# Quantum Monte Carlo in Classical Phase Space. Mean Field and Exact Results for a One Dimensional Harmonic Crystal

## TL;DR

This paper applies Monte Carlo simulations in classical phase space to a one-dimensional quantum harmonic crystal, analyzing symmetrization effects and comparing mean field approximations with exact results across temperature regimes.

## Contribution

It introduces a phase space Monte Carlo method for quantum crystals and evaluates mean field approximations against exact solutions, including a novel cluster mean field approach.

## Key findings

- High-temperature singlet mean field is very accurate.
- Pair mean field improves accuracy at lower temperatures.
- Cluster mean field accounts for non-commutativity and is extendable to 3D.

## Abstract

Monte Carlo simulations are performed in classical phase space for a one-dimensional quantum harmonic crystal. Symmetrization effects for spinless bosons and fermions are quantified. The algorithm is tested for a range of parameters against exact results that use 20,000 energy levels. It is shown that the singlet mean field approximation is very accurate at high temperatures, and that the pair mean field approximation gives a systematic improvement in the intermediate and low temperature regime. The latter is derived from a cluster mean field approximation that accounts for the non-commutativity of position and momentum, and that can be applied in three dimensions.