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arXiv:2003.12084 · Physical review. B./Physical review. B

Exact solution and correlations of a dimer model on the checkerboard lattice

Julia Wildeboer, Zohar Nussinov, Alexander Seidel

Published Jul 6, 2020• 6 citations
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Abstract

We present analytic results for a special dimer model on the nonbipartite and nonplanar checkerboard lattice that does not allow for parallel dimers surrounding diagonal links. We exactly calculate the number of closed packed dimer coverings on finite checkerboard lattices under periodic boundary conditions and determine all dimer-dimer correlations. The latter are found to vanish beyond a certain distance. We find that this solvable model, despite being nonplanar, is in close kinship with well-known paradigm-setting planar counterparts that allow exact mappings to ${\mathbb{Z}}_{2}$ lattice gauge theory.

DOI
10.1103/physrevb.102.020401

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We present analytic results for a special dimer model on the nonbipartite and nonplanar checkerboard lattice that does not allow for parallel dimers surrounding diagonal links. We exactly calculate the number of closed packed dimer coverings on finite checkerboard lattices under periodic boundary conditions and determine all dimer-dimer correlations. The latter are found to vanish beyond a certain distance. We find that this solvable model, despite being nonplanar, is in close kinship with well-known paradigm-setting planar counterparts that allow exact mappings to Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theory.
We present the exact closed-form expression for the partition function of a dimer model on a generalized finite checkerboard rectangular lattice under periodic boundary conditions. We investigate three different sets of dimer weights, each with different critical behaviors. We then consider different limits for the model on the three lattices. In one limit, the model for each of the three lattices is reduced to the dimer model on a rectangular lattice, which belongs to the c=-2 universality class. In another limit, two of the lattices reduce to the anisotropic Kasteleyn model on a honeycomb lattice, the universality class of which is given by c=1. The result that the dimer model on a generalized checkerboard rectangular lattice can manifest different critical behaviors is consistent with early studies in the thermodynamic limit and also provides insight into corrections to scaling arising from the finite-size versions of the model.
Exactly solvable non-planar Z2\mathbb{Z}_{2}Z2​ dimer liquids on checkerboard and ruby lattices
We generalize an influential framework for exactly solvable quantum dimer models with dual Ising gauge theory descriptions realizing the Z2\mathbb{Z}_{2}Z2​ topological phase from trivalent to tetravalent parent lattices. The resulting quantum dimer models live on crossed-medial lattices and possess the remarkable feature of having crossed plaquettes, leading to transition graphs with generically intersecting loops. This non-planar structure runs counter to established templates for analytically tractable dimer liquids. As the simplest realization of the construction, we introduce an exactly solvable quantum dimer model on the checkerboard lattice that allows an exact mapping to the toric code, thus providing a particularly direct connection between the latter and Anderson's short range resonating valence bond paradigm. We further show that a corresponding crossed ruby lattice construction, dual to an Ising gauge theory on the kagome lattice, naturally falls within the same framework. More generally, the construction gives rise to a broad class of exactly solvable crossed-medial quantum dimer models and admits natural iteration, generating cascades of solvable Z2\mathbb{Z}_{2}Z2​ topological dimer liquids beyond the standard planar setting. We furthermore extend Kasteleyn methods to the relevant non-planar graphs, enabling controlled wave-function deformations away from the commuting-projector points while retaining efficient evaluation of correlation functions.
Using the tensor network approach, we investigate the monomer-dimer models on a checkerboard lattice, in which there are interactions (with strength vvv) between the parallel dimers on half of the plaquettes. For the fully packed interacting dimer model, we observe a Kosterlitz-Thouless (KT) transition between the low-temperature symmetry breaking and the high-temperature critical phases; for the doped monomer-dimer case with finite chemical potential $\ensuremath{\mu}$, we also find an order-disorder phase transition which is of second order instead. We use the boundary matrix product state approach to detect the KT and second-order phase transitions and obtain the phase diagrams $v\text{\ensuremath{-}}T$ and $\ensuremath{\mu}\text{\ensuremath{-}}T$. Moreover, for the noninteracting monomer-dimer model (setting $\ensuremath{\mu}=\ensuremath{\nu}=0$), we get an extraordinarily accurate determination of the free energy per site (negative of the monomer-dimer constant h2{h}_{2}h2​) as $f=\ensuremath{-}0.662\phantom{\rule{0.16em}{0ex}}798\phantom{\rule{0.16em}{0ex}}972\phantom{\rule{0.16em}{0ex}}833\phantom{\rule{0.16em}{0ex}}746$ with the dimer density n=0.638123109228547n=0.638\phantom{\rule{0.16em}{0ex}}123\phantom{\rule{0.16em}{0ex}}109\phantom{\rule{0.16em}{0ex}}228\phantom{\rule{0.16em}{0ex}}547n=0.638123109228547, both of 15 correct digits.
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 classical hard-core dimer models on the square lattice with links extending beyond nearest-neighbors. Numerically, using a directed-loop Monte Carlo algorithm, we find that, in the presence of longer dimers preserving the bipartite graph structure, algebraic correlations persist. While the confinement exponent for monomers drifts, the leading decay of dimer correlations remains 1∕r21∕{r}^{2}1∕r2, although the logarithmic peaks present in the dimer structure factor of the nearest-neighbor model vanish. By contrast, an arbitrarily small fraction of next-nearest-neighbor dimers leads to the onset of exponential dimer correlations and deconfinement. We discuss these results in the framework of effective theories, and provide an approximate but accurate analytical expression for the dimer correlations.
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.