Equivalence of Bose-Einstein condensation and symmetry breaking

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.

An error in my summary of previous results (the definition of F') is corrected. The correction is to be done also in the PRL

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• Rigorous Upper Bound on the Critical Temperature of Dilute Bose Gases
• Bose-Einstein Condensation and Spontaneous Symmetry Breaking
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• Landau functions for non-interacting bosons
• Variational wave functions for homogenous Bose systems
• A possible mechanism of concurring diagonal and off-diagonal long-range order for soft interactions