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The point with zero momentum in the spectrum of acoustic phonon corresponds to a translation of the crystal.

Is there an interpretation like this of the Bogolyubov sound in BEC?

I have read that in BEC, the order parameter (a complex scalar) has a definite phase, which is a breaking of gauge symmetry. But I know little about gauge other than that in electromagnetism we can use different electromagnetic potential to obtain the same equation of motion.

Besides, I am unclear about the physical picture of BEC. Does the many-body wavefuction of the system in BEC have a definite phase?

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