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Improved Sweeping Cluster Algorithm for Quantum Dimer Model

Zheng Yan

Published Nov 16, 2020• 3 citations
OpenAlex

Abstract

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.

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quant-pharXiv:2610.03893•Oct 6, 2026•0 citations
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.
Local constraint is closely related to the gauge field, so constrained models are usually effective low energy descriptions and important in condensed matter physics. On the other hand, local restriction hinders the application of numerical algorithms. In addition to the computational difficulties of the constraints, the various topological sectors which cannot be connected through local operators are also one of the key computational difficulties. Taking a quantum dimer model as an example in this paper, we construct a global scheme based on a sweeping cluster Monte Carlo method, which can sample among different topological sectors. In principle, this method can be generalized to other 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 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.
The quantum dimer model is a low-energy effective model for many magnetic systems (materials) that are candidates for quantum spin liquids. It has strict local constraints which is described by gauge field theory. However, since constraints hinder the application of numerical algorithms, the phase diagrams of quantum dimer models are still controversial, even on a square lattice. The core controversy is whether the mixed state exists due to the restriction. In this paper, we give strong evidence to solve this dispute. With our sweeping cluster quantum Monte Carlo method, we studied the phase diagram of a large parameter region by introducing the definition of the pair correlation function and other supporting evidence to distinguish the mixed phase from the columnar phase with high precision. In particular, we find that the ground state belongs to the mixed phase for a vast parameter region.
The sign problem remains a major obstacle to extending the success of Monte Carlo methods to generic quantum many-body systems. A powerful alternative relies on cluster expansions, which provide convergent representations of free energies and related quantities, with applications ranging from tensor network contractions to quantum dynamics and circuits. Within their convergence regime, these expansions often yield deterministic polynomial-time algorithms. However, such methods require exhaustive enumeration of connected clusters, leading to a computational overhead of (Nε−1)O(log⁡Δ)(N\varepsilon^{-1})^{\mathcal{O}(\logΔ)}(Nε−1)O(logΔ), where NNN is the system size, ε\varepsilonε the desired precision, and ΔΔΔ the interaction-graph degree. This becomes particularly severe in long-range systems, where Δ≡NΔ\equiv NΔ≡N, resulting in quasi-polynomial runtimes. Here, we introduce a Markov chain algorithm that samples cluster expansions directly in the space of connected clusters, replacing deterministic enumeration by Monte Carlo sampling. Because sampling occurs in an abstract cluster space rather than a physical configuration space, the conventional exponential sign problem is absent within the convergence regime. For general polymer models, we establish an FPRAS with runtime O~(N4ε−3)\tilde{\mathcal{O}}(N^4\varepsilon^{-3})O~(N4ε−3). For quantum partition functions at temperatures β≤βsampling≡O(1)β\leqβ_{\mathrm{sampling}}\equiv\mathcal{O}(1)β≤βsampling​≡O(1), this improves to O~(N3+1/Dε−3)\tilde{\mathcal{O}}(N^{3+1/D}\varepsilon^{-3})O~(N3+1/Dε−3), where DDD is the lattice dimension. The framework directly applies to long-range quantum systems with interactions ∝1/rα\propto1/r^α∝1/rα, including 222-local models with α>2Dα>2Dα>2D.