In the theory of Bose-Einstein condensation, one way to define the order parameter is by using the concept of spontaneous symmetry breaking. One says that, below the critical temperature, the condensate aquires a well-defined phase by spontaneously breaking a U(1) symmetry. This is analogous to the technique used to define the classical electric field, $E_{\text{class}}(r,t) = \langle\hat{E}(r,t)\rangle$, where $\langle\hat{E}(r,t)\rangle$ is the quantum mechanical electric field operator in terms of the standard creation and annihilation operators. So, in a similar fashion, we say that
$$\Psi(r,t) = \langle\hat{\Psi}(r,t)\rangle$$ where $\hat{\Psi}(r,t)$ is the Bose field operator.
I see two problems with this approach however. One is that while superposition of states corresponding to different photon numers can exist in nature, the same cannot be said about atoms however, as one cannot create or destroy them!
The second point is more technical, and presented in these notes, on page 87.
if
$\langle\hat{\psi}\rangle(t=0)\neq 0$ the state of the system necessarily
involves a coherent superposition of states with different total number
of particles; such a state cannot be stationary
(as states with different
number of particles have also different energies) and it experiences
a phase collapse $\langle\hat{\psi}\rangle(t)\rightarrow 0$ making
the description of the evolution of the system more involved.
So, my question is, is it correct to think of BEC as a $U(1)$ symmetry breaking transition? In particular, is this the only way of explaing things like the zero sound mode (a Goldstone mode due to this symmetry breaking). Or can one avoid this concept altogether?
Best Answer
One can avoid the concept of symmetry breaking in this context, to avoid "non-conservation of the particle number".
People have devised way to do that, see for example http://arxiv.org/pdf/cond-mat/0105058v1.pdf. However, all these approaches gives the same results than standard Bogoliubov-like methods in the thermodynamic limit. This is not too surprising, given that in the thermodynamic limit, the fluctuations of the total number of particles vanishes (much in the same way that the equivalence of ensemble insure that canonical and grand-canonical ensembles give the same results). These approaches only make things more complicated for the fun of it (or consistency, depending on your community...).