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

arXiv:cond-mat/9904343 · Physical Review Letters

Triangular Trimers on the Triangular Lattice: An Exact Solution

Alain Verberkmoes, Bernard Nienhuis

Published Nov 15, 1999• 47 citations
arXivHTMLPDF

Abstract

A model, consisting of triangular trimers on the triangular lattice is presented. In analogy to the dimer problem, these particles cover the lattice completely without overlap. The model has a honeycomb structure of hexagonal cells separated by rigid domain walls. The transfer matrix can be diagonalized by a Bethe Ansatz with two types of particles. This leads to an exact expression for the entropy on a two-dimensional subset of the parameter space.

DOI
10.1103/physrevlett.83.3986

Related papers

Bethe ansatz solution of triangular trimers on the triangular lattice

Alain Verberkmoes, Bernard Nienhuis

Recently, a model consisting of triangular trimers covering the triangular lattice was introduced and its exact free energy given. In this paper we present the complete calculation leading to this exact result. The solution involves a coordinate Bethe ansatz with two kinds of particles. It is similar to that of the square-triangle random tiling model by Widom and Kalugin. The connection of the trimer model with related solvable models is discussed.

arXiv:cond-mat/9909068•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•May 24, 2001•11 citations

Lattice statistics in three dimensions: Solution of layered dimer and layered domain wall models

Vladislav Popkov, Doochul Kim, H. Y. Huang +1 more

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.

arXiv:cond-mat/9703065•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Oct 1, 1997•4 citations

Phase diagram of the hexagonal lattice quantum dimer model

Roderich Moessner, Shivaji L. Sondhi, Premala Chandra

We discuss the phase diagram of the quantum dimer model on the hexagonal (honeycomb) lattice. In addition to the columnar and staggered valence-bond solids which have been discussed in previous work, we establish the existence of a plaquette valence-bond solid. The transition between the plaquette and columnar phases at $v/t=\ensuremath{-}0.2\ifmmode\pm\else\textpm\fi{}0.05$ is argued to be first order. We note that this model should describe valence-bond-dominated phases of frustrated Heisenberg models on the hexagonal lattice and discuss its relation to recent exact diagonalization work by Fouet et al. on the ${J}_{1}\ensuremath{-}{J}_{2}$ model on the same lattice. Our results also shed light on the properties of the transverse field Ising antiferromagnet on the triangular lattice and the classical Ising antiferromagnet on the stacked triangular lattice, which are related to dimer models by duality.

arXiv:cond-mat/0106288•Physical review. B, Condensed matter•Sep 19, 2001•153 citations

Interacting dimers on the honeycomb lattice: an exact solution of the five-vertex model

H. Y. Huang, Fa Yueh Wu, Kunz, H. +1 more

The problem of close-packed dimers on the honeycomb lattice was solved by Kasteleyn in 1963. Here we extend the solution to include interactions between neighboring dimers in two spatial lattice directions. The solution is obtained by using the method of Bethe ansatz and by converting the dimer problem into a five-vertex problem. The complete phase diagram is obtained and it is found that a new frozen phase, in which the attracting dimers prevail, arises when the interaction is attractive. For repulsive dimer interactions a new first-order line separating two frozen phases occurs. The transitions are continuous and the critical behavior in the disorder regime is found to be the same as in the case of noninteracting dimers characterized by a specific heat exponent α = 12.

•Jan 1, 1996•18 citations

Ising Model on a Triangular Lattice with Three-spin Interactions. I. Free Energy and Correlation Length

R. J. Baxter

Following the demonstration in Part I that the Ising mogel with three-spin interactions on a triangular lattice is equivalent to a colouring problem on a hexagonal lattice, and that a generalized Bethe ansatz can be used to obtain equations for the eigenv

Australian Journal of Physics•Jun 1, 1974•52 citations

EXACT BOUNDARY MAGNETIZATION OF THE LAYERED ISING MODEL ON TRIANGULAR AND HONEYCOMB LATTICES

Alessandro Pelizzola

In the present paper we extend a previous exact calculation of the boundary magnetization of the square lattice layered Ising model, based on a transfer-matrix effective-field technique, to the triangular lattice. The result is then further extended to the honeycomb lattice by means of a star-triangle transformation. Finally, the (11) boundary of the square lattice is analyzed as a particular case of the triangular and honeycomb lattices.

Modern Physics Letters B•Feb 28, 1996•2 citations
A model, consisting of triangular trimers on the triangular lattice is presented. In analogy to the dimer problem, these particles cover the lattice completely without overlap. The model has a honeycomb structure of hexagonal cells separated by rigid domain walls. The transfer matrix can be diagonalized by a Bethe Ansatz with two types of particles. This leads to an exact expression for the entropy on a two-dimensional subset of the parameter space.
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.
We discuss the phase diagram of the quantum dimer model on the hexagonal (honeycomb) lattice. In addition to the columnar and staggered valence-bond solids which have been discussed in previous work, we establish the existence of a plaquette valence-bond solid. The transition between the plaquette and columnar phases at $v/t=\ensuremath{-}0.2\ifmmode\pm\else\textpm\fi{}0.05$ is argued to be first order. We note that this model should describe valence-bond-dominated phases of frustrated Heisenberg models on the hexagonal lattice and discuss its relation to recent exact diagonalization work by Fouet et al. on the ${J}_{1}\ensuremath{-}{J}_{2}$ model on the same lattice. Our results also shed light on the properties of the transverse field Ising antiferromagnet on the triangular lattice and the classical Ising antiferromagnet on the stacked triangular lattice, which are related to dimer models by duality.
The problem of close-packed dimers on the honeycomb lattice was solved by Kasteleyn in 1963. Here we extend the solution to include interactions between neighboring dimers in two spatial lattice directions. The solution is obtained by using the method of Bethe ansatz and by converting the dimer problem into a five-vertex problem. The complete phase diagram is obtained and it is found that a new frozen phase, in which the attracting dimers prevail, arises when the interaction is attractive. For repulsive dimer interactions a new first-order line separating two frozen phases occurs. The transitions are continuous and the critical behavior in the disorder regime is found to be the same as in the case of noninteracting dimers characterized by a specific heat exponent α = 12.