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

## TL;DR

This paper studies how symmetry breaking relates to Bose-Einstein condensation in a system of interacting bosons at specific temperatures.

## Contribution

The paper establishes a direct link between spontaneous U(1) symmetry breaking and Bose-Einstein condensation in the mean-field Bose gas.

## Key findings

- Spontaneous U(1) symmetry breaking occurs when the system exhibits Bose-Einstein condensation.
- The one-particle density matrix of the Gibbs state has a macroscopic eigenvalue under these conditions.

## 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 \documentclass[12pt]{minimal}
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\begin{document}$$N^{-1} v(x)$$\end{document}N-1v(x). The system is studied at inverse temperatures of order \documentclass[12pt]{minimal}
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\begin{document}$$N^{-2/3}$$\end{document}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.