Enhancing the ergodicity of Worldvolume HMC via embedding generalized thimble HMC

TL;DR

This paper introduces an embedding of generalized thimble HMC into Worldvolume HMC to improve ergodicity and efficiency in simulating larger lattice systems, specifically applied to the doped Hubbard model.

Contribution

It proposes a novel combined algorithm that embeds GT-HMC into WV-HMC, enhancing ergodic exploration and enabling larger system simulations with controlled errors.

Findings

The combined algorithm agrees with standalone methods within statistical errors.

It allows simulations on larger lattices, exemplified by an 8x8 doped Hubbard model.

Feasibility is demonstrated through extrapolation to zero Trotter step at fixed temperature.

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.