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arXiv:2106.09674

The Breaking of Geometric Constraint of Classical Dimers on the Square Lattice

Hongxu Yao, Jiaze Li, Jintao Hou

Published Jun 17, 2021• 0 citations
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Abstract

We study a model of two-dimensional classical dimers on the square lattice with strong geometric constraints (there is exactly one bond with the nearest point for every point in the lattice). This model corresponds to the quantum dimer model suggested by D.S. Rokhsar and S.A. Kivelson (1988). We use the directed-loop algorithm to show the system undergoes a Berezinskii-Kostelitz Thousless transition (BKT transition) in finite temperatures. After that, if we destroy the geometric constraint of dimers, the topological transition will transfer to a quasi one-order transition. For the dimer updates, we also introduce a new cluster updating algorithm called the edged cluster algorithm. By this method, we succeed in rapidly traversing the winding (topological) sections uniformly and widening the effective matrical ensemble to include more topological sections.

DOI
10.48550/arxiv.2106.09674

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We study a model of two-dimensional classical dimers on the square lattice with strong geometric constraints (there is exactly one bond with the nearest point for every point in the lattice). This model corresponds to the quantum dimer model suggested by D.S. Rokhsar and S.A. Kivelson (1988). We use the directed-loop algorithm to show the system undergoes a Berezinskii-Kostelitz Thousless transition (BKT transition) in finite temperatures. After that, if we destroy the geometric constraint of dimers, the topological transition will transfer to a quasi one-order transition. For the dimer updates, we also introduce a new cluster updating algorithm called the edged cluster algorithm. By this method, we succeed in rapidly traversing the winding (topological) sections uniformly and widening the effective matrical ensemble to include more topological sections.
We present a quantum Monte Carlo investigation of the finite-temperature phase diagram of the quantum dimer model on the square lattice. We use the sweeping cluster algorithm, which allows the exact implementation of the dimer constraint, supplemented with an equal-time directed loop move that allows sampling the winding sectors. We find a high-temperature critical phase with power-law correlations that extends down to the Rokhsar-Kivelson point, in the vicinity of which a reentrance effect in the lines of constant exponent is found. For small values of the kinetic energy strength, we find finite-temperature transitions to ordered states (columnar and staggered) which match those of interacting classical dimer models.
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.
The quantum dimer model has been continuously discussed both theoretically and numerically.Numerical methods have been developed on the quantum dimer model, but efficiency has been an essential problem due to the strong geometry constraint of the model.Recently, sweeping cluster update based on stochastic series expansion has shown power on quantum Monte Carlo on quantum dimer model.In this thesis, we propose improvements to the sweeping cluster update algorithm, which can reduce the time complexity by smartly choosing the next operator to update on.We also show a general framework to extend the current method to a broader range of models, including dimermonomer and monomer-monomer operators.As a benchmark of our algorithm, we simulate the quantum dimer model at finite temperatures in a wide region of parameters.By measuring several order parameters by the quantum Monte Carlo and using the finitescaling analysis, the phase boundary of the quantum dimer model on the square lattice can be estimated numerically, and the phase diagram in V /J < 1 region is determined accurately.We find evidence to support the expectation of previous research that the phase transition at finite temperature of quantum dimer model shows similar behavior to the Kosterlitz-Thouless transition in the classical dimer model.The intermediate plaquette phase may not exist, and the columnar phase in the classical model extends up to the Rokhsar-Kivelson point.iii This thesis is supported by Prof. Synge Todo and Prof. Hidemaro Suwa.I greatly appreciate the discussions we had during my doctoral years with them.Many ideas have been discussed in the meetings and then applied to the model for interesting numerical results.I also appreciated
Quantum dimer models~(QDMs) featured by strong geometric constraint are effective low energy descriptions of many quantum spin systems. The geometric restriction described by local gauge field, hinders the application of numerical algorithms. Before sweeping cluster method was applied in world-line quantum Monte Carlo (QMC) algorithm, there is only projector QMC which obey the constraints and could be used for calculation on QDMs. However, the projector QMC for QDMs has some drawbacks, e.g., it is not effective when the parameter interval away from Rokhsar-Kivelson (RK) point. That's because the projector method still lacks a cluster update to improve its efficiency. Although sweeping cluster algorithm improves the update for these systems, it also only works in one winding (topological) sector. In this paper, we improve the sweeping cluster method to sample in different winding sectors.
The quantum dimer and loop models attract great attention, partially because the fundamental importance in the phases and phase transitions emerging in these prototypical constrained systems, and partially due to their intimate relevance toward the on-going experiments on Rydberg atom arrays in which the blockade mechanism naturally enforces the local constraint. Here we show, by means of the sweeping cluster quantum Monte Carlo method, the complete ground state phase diagram of the fully packed quantum loop model on the square lattice. We find between the lattice nematic (LN) phase with strong dimer attraction and the staggered phase (SP) with strong dimer repulsion, there emerges a resonating plaquette (RP) phase with off-diagonal translational symmetry breaking. Such a quantum phase is separated from the LN via a first order transition and from the SP by the famous Rokhsar-Kivelson point. Our renormalization group analysis reveals the different flow directions, fully consistent with the order parameter histogram in Monte Carlo simulations. The realization and implication of our phase diagram in Rydberg experiments are proposed.
Using exact diagonalizations and Green's function Monte Carlo simulations, we have studied the zero-temperature properties of the quantum dimer model on the triangular lattice on clusters with up to 588 sites. A detailed comparison of the properties in different topological sectors as a function of the cluster size and for different cluster shapes has allowed us to identify different phases, to show explicitly the presence of topological degeneracy in a phase close to the Rokhsar-Kivelson point, and to understand finite-size effects inside this phase. The nature of the various phases has been further investigated by calculating dimer-dimer correlation functions. The present results confirm and complement the phase diagram proposed by Moessner and Sondhi on the basis of finite-temperature simulations [Phys. Rev. Lett. 86, 1881 (2001)].