Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2507.23748

Applying the Worldvolume Hybrid Monte Carlo method to the Hubbard model away from half filling

Masafumi Fukuma, Yusuke Namekawa

Published Jul 31, 2025• 0 citations
arXivHTMLPDF

Abstract

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 $6 \times 6$ and $8 \times 8$ at temperature $T/t = 1/6.4 \simeq 0.156$ and interaction strength $U/t = 8.0$, using Trotter number $N_t = 20$ (Trotter step $ε= 0.32$). 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 $O(N^3)$, where $N$ 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 $O(N^2)$ at the expense of careful parameter tuning, will be discussed in a separate publication.

DOI
10.48550/arxiv.2507.23748

Related papers

Analyzing the two-dimensional doped Hubbard model with the Worldvolume HMC method

Yusuke Namekawa, Masafumi Fukuma

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 $8 \times 8$ lattice at temperature $T/t = 1/6.4 \approx 0.156$ and interaction strength $U/t = 8.0$ ($t$ is the hopping amplitude), for which the standard determinant quantum Monte Carlo fails.

International Symposium on Lattice Field Theory (LATTICE)•May 30, 2026•0 citations

Enhancing the ergodicity of Worldvolume HMC via embedding generalized thimble HMC

Masafumi Fukuma, Yusuke Namekawa

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.

arXiv:2508.02659•Aug 4, 2025•0 citations

Analyzing the two-dimensional doped Hubbard model with the Worldvolume HMC method

Masafumi Fukuma, Yusuke Namekawa

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 $8 \times 8$ lattice at temperature $T/t = 1/6.4 \approx 0.156$ and interaction strength $U/t = 8.0$ ($t$ is the hopping amplitude), for which the standard determinant quantum Monte Carlo fails.

arXiv:2605.14965•May 14, 2026•0 citations

Taming the sign problem of the finite density Hubbard model via Lefschetz thimbles

Maksim Ulybyshev, Christopher Winterowd, Savvas Zafeiropoulos

We study the sign problem in the Hubbard model on the hexagonal lattice away from half-filling using the Lefschetz thimbles method. We identify the saddle points, reduce their amount, and perform quantum Monte Carlo (QMC) simulations using the holomorphic gradient flow to deform the integration contour in complex space. Finally, the results are compared against exact diagonalization (ED). We show that the sign problem can be substantially weakened, even in the regime with low temperature and large chemical potential, where standard QMC techniques exhibit an exponential decay of the average sign.

arXiv:1906.02726•Jun 6, 2019•12 citations

Applying the tempered Lefschetz thimble method to the Hubbard model away from half filling

Masafumi Fukuma, Nobuyuki Matsumoto, Naoya Umeda

The tempered Lefschetz thimble method is a parallel-tempering algorithm toward solving the numerical sign problem. It uses the flow time of the gradient flow as a tempering parameter and is expected to tame both the sign and multimodal problems simultaneously. In this paper, we further develop the algorithm so that the expectation values can be estimated precisely with a criterion ensuring global equilibrium and the sufficiency of the sample size. To demonstrate that this algorithm works well, we apply it 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 numerical results agree nicely with exact values.

arXiv:1906.04243•Physical review. D/Physical review. D.•Dec 17, 2019•41 citations

Lefschetz thimble Monte Carlo for many body theories: application to the repulsive Hubbard model away from half filling

Mukherjee, Abhishek, M. Cristoforetti

Recently, a new method, based on stochastic integration on the surfaces of steepest descent of the action, was introduced to tackle the sign problem in quantum field theories. We show how this method can be used in many body theories to perform fully non-perturbative calculations of quantum corrections about mean field solutions. We discuss an explicit algorithm for implementing our method, and present results for the repulsive Hubbard model away from half-filling at intermediate temperatures. Our results are consistent with those from the recent state of the art cluster dynamical mean field theory calculations.

arXiv:1403.5680•Mar 22, 2014•17 citations
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.
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.
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 study the sign problem in the Hubbard model on the hexagonal lattice away from half-filling using the Lefschetz thimbles method. We identify the saddle points, reduce their amount, and perform quantum Monte Carlo (QMC) simulations using the holomorphic gradient flow to deform the integration contour in complex space. Finally, the results are compared against exact diagonalization (ED). We show that the sign problem can be substantially weakened, even in the regime with low temperature and large chemical potential, where standard QMC techniques exhibit an exponential decay of the average sign.
The tempered Lefschetz thimble method is a parallel-tempering algorithm toward solving the numerical sign problem. It uses the flow time of the gradient flow as a tempering parameter and is expected to tame both the sign and multimodal problems simultaneously. In this paper, we further develop the algorithm so that the expectation values can be estimated precisely with a criterion ensuring global equilibrium and the sufficiency of the sample size. To demonstrate that this algorithm works well, we apply it 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 numerical results agree nicely with exact values.
Recently, a new method, based on stochastic integration on the surfaces of steepest descent of the action, was introduced to tackle the sign problem in quantum field theories. We show how this method can be used in many body theories to perform fully non-perturbative calculations of quantum corrections about mean field solutions. We discuss an explicit algorithm for implementing our method, and present results for the repulsive Hubbard model away from half-filling at intermediate temperatures. Our results are consistent with those from the recent state of the art cluster dynamical mean field theory calculations.