Implementation of the HMC algorithm on the tempered Lefschetz thimble method

TL;DR

This paper implements the hybrid Monte Carlo algorithm within the tempered Lefschetz thimble method to improve computational efficiency and handle zeros of fermion determinants, demonstrated on the Hubbard model.

Contribution

The paper introduces a novel implementation of HMC within TLTM, enabling efficient large-scale computations and addressing fermion determinant zeros.

Findings

HMC reduces autocorrelation times compared to Metropolis

Implementation works correctly on the Hubbard model

Significantly less computational time required

Abstract

The tempered Lefschetz thimble method (TLTM) is a parallel-tempering algorithm towards solving the numerical sign problem, where the system is tempered by the antiholomorphic gradient flow to tame both the sign and ergodicity problems simultaneously. In this paper, we implement the hybrid Monte Carlo (HMC) algorithm for transitions on each flowed surface, expecting that this implementation on TLTM will give a useful framework for future computations of large-scale systems including fermions. Although the use of HMC in Lefschetz thimble methods has been proposed so far, our crucial achievement here is that HMC is implemented on TLTM so as to work within the parallel-tempering algorithm in TLTM, especially by developing an algorithm to handle zeros of fermion determinants in the course of the molecular-dynamics process. We confirm that the algorithm works correctly by applying it to the sign problem of the Hubbard model on a small lattice, for which the TLTM is known to work with the Metropolis algorithm. We show that the use of HMC significantly reduces the autocorrelation times with less computational times compared to the Metropolis algorithm.