A Supersymmetric Approach to Excited States via Quantum Monte Carlo

TL;DR

This paper introduces a supersymmetric quantum Monte Carlo method that accurately determines excitation energies by leveraging isospectral Hamiltonians, avoiding node problems, and applicable to various 1D potentials.

Contribution

It proposes a novel SUSY-based approach within quantum Monte Carlo to compute excited states, overcoming traditional node issues and extendable to higher dimensions.

Findings

Successfully applied to tunneling states in a double-well potential

Accurately reconstructs spectra and states of Schrödinger equations

Avoids node problems in Monte Carlo calculations

Abstract

We present here a supersymmetric (SUSY) approach for determining excitation energies within the context of a quantum Monte Carlo scheme. By using the fact that SUSY quantum mechanics gives rises to a series of isospectral Hamiltonians, we show that Monte Carlo ground-state calculations in the SUSY partners can be used to reconstruct accurately both the spectrum and states of an arbitrary Schr\"odinger equation. Since the ground-state of each partner potential is node-less, we avoid any ``node''-problem typically associated with the Monte Carlo technique. While we provide an example of using this approach to determine the tunneling states in a double-well potential, the method is applicable to any 1D potential problem. We conclude by discussing the extension to higher dimensions.