Bosons may succumb to a Bose-Einstein condensation at a certain critical temperature $T_c$, thus entering the BEC phase.
The only thing I know about the BEC is that since we are talking about bosons it is possible for all of them to occupy the same ground state. As the ground state is the energetically lowest one it makes sense that given low enough temperature most of the bosons will occupy the ground state.
Why however is it a phase transition? For the transition to the BEC to be a proper phase transition we need an order parameter that suddenly becomes non-zero when crossing into a certain realm of temperatures. Also we need a broken symmetry below the critical temperature. I have no idea what either of those are in the BEC case!
Related: How can one experimentally show that the magnons in a ferromagnet have formed a Bose-Einstein condensate? I guess probing the solid with neutrons would show a big increase in intensity for low-energy magnons. However how can we for sure tell that we are in a BEC state and not just at low temperatures?
Best Answer
This is a very good question. It turns out that the phase transition occurs precisely when the chemical potential becomes equal to zero (assuming that the ground state energy is at zero). The order parameter in the BEC is the "macroscopic wave function" or rather the square root of the single-particle reduced density matrix. The broken symmetry is usually said to be gauge symmetry. There is a very good discussion of these points in Quantum Liquids by Leggett where he also briefly discusses the difficulty in thinking about gauge symmetry breaking. Though this is the mainstream scientific opinion, Leggett disagrees.
I'm sorry I cannot be of more help with regard to your question on magnons.