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Physical review. B, Condensed matter

Equivalences between a dimer, a vertex, and a spin model

Roberto Nicolau Onody, V. Kurak

Published Sep 1, 1988• 2 citations
Publisher

Abstract

It is shown that a recently introduced anisotropic dimer model with domain-wall behavior can be mapped into a staggered free-fermion six-vertex model. The associated spin model with ferromagnetic and antiferromagnetic couplings and interactions between first and diagonal neighbors on a square lattice is also given.
DOI
10.1103/physrevb.38.5061

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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 .
Abstract The dimer model on a strip is considered as a Yang–Baxter integrable six vertex model at the free-fermion point with crossing parameter and quantum group invariant boundary conditions. A one-to-many mapping of vertex onto dimer configurations allows for the solution of the free-fermion model to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by compared to their usual orientation. In a suitable gauge, the dimer model is described by the Temperley–Lieb algebra with loop fugacity . It follows that the model is exactly solvable in geometries of arbitrary finite size. We establish and solve transfer matrix inversion identities on the strip with arbitrary finite width N . In the continuum scaling limit, in sectors with magnetization S z , we obtain the conformal weights where . We further show that the corresponding finitized characters decompose into sums of q -Narayana numbers or, equivalently, skew q -binomials. 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 degrees of freedom. We argue that, in the continuum scaling limit, there exist nontrivial Jordan blocks of rank 2 in the Virasoro dilatation operator L 0 . This confirms that, with quantum group invariant boundary conditions, the dimer model gives rise to a logarithmic conformal field theory with central charge c = −2, minimal conformal weight and effective central charge . Our analysis of the structure of the ensuing rank 2 modules indicates that the familiar staggered c = −2 modules appear as submodules.
Analyses are given for two three-dimensional lattice systems: A system of close-packed dimers placed in layers of honeycomb lattices and a layered triangular-lattice interacting domain wall model, both with nontrivial interlayer interactions. We show that both models are equivalent to a five-vertex model on the square lattice with interlayer vertex-vertex interactions. Using the method of Bethe ansatz, a closed-form expression for the free energy is obtained and analyzed. We deduce the exact phase diagram and determine the nature of the phase transitions determined as a function of the strength of the interlayer interaction.
Motivated by recent interest in 2+1 dimensional quantum dimer models, we revisit Fisher's mapping of two-dimensional Ising models to hardcore dimer models. First, we note that the symmetry breaking transition of the ferromagnetic Ising model maps onto a non-symmetry breaking transition in dimer language---instead it becomes a deconfinement transition for test monomers. Next, we introduce a modification of Fisher's mapping in which a second dimer model, also equivalent to the Ising model, is defined on a generically different lattice derived from the dual. In contrast to Fisher's original mapping, this enables us to reformulate frustrated Ising models as dimer models with positive weights and we illustrate this by providing a new solution of the fully frustrated Ising model on the square lattice. Finally, by means of the modified mapping we show that a large class of three-dimensional Ising models are precisely equivalent, in the time continuum limit, to particular quantum dimer models. As Ising models in three dimensions are dual to Ising gauge theories, this further yields an exact map between the latter and the quantum dimer models. The paramagnetic phase in Ising language maps onto a deconfined, topologically ordered phase in the dimer models. Using this set of ideas, we also construct an exactly soluble quantum eight vertex model.