Local Variational Principle

TL;DR

The paper introduces a Local Variational Principle that provides improved bounds for quantum systems at finite temperature, outperforming traditional methods especially at low temperatures, with potential for systematic enhancement.

Contribution

It generalizes the Gibbs-Bogoliubov-Feynman inequality for spinless particles and demonstrates its advantages over existing variational methods in quantum statistical mechanics.

Findings

Provides a pointwise lower bound for the density matrix.

Produces groundstate energies better than the Rayleigh-Ritz method.

Performs better than centroid path and Gibbs-Bogoliubov-Feynman methods at low temperatures.

Abstract

A generalization of the Gibbs-Bogoliubov-Feynman inequality for spinless particles is proven and then illustrated for the simple model of a symmetric double-well quartic potential. The method gives a pointwise lower bound for the finite-temperature density matrix and it can be systematically improved by the Trotter composition rule. It is also shown to produce groundstate energies better than the ones given by the Rayleigh-Ritz principle as applied to the groundstate eigenfunctions of the reference potentials. Based on this observation, it is argued that the Local Variational Principle performs better than the equivalent methods based on the centroid path idea and on the Gibbs-Bogoliubov-Feynman variational principle, especially in the range of low temperatures.