Analytical evaluations of the Path Integral Monte Carlo thermodynamic and Hamiltonian energies for the harmonic oscillator

TL;DR

This paper analytically evaluates thermodynamic and Hamiltonian energies for the harmonic oscillator using Path Integral Monte Carlo, demonstrating high-order convergence and highlighting limitations of the primitive approximation in simulations.

Contribution

It introduces an analytical approach to evaluate energies using a universal propagator and achieves twelfth-order optimization with minimal beads.

Findings

Hamiltonian energy can be optimized to twelfth order.

Primitive approximation converges slowly for thermodynamic energy.

High-order propagators improve efficiency in PIMC simulations.

Abstract

By use of the recently derived $universal$ discrete imaginary-time propagator of the harmonic oscillator, both thermodynamic and Hamiltonian energies can be given analytically, and evaluated numerically at each imaginary time step, for $any$ short-time propagator. This work shows that, using only currently known short-time propagators, the Hamiltonian energy can be optimized to the twelfth order, converging to the ground state energy of the harmonic oscillator in as few as three beads. This study makes it absolutely clear that the widely used second-order primitive approximation propagator, when used in computing the thermodynamic energy, converges extremely slowly with increasing number of beads.