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Improvement of numerical algorithm for quantum dimer models

Ruixiao Cao

Published Sep 22, 2023• 0 citations

Abstract

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

DOI: 10.15083/0002012194

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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.

A continuous-time formulation of the diffusion Monte Carlo method for lattice models is presented. In its simplest version, without the explicit use of trial wave functions for importance sampling, the method is an excellent tool for investigating quantum lattice models in parameter regions close to generalized Rokhsar-Kivelson points. This is illustrated by showing results for the quantum dimer model on both triangular and square lattices. The potential energy of two test monomers as a function of their separation is computed at zero temperature. The existence of deconfined monomers in the triangular lattice is confirmed. The method allows also the study of dynamic monomers. A finite fraction of dynamic monomers is found to destroy the confined phase on the square lattice when the hopping parameter increases beyond a finite critical value. The phase boundary between the monomer confined and deconfined phases is obtained.

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.

Quantum spin systems with strong geometric restrictions give rise to rich quantum phases such as valence bond solids and spin liquid states. However, the geometric restrictions often hamper the application of sophisticated numerical approaches. Based on the stochastic series expansion method, we develop an efficient and exact quantum Monte Carlo ``sweeping cluster'' algorithm which automatically satisfies the geometrical restrictions. Here we use the quantum dimer model as a benchmark to demonstrate the reliability and power of this algorithm. Comparing to existing numerical methods, we can obtain higher accuracy results for a wider parameter region and much more substantial system sizes.

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.