Fast Estimator of jacobians in Monte Carlo Integration on Lefschetz Thimbles

TL;DR

This paper introduces two efficient estimators for Jacobians in the Lefschetz thimble approach to Monte Carlo integration, significantly reducing computational costs from cubic to linear in volume, with analytical and numerical validation.

Contribution

It proposes novel Jacobian estimators that lower computational complexity from O(V^3) to O(V) in Lefschetz thimble Monte Carlo methods.

Findings

Estimators reduce computational cost significantly.

Analytical regimes where estimators are effective.

Numerical validation in two models.

Abstract

A solution to the sign problem is the so-called "Lefschetz thimble approach" where the domain of integration for field variables in the path integral is deformed from the real axis to a sub-manifold in the complex space. For properly chosen sub-manifolds ("thimbles") the sign problem disappears or is drastically alleviated. The parametrization of the thimble by real coordinates require the calculation of a jacobian with a computational cost of order O(V^3), where V is proportional to the spacetime volume. In this note we propose two estimators for this jacobian with a computational cost of order O(V). We discuss analytically the regimes where we expect the estimator to work and show numerical examples in two different models.