Hamilton dynamics for the Lefschetz thimble integration akin to the complex Langevin method

TL;DR

This paper introduces a Hamiltonian dynamics approach to Lefschetz thimble integration, connecting it with supersymmetric quantum mechanics and complex Langevin methods to address the sign problem.

Contribution

It presents a novel framework linking Lefschetz thimbles to supersymmetric Hamiltonian dynamics, enabling numerical construction of wave-functions for complex integration.

Findings

Wave-functions are well-localized on a grid in a toy model

The approach confirms the identification of steepest descent cycles as ground states

The method offers a new perspective on complexified path integrals

Abstract

The Lefschetz thimble method, i.e., the integration along the steepest descent cycles, is an idea to evade the sign problem by complexifying the theory. We discuss that such steepest descent cycles can be identified as ground-state wave-functions of a supersymmetric Hamilton dynamics, which is described with a framework akin to the complex Langevin method. We numerically construct the wave-functions on a grid using a toy model and confirm their well-localized behavior.