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arXiv:2508.02659

Enhancing the ergodicity of Worldvolume HMC via embedding generalized thimble HMC

Masafumi Fukuma, Yusuke Namekawa

Published Aug 4, 2025• 0 citations
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Abstract

The Worldvolume Hybrid Monte Carlo (WV-HMC) method [arXiv:2012.08468] is an efficient and versatile algorithm that mitigates the sign problem while resolving the ergodicity issues inherent in Lefschetz-thimble approaches. We focus on cases where the maximum flow time can be kept small, such as when applying WV-HMC to the doped Hubbard model utilizing a redundant, nonphysical parameter. An optimal choice of this parameter significantly reduces the sign problem on the original integration surface. This allows for small flow times, thereby enabling the simulation of larger system sizes at a modest computational cost. However, when the worldvolume reduces to a thin layer, phase-space exploration becomes inefficient, and ergodicity problems may reemerge. To address this limitation in WV-HMC, we propose embedding generalized thimble HMC (GT-HMC) into the WV-HMC framework. GT-HMC performs updates on a single deformed surface at a fixed flow time. Despite its inherent ergodicity issues at the zeros of the Boltzmann weight, GT-HMC efficiently explores the allowed region and typically permits larger molecular dynamics step sizes than WV-HMC. Consequently, it is highly effective in regions where ergodicity issues are less severe. We prove that GT-HMC can be consistently embedded within WV-HMC and confirm that the standalone and combined algorithms agree within statistical errors for the two-dimensional doped Hubbard model on an $8 \times 8$ lattice. This combined algorithm enables simulations on larger spacetime lattices. We demonstrate the feasibility of this approach by extrapolating the number and energy densities to the zero Trotter step limit at fixed temperature $T/t = 1/6.4\simeq 0.156$ and repulsive interaction $U/t = 8.0$. Even with modest sample sizes, we achieve controlled statistical errors across the entire range of the chemical potential.

DOI
10.48550/arxiv.2508.02659

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The Worldvolume Hybrid Monte Carlo (WV-HMC) method [arXiv:2012.08468] is an efficient and versatile algorithm that mitigates the sign problem while resolving the ergodicity issues inherent in Lefschetz-thimble approaches. We focus on cases where the maximum flow time can be kept small, such as when applying WV-HMC to the doped Hubbard model utilizing a redundant, nonphysical parameter. An optimal choice of this parameter significantly reduces the sign problem on the original integration surface. This allows for small flow times, thereby enabling the simulation of larger system sizes at a modest computational cost. However, when the worldvolume reduces to a thin layer, phase-space exploration becomes inefficient, and ergodicity problems may reemerge. To address this limitation in WV-HMC, we propose embedding generalized thimble HMC (GT-HMC) into the WV-HMC framework. GT-HMC performs updates on a single deformed surface at a fixed flow time. Despite its inherent ergodicity issues at the zeros of the Boltzmann weight, GT-HMC efficiently explores the allowed region and typically permits larger molecular dynamics step sizes than WV-HMC. Consequently, it is highly effective in regions where ergodicity issues are less severe. We prove that GT-HMC can be consistently embedded within WV-HMC and confirm that the standalone and combined algorithms agree within statistical errors for the two-dimensional doped Hubbard model on an 8×88 \times 88×8 lattice. This combined algorithm enables simulations on larger spacetime lattices. We demonstrate the feasibility of this approach by extrapolating the number and energy densities to the zero Trotter step limit at fixed temperature T/t=1/6.4≃0.156T/t = 1/6.4\simeq 0.156T/t=1/6.4≃0.156 and repulsive interaction U/t=8.0U/t = 8.0U/t=8.0. Even with modest sample sizes, we achieve controlled statistical errors across the entire range of the chemical potential.
The Worldvolume Hybrid Monte Carlo (WV-HMC) method [arXiv:2012.08468] is an efficient algorithm for addressing the numerical sign problem at moderate computational cost. It mitigates the sign problem while avoiding the ergodicity issues inherent in approaches based on Lefschetz thimbles. In this study, we apply WV-HMC to the two-dimensional Hubbard model doped away from half filling, which is known to suffer from a severe sign problem. We compute the number density and the energy density on lattices of size 6×66 \times 66×6 and 8×88 \times 88×8 at temperature T/t=1/6.4≃0.156T/t = 1/6.4 \simeq 0.156T/t=1/6.4≃0.156 and interaction strength U/t=8.0U/t = 8.0U/t=8.0, using Trotter number Nt=20N_t = 20Nt​=20 (Trotter step ε=0.32ε= 0.32ε=0.32). Our results demonstrate that WV-HMC remains effective even in parameter regimes where standard (non-thimble) determinant quantum Monte Carlo methods fail. In this work, fermion matrix inversions are performed using direct solvers, leading to a computational cost of O(N3)O(N^3)O(N3), where NNN denotes the number of degrees of freedom and is proportional to the spacetime lattice volume. An alternative algorithm employing pseudofermions and iterative solvers, which reduces the cost to O(N2)O(N^2)O(N2) at the expense of careful parameter tuning, will be discussed in a separate publication.
We apply the Worldvolume Hybrid Monte Carlo (WV-HMC) method [arXiv:2012.08468] to the two-dimensional Hubbard model, which is known to suffer from a severe sign problem when the system is doped (away from half filling). We show that the method predicts physical observables with controlled statistical errors on an 8×88 \times 88×8 lattice at temperature T/t=1/6.4≈0.156T/t = 1/6.4 \approx 0.156T/t=1/6.4≈0.156 and interaction strength U/t=8.0U/t = 8.0U/t=8.0 (ttt is the hopping amplitude), for which the standard determinant quantum Monte Carlo fails.
We apply the Worldvolume Hybrid Monte Carlo (WV-HMC) method [arXiv:2012.08468] to the two-dimensional Hubbard model, which is known to suffer from a severe sign problem when the system is doped (away from half filling). We show that the method predicts physical observables with controlled statistical errors on an 8×88 \times 88×8 lattice at temperature T/t=1/6.4≈0.156T/t = 1/6.4 \approx 0.156T/t=1/6.4≈0.156 and interaction strength U/t=8.0U/t = 8.0U/t=8.0 (ttt is the hopping amplitude), for which the standard determinant quantum Monte Carlo fails.
The Hubbard model arises naturally when electron-electron interactions are added to the tight-binding descriptions of many condensed matter systems. For instance, the two-dimensional Hubbard model on the honeycomb lattice is central to the ab initio description of the electronic structure of carbon nanomaterials, such as graphene. Such low-dimensional Hubbard models are advantageously studied with Markov chain Monte Carlo methods, such as hybrid Monte Carlo (HMC). HMC is the standard algorithm of the lattice gauge theory community, as it is well suited to theories of dynamical fermions. As HMC performs continuous, global updates of the lattice degrees of freedom, it provides superior scaling with system size relative to local updating methods. A potential drawback of HMC is its susceptibility to ergodicity problems due to so-called exceptional configurations, for which the fermion operator cannot be inverted. Recently, ergodicity problems were found in some formulations of HMC simulations of the Hubbard model. Here, we address this issue directly and clarify under what conditions ergodicity is maintained or violated in HMC simulations of the Hubbard model. We study different lattice formulations of the fermion operator and provide explicit, representative calculations for small systems, often comparing to exact results. We show that a fermion operator can be found which is both computationally convenient and free of ergodicity problems.
The tempered Lefschetz thimble method (TLTM) is a parallel-tempering algorithm towards solving the numerical sign problem. It tames both the sign and ergodicity problems simultaneously by tempering the system with the flow time of continuous deformations of the integration region. In this article, after reviewing the basics of the TLTM, we explain a new algorithm within the TLTM that enables us to estimate the expectation values precisely with a criterion ensuring global equilibrium and the sufficiency of the sample size. To demonstrate the effectiveness of the algorithm, we apply the TLTM to the quantum Monte Carlo simulation of the Hubbard model away from half filling on a two-dimensional lattice of small size, and show that the obtained numerical results agree nicely with exact values.
A sensible application of the hybrid Monte Carlo (HMC) method is often hindered by the presence of large—or even infinite—potential barriers. These potential barriers separate the configuration space into distinct sectors and can lead to ergodicity violations that bias measurements. In this work, we address this problem by augmenting HMC with a multiplicative Metropolis-Hastings update in a “radial direction” of the fields which enables crossing the potential barriers and ensures ergodicity of the sampling algorithm at comparably low computational cost. We demonstrate the algorithm on a simple toy model and show how it can be applied to the fermionic Hubbard model describing physics ranging from an exactly solvable two-site system to the C 20 H 12 perylene molecule. Our numerical results show that the radial updates successfully remove ergodicity violations, while simultaneously reducing autocorrelation times.