A note on spontaneous symmetry breaking in the mean-field Bose gas

TL;DR

This paper investigates the conditions under which spontaneous symmetry breaking occurs in a three-dimensional homogeneous Bose gas, establishing a direct link between symmetry breaking and Bose--Einstein condensation at critical temperature scales.

Contribution

It proves that spontaneous $U(1)$ symmetry breaking in the mean-field Bose gas occurs if and only if Bose--Einstein condensation is present, clarifying the relationship between these phenomena.

Findings

Spontaneous symmetry breaking occurs if and only if Bose--Einstein condensation is present.

The study is conducted at temperature scales of order $N^{-2/3}$.

The one-particle density matrix has a macroscopic eigenvalue when symmetry breaking occurs.

Abstract

We consider the homogeneous Bose gas in the three-dimensional unit torus, where $N$ particles interact via a two-body potential of the form $N^{-1} v(x)$. The system is studied at inverse temperatures of order $N^{-2/3}$, which corresponds to the temperature scale of the Bose--Einstein condensation phase transition. We show that spontaneous $U(1)$ symmetry breaking occurs if and only if the system exhibits Bose--Einstein condensation in the sense that the one-particle density matrix of the Gibbs state has a macroscopic eigenvalue.