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arXiv:2501.19402 · Letters in Mathematical Physics

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

Andreas Deuchert, Phan Thành Nam, Marcin Napiórkowski

Published Oct 14, 2025• 0 citations
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Abstract

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 $$N^{-1} v(x)$$ N - 1 v ( x ) . The system is studied at inverse temperatures of order $$N^{-2/3}$$ 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.

DOI
10.1007/s11005-025-01996-z

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We consider a homogeneous Bose gas of particles with an attractive interaction in an elongated three-dimensional box with periodic boundary conditions. Mean-field theory predicts for this system a spontaneous breaking of the translational symmetry at a certain value of the interaction strength. We show that at this point a second-order quantum phase transition occurs. We investigate the system in the vicinity of the critical point using Bogoliubov theory and a continuous description, that allows us to analyze quantum fluctuations in the system even when the Bogoliubov approach breaks down.
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We consider the homogeneous mean-field Bose gas at temperatures proportional to the critical temperature of its Bose-Einstein condensation phase transition. We prove a trace norm approximation for the grand canonical Gibbs state in terms of a reference state, which is given by a convex combination of products of coherent states and Gibbs states associated with certain temperature-dependent Bogoliubov Hamiltonians. The convex combination is expressed as an integral over a Gibbs distribution of a one-mode Φ4Φ^4Φ4-theory describing the condensate. This result justifies an analogue of Lee and Yang's extension of Bogoliubov theory to positive temperatures, and it allows us to derive various limiting distributions for the number of particles in the condensate, as well as precise formulas for the one- and two-particle density matrices of the Gibbs state. Key ingredients of our proof, which are of independent interest, include two novel abstract correlation inequalities. The proof of one of them is based on an application of an infinite-dimensional version of Stahl's theorem.