Equivalence of Bose-Einstein condensation and symmetry breaking

TL;DR

This paper demonstrates that Bogoliubov's approximation accurately predicts symmetry breaking in Bose systems, establishing an equivalence between Bose-Einstein condensation and symmetry breaking fields under broad conditions.

Contribution

It proves that Bogoliubov's approximation implies the equality of certain expectation values related to symmetry breaking, extending Ginibre's results to a broader context.

Findings

Bogoliubov's approximation implies symmetry breaking equivalence.

The equality holds for superstable pair interactions.

The proof uses convexity inequalities, simplifying previous approaches.

Abstract

Based on a classic paper by Ginibre [Commun. Math. Phys. {\bf 8} 26 (1968)] it is shown that whenever Bogoliubov's approximation, that is, the replacement of a_0 and a_0^* by complex numbers in the Hamiltonian, asymptotically yields the right pressure, it also implies the asymptotic equality of |< a_0>|^2/V and < a_0^*a_0>/V in symmetry breaking fields, irrespective of the existence or absence of Bose-Einstein condensation. Because the former was proved by Ginibre to hold for absolutely integrable superstable pair interactions, the latter is equally valid in this case. Apart from Ginibre's work, our proof uses only a simple convexity inequality due to Griffiths.