Path Integral Monte Carlo on a Sphere

TL;DR

This paper employs path integral Monte Carlo methods to study quantum many-body fluids on a spherical surface, analyzing thermodynamic, structural, and superfluid properties, and exploring effects of curvature and quantum statistics.

Contribution

It introduces a numerical approach to simulate quantum fluids on curved surfaces, including fermions and anyons, and investigates curvature effects on their properties.

Findings

Superfluid fraction behavior matches universal jump predictions.

Curvature influences particle dynamics and thermodynamic properties.

Restricted path integral approximates interacting fermion and anyon systems.

Abstract

We solve numerically exactly a simple toy model to quantum general relativity or more properly to path integral on a curved space. We consider the thermal equilibrium of a quantum many body problem on the sphere, the surface of constant positive curvature. We use path integral Monte Carlo to measure the kinetic energy, the internal energy and the static structure of a bosons, fermions and anyons fluid at low temperatures on the sphere. For bosons we also measure the superfluid fraction and compare its behavior at the critical temperature with the universal jump predicted by Nelson and Kosterlitz in flat space in the thermodynamic limit at the superfluid phase transition. For fermions and anyons it is necessary to use the restricted path integral recipe in order to overcome the sign problem. Even if this recipe is exact for the non interacting fluid it reduces to just an approximation for an interacting system. And we make the example of the electron gas at low temperature. Snapshots of the many body path configuration during the evolution of the computer experiment show that the ``speed'' of the single particle path near the poles slows down as a consequence of the ``hairy ball theorem'' of Poincar\'e. The influence of curvature on the thermodynamic and structural properties of the many body fluid is also studied.