Multi-particle quantum systems within the Worldline Monte Carlo formalism

TL;DR

This paper extends the Worldline Monte Carlo method to simulate multi-particle quantum systems, accurately estimating ground state energies and demonstrating favorable computational scaling compared to traditional methods.

Contribution

It introduces a novel extension of the Worldline Monte Carlo approach for multi-particle systems, including interaction simulation and benchmarking against exact solutions.

Findings

Accurate ground state energy estimations for multi-particle systems.

Favorable computational complexity scaling.

Validation against numerically exact diagonalisation results.

Abstract

We extend the Worldline Monte Carlo approach to computationally simulating the Feynman path integral of non-relativistic multi-particle quantum-mechanical systems. We show how to generate an arbitrary number of worldlines distributed according to the (free) kinetic part of the multi-particle quantum dynamics and how to simulate interactions between worldlines in the ensemble. We test this formalism with two- and three-particle quantum mechanical systems, with both long range Coulomb-like interactions between the particles and external fields acting separately on the particles, in various spatial dimensionality. We extract accurate estimations of the ground state energy of these systems using the late-time behaviour of the propagator, validating our approach with numerically exact solutions obtained via straightforward diagonalisation of the Hamiltonian. Systematic benchmarking of the new approach, presented here for the first time, shows that the computational complexity of Wordline Monte Carlo scales more favourably with respect to standard numerical alternatives. The method, which is general, numerically exact, and computationally not intensive, can easily be generalised to relativistic systems.