A study of symmetry breaking in a relativistic Bose gas using the contraction algorithm

TL;DR

This paper investigates symmetry breaking in a relativistic Bose gas at finite density using Lefschetz thimbles to address the sign problem, analyzing contributions of different thimbles in various phases.

Contribution

It applies the Lefschetz thimble approach to study symmetry breaking and thimble contributions in a relativistic Bose gas, highlighting the role of sub-dominant thimbles.

Findings

Sub-dominant thimbles contribute significantly only at small explicit symmetry breaking.

In the thermodynamic limit, sub-dominant contributions are negligible.

Symmetry breaking effects depend on the phase and volume effects.

Abstract

A relativistic Bose gas at finite density suffers from a sign problem that makes direct numerical simulations not feasible. One possible solution to the sign problem is to re-express the path integral in terms of Lefschetz thimbles. Using this approach we study the relativistic Bose gas both in the symmetric phase (low-density) and the spontaneously broken phase (high-density). In the high-density phase we break explicitly the symmetry and determine the dependence of the order parameter on the breaking. We study the relative contributions of the dominant and sub-dominant thimbles in this phase. We find that the sub-dominant thimble only contributes substantially when the explicit symmetry breaking is small, a regime that is dominated by finite volume effects. In the regime relevant for the thermodynamic limit, this contribution is negligible.