Path integrals in the theory of condensed helium

TLDR

Feynman's path‑integral approach applied to superfluid $^{4}$He demonstrates that the thermodynamics of Bose systems are exactly equivalent to those of a peculiar interacting classical ring‑polymer system. This review introduces the boson superfluid picture and illustrates how superfluidity and Bose condensation emerge within it. By mapping bosons to classical ring polymers, the author extends Monte Carlo techniques, refines density matrices, develops efficient sampling and estimators, and contrasts the resulting path‑integral Monte Carlo method with other quantum Monte Carlo approaches. Simulations reproduce experimental measurements of pair correlations, superfluid density, energy, and momentum distribution in liquid and solid helium with excellent agreement.

Abstract

One of Feynman's early applications of path integrals was to superfluid $^{4}\mathrm{He}$. He showed that the thermodynamic properties of Bose systems are exactly equivalent to those of a peculiar type of interacting classical "ring polymer." Using this mapping, one can generalize Monte Carlo simulation techniques commonly used for classical systems to simulate boson systems. In this review, the author introduces this picture of a boson superfluid and shows how superfluidity and Bose condensation manifest themselves. He shows the excellent agreement between simulations and experimental measurements on liquid and solid helium for such quantities as pair correlations, the superfluid density, the energy, and the momentum distribution. Major aspects of computational techniques developed for a boson superfluid are discussed: the construction of more accurate approximate density matrices to reduce the number of points on the path integral, sampling techniques to move through the space of exchanges and paths quickly, and the construction of estimators for various properties such as the energy, the momentum distribution, the superfluid density, and the exchange frequency in a quantum crystal. Finally the path-integral Monte Carlo method is compared to other quantum Monte Carlo methods.