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arXiv:0705.4550 · Physical Review E

Criticality of a classical dimer model on the triangular lattice

Fabien Trousselet, Pierre Pujol, Fabien Alet, Didier Poilblanc

Published Oct 19, 2007• 10 citations
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Abstract

We consider a classical interacting dimer model which interpolates between the square lattice case and the triangular lattice case by tuning a chemical potential in the diagonal bonds. The interaction energy simply corresponds to the number of plaquettes with parallel dimers. Using transfer matrix calculations, we find in the anisotropic triangular case a succession of different physical phases as the interaction strength is increased: a short-range disordered liquid dimer phase at low interactions, then a critical phase similar to the one found for the square lattice, and finally a transition to an ordered columnar phase for large interactions. Our results indicate that criticality and nonbipartiteness are compatible in a dimer model. For the isotropic triangular case, we have indications that the system undergoes a first-order phase transition to an ordered phase, without appearance of an intermediate critical phase.

DOI
10.1103/physreve.76.041125

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We consider a classical interacting dimer model which interpolates between the square lattice case and the triangular lattice case by tuning a chemical potential in the diagonal bonds. The interaction energy simply corresponds to the number of plaquettes with parallel dimers. Using transfer matrix calculations, we find in the anisotropic triangular case a succession of different physical phases as the interaction strength is increased: a short-range disordered liquid dimer phase at low interactions, then a critical phase similar to the one found for the square lattice, and finally a transition to an ordered columnar phase for large interactions. Our results indicate that criticality and nonbipartiteness are compatible in a dimer model. For the isotropic triangular case, we have indications that the system undergoes a first-order phase transition to an ordered phase, without appearance of an intermediate critical phase.
The plaquette phase of the square-lattice quantum dimer model is studied using a continuous-time reptation quantum Monte Carlo method for lattices of sizes up to $48\ifmmode\times\else\texttimes\fi{}48$ sites. We determine the location of the phase transition between the columnar and plaquette phases to occur at ${V}_{c}∕J=0.60\ifmmode\pm\else\textpm\fi{}0.05$ which is significantly larger than inferred from previous exact diagonalization studies on smaller lattices. Off diagonal correlation functions are obtained. They exhibit long-range order in the plaquette phase but not at the Rokhsar-Kivelson point. We also observe significant finite-size corrections to scaling for the transition between the plaquette phase and the critical resonating valence bond liquid. This study demonstrates the importance of understanding finite-size effects when considering critical properties of the square-lattice quantum dimer model.
We study a model of close-packed dimers on the square lattice with a nearest neighbor interaction between parallel dimers. This model corresponds to the classical limit of quantum dimer models [D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988)]. By means of Monte Carlo and transfer matrix calculations, we show that this system undergoes a Kosterlitz-Thouless transition separating a low temperature ordered phase where dimers are aligned in columns from a high temperature critical phase with continuously varying exponents. This is understood by constructing the corresponding Coulomb gas, whose coupling constant is computed numerically. We also discuss doped models and implications on the finite-temperature phase diagram of quantum dimer models.
Abstract The critical behavior of a dimer model with an interaction favoring parallel dimers in each plaquette of the square lattice is studied numerically using the corner transfer matrix renormalization group algorithm. The critical exponents are known to depend on the chemical potential of vacancies, or monomers. At large average density of the latter, the phase transition becomes the first-order. We compute the scaling dimensions of both the order parameter and temperature in the second-order regime and compare them with the conjecture that the critical behavior is the same as the Ashkin–Teller model on its self-dual critical line.
We study the spin-1/2 and spin-1 [Formula: see text] Heisenberg antiferromagnets (HAFs) on an infinite, anisotropic, two-dimensional triangular lattice, using the coupled cluster method. With respect to an underlying square-lattice geometry the model contains antiferromagnetic (J1 > 0) bonds between nearest neighbours and competing [Formula: see text] bonds between next-nearest-neighbours across only one of the diagonals of each square plaquette, the same diagonal in each square. In a topologically equivalent triangular-lattice geometry the model has two sorts of nearest-neighbour bonds, with [Formula: see text] bonds along parallel chains and with J1 bonds providing an interchain coupling. The model thus interpolates between an isotropic HAF on the square lattice at one extreme (κ = 0) and a set of decoupled chains at the other (κ → ∞), with the isotropic HAF on the triangular lattice in between at κ = 1. For the spin-1/2 [Formula: see text] model, we find a weakly first-order (or possibly second-order) quantum phase transition from a Néel-ordered state to a helical state at a first critical point at κc1 = 0.80 ± 0.01, and a second critical point at κc2 = 1.8 ± 0.4 where a first-order transition occurs between the helical state and a collinear stripe-ordered state. For the corresponding spin-1 model we find an analogous transition of the second-order type at κc1 = 0.62 ± 0.01 between states with Néel and helical ordering, but we find no evidence of a further transition in this case to a stripe-ordered phase.
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.
We discuss phase transitions and the phase diagram of a classical dimer model with anisotropic interactions defined on a square lattice. For the attractive region, the perturbation of the orientational order parameter introduced by the anisotropy causes the Berezinskii-Kosterlitz-Thouless transitions from a dimer-liquid to columnar phases. According to the discussion by Nomura and Okamoto for a quantum-spin chain system [J. Phys. A 27, 5773 (1994)], we proffer criteria to determine transition points and also universal level-splitting conditions. Subsequently, we perform numerical diagonalization calculations of the nonsymmetric real transfer matrices up to linear dimension specified by L=20 and determine the global phase diagram. For the repulsive region, we find the boundary between the dimer-liquid and the strong repulsion phases. Based on the dispersion relation of the one-string motion, which exhibits a twofold "zero-energy flat band" in the strong repulsion limit, we give an intuitive account for the property of the strong repulsion phase.