable as a six-vertex model at the free-fermion point with crossing parameter λ = π 2 . A one-to-many mapping of vertices onto dimer configurations allows the free-fermion solutions to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by 45 ∘ compared to their usual orientation. This dimer model is exactly solvable in geometries of arbitrary finite size. In this paper, we establish and solve inversion identities for Dimers with periodic boundary conditions on the cylinder. In the particle representation, the local face tile operators give a representation of the fermion algebra, and the fermion particle trajectories play the role of nonlocal (logarithmic) degrees of freedom. In a suitable gauge, the dimer model is described by the Temperley–Lieb algebra with loop fugacity β = 2 cos ⁡ λ = 0 . At the isotropic point, the exact solution allows for the explicit counting of 45 ∘ rotated dimer configurations on a periodic M × N rectangular lattice. We show that the modular invariant partition function on the torus is the same as that of symplectic fermions and critical dense polymers. We also show that nontrivial Jordan cells appear for the dimer Hamiltonian on the strip with vacuum boundary conditions. We therefore argue that, in the continuum scaling limit, the dimer model gives rise to a logarithmic conformal field theory with central charge c = − 2 , minimal conformal weight Δ m i n = − 1 / 8 and effective central charge c e f f = 1 .

arXiv:1612.09477•Journal of Physics A Mathematical and Theoretical•Sep 29, 2017•8 citations

Classical dimers on the triangular lattice

Paul Fendley, Roderich Moessner, Shivaji L. Sondhi

We study the classical hard-core dimer model on the triangular lattice. Following Kasteleyn's fundamental theorem on planar graphs, this problem is soluble using Pfaffians. This model is particularly interesting for, unlike the dimer problems on the bipartite square and hexagonal lattices, its correlations are short ranged with a correlation length of less than one lattice constant. We compute the dimer-dimer and monomer-monomer correlators, and find that the model is deconfining: the monomer-monomer correlator falls off exponentially to a constant value $0.1494\dots{},$ only slightly below the nearest-neighbor value of $1/6.$ We also consider the anisotropic triangular lattice model in which the square lattice is perturbed by diagonal bonds of one orientation and small fugacity. We show that the model becomes noncritical immediately and that this perturbation is equivalent to adding a mass term to each of two Majorana fermions that are present in the long wavelength limit of the square lattice problem.

arXiv:cond-mat/0206159•Physical review. B, Condensed matter•Dec 13, 2002•141 citations

Classical dimers with aligning interactions on the square lattice

Fabien Alet, Yacine Ikhlef, Jesper Lykke Jacobsen +2 more

We present a detailed study of a model of close-packed dimers on the square lattice with an interaction between nearest-neighbor dimers. The interaction favors parallel alignment of dimers, resulting in a low-temperature crystalline phase. With large-scale Monte Carlo and transfer matrix calculations, we show that the crystal melts through a Kosterlitz-Thouless phase transition to give rise to a high-temperature critical phase, with algebraic decays of correlations functions with exponents that vary continuously with the temperature. We give a theoretical interpretation of these results by mapping the model to a Coulomb gas, whose coupling constant and associated exponents are calculated numerically with high precision. Introducing monomers is a marginal perturbation at the Kosterlitz-Thouless transition and gives rise to another critical line. We study this line numerically, showing that it is in the Ashkin-Teller universality class, and terminates in a tricritical point at finite temperature and monomer fugacity. In the course of this work, we also derive analytic results relevant to the noninteracting case of dimer coverings, including a Bethe ansatz (at the free fermion point) analysis, a detailed discussion of the effective height model, and a free field analysis of height fluctuations.

arXiv:cond-mat/0607747•Physical Review E•Oct 26, 2006•106 citations

Tensor renormalization group approach to classical dimer models

Krishanu Roychowdhury, Ching-Yu Huang

We analyze classical dimer models on a square and a triangular lattice using a tensor network representation of the dimers. The correlation functions are numerically calculated using the recently developed ``tensor renormalization group'' (TRG) technique. The partition function for the dimer problem can be calculated exactly by the Pfaffian method, which is used here as a platform for comparing the numerical results. The TRG approach turns out to be a powerful tool for describing gapped systems with exponentially decaying correlations very efficiently due to its fast convergence. This is the case for the dimer model on the triangular lattice. However, the convergence becomes very slow and unstable in the case of the square lattice where the model has algebraically decaying correlations. We highlight these aspects with numerical simulations and critically appraise the robustness of the TRG approach by contrasting the results for small and large system sizes against the exact calculations. Furthermore, we benchmark our TRG results with the classical Monte Carlo method.

arXiv:1412.1936•Physical Review B•May 14, 2015•7 citations

Derivation of height field theory for the two-dimensional classical dimer model from a Grassmann-integral representation

Stephen Powell

The classical dimer model on bipartite lattices hosts a Coulomb phase, characterized by algebraic correlations and topological order. Its long-wavelength properties can be described by the fluctuations of a vector field with zero divergence, which, in two dimensions, is equivalent to a continuum height model. We show how this field theory can be derived constructively for both square and honeycomb lattices, starting from an exact representation of the dimer model in terms of Grassmann integrals. Taking the continuum limit gives a massless Dirac fermion in two-dimensional Euclidean space, which, using bosonization in the path integral representation, maps exactly to the well-known height field theory, incorporating the relationship between its boundary discontinuity (or "tilt") and the flux. By including source terms coupling to the flux density and the local valence-bond-solid order parameter, which we show are the only ones required to describe the asymptotic long-distance correlations, we derive expressions for the dimer observables in terms of the height.

arXiv:2606.17154•Jun 15, 2026•0 citations
In this present 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 ıt via the so-called height mapping which will be extensively used to interpret exact results. 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.
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 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 predicts 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.
Abstract It is shown that Dimers is Yang–Baxter integrable as a six-vertex model at the free-fermion point with crossing parameter λ = π 2 . A one-to-many mapping of vertices onto dimer configurations allows the free-fermion solutions to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by 45 ∘ compared to their usual orientation. This dimer model is exactly solvable in geometries of arbitrary finite size. In this paper, we establish and solve inversion identities for Dimers with periodic boundary conditions on the cylinder. In the particle representation, the local face tile operators give a representation of the fermion algebra, and the fermion particle trajectories play the role of nonlocal (logarithmic) degrees of freedom. In a suitable gauge, the dimer model is described by the Temperley–Lieb algebra with loop fugacity β = 2 cos ⁡ λ = 0 . At the isotropic point, the exact solution allows for the explicit counting of 45 ∘ rotated dimer configurations on a periodic M × N rectangular lattice. We show that the modular invariant partition function on the torus is the same as that of symplectic fermions and critical dense polymers. We also show that nontrivial Jordan cells appear for the dimer Hamiltonian on the strip with vacuum boundary conditions. We therefore argue that, in the continuum scaling limit, the dimer model gives rise to a logarithmic conformal field theory with central charge c = − 2 , minimal conformal weight Δ m i n = − 1 / 8 and effective central charge c e f f = 1 .
We study the classical hard-core dimer model on the triangular lattice. Following Kasteleyn's fundamental theorem on planar graphs, this problem is soluble using Pfaffians. This model is particularly interesting for, unlike the dimer problems on the bipartite square and hexagonal lattices, its correlations are short ranged with a correlation length of less than one lattice constant. We compute the dimer-dimer and monomer-monomer correlators, and find that the model is deconfining: the monomer-monomer correlator falls off exponentially to a constant value 0.1494…,0.1494\dots{},0.1494…, only slightly below the nearest-neighbor value of 1/6.1/6.1/6. We also consider the anisotropic triangular lattice model in which the square lattice is perturbed by diagonal bonds of one orientation and small fugacity. We show that the model becomes noncritical immediately and that this perturbation is equivalent to adding a mass term to each of two Majorana fermions that are present in the long wavelength limit of the square lattice problem.
We present a detailed study of a model of close-packed dimers on the square lattice with an interaction between nearest-neighbor dimers. The interaction favors parallel alignment of dimers, resulting in a low-temperature crystalline phase. With large-scale Monte Carlo and transfer matrix calculations, we show that the crystal melts through a Kosterlitz-Thouless phase transition to give rise to a high-temperature critical phase, with algebraic decays of correlations functions with exponents that vary continuously with the temperature. We give a theoretical interpretation of these results by mapping the model to a Coulomb gas, whose coupling constant and associated exponents are calculated numerically with high precision. Introducing monomers is a marginal perturbation at the Kosterlitz-Thouless transition and gives rise to another critical line. We study this line numerically, showing that it is in the Ashkin-Teller universality class, and terminates in a tricritical point at finite temperature and monomer fugacity. In the course of this work, we also derive analytic results relevant to the noninteracting case of dimer coverings, including a Bethe ansatz (at the free fermion point) analysis, a detailed discussion of the effective height model, and a free field analysis of height fluctuations.
We analyze classical dimer models on a square and a triangular lattice using a tensor network representation of the dimers. The correlation functions are numerically calculated using the recently developed ``tensor renormalization group'' (TRG) technique. The partition function for the dimer problem can be calculated exactly by the Pfaffian method, which is used here as a platform for comparing the numerical results. The TRG approach turns out to be a powerful tool for describing gapped systems with exponentially decaying correlations very efficiently due to its fast convergence. This is the case for the dimer model on the triangular lattice. However, the convergence becomes very slow and unstable in the case of the square lattice where the model has algebraically decaying correlations. We highlight these aspects with numerical simulations and critically appraise the robustness of the TRG approach by contrasting the results for small and large system sizes against the exact calculations. Furthermore, we benchmark our TRG results with the classical Monte Carlo method.
The classical dimer model on bipartite lattices hosts a Coulomb phase, characterized by algebraic correlations and topological order. Its long-wavelength properties can be described by the fluctuations of a vector field with zero divergence, which, in two dimensions, is equivalent to a continuum height model. We show how this field theory can be derived constructively for both square and honeycomb lattices, starting from an exact representation of the dimer model in terms of Grassmann integrals. Taking the continuum limit gives a massless Dirac fermion in two-dimensional Euclidean space, which, using bosonization in the path integral representation, maps exactly to the well-known height field theory, incorporating the relationship between its boundary discontinuity (or "tilt") and the flux. By including source terms coupling to the flux density and the local valence-bond-solid order parameter, which we show are the only ones required to describe the asymptotic long-distance correlations, we derive expressions for the dimer observables in terms of the height.