I am teaching myself Quantum Statistical Mechanics (targetting semiconductor and condensed matter physics) just for passion and fun. I'm following with great care the book by Abrikosov/Gorkov/Dzyaloshinskii, currently studying the T = 0 K scenario. There, phonons are very briefly discussed and they say the chemical potential $\mu$ of a system of phonons is zero. I'd like to understand a few conceptual statements related to this. Of course, I have carried out a careful search over the internet to make sure I needed to ask you (by the way, I have no one to speak to about this).
In particular, I have read Post1, Post2, and some others related to photons, but I am not yet satisfied. There it is discussed that non-conservation of the particle (phonon, in this case) number should reflect in the thermodynamic state function not depending on such macroscopic variable N (average number of particles), hence vanishing the corresponding derivative, which is the chemical potential. In fact, the chemical potential is a Lagrange multiplier to enforce N-conservation upon application of the variational principle, so it makes sense that it must be zero if such a condition is absent. In particular, as far as I understand, at absolute zero, the Lagrange multiplier enforces that the ground state (g.s.) of the many-body system possesses a well-defined average number of particles. What I do not get is why Abrikosov (and many others) start by the argument "phonon number is undetermined" or similar. So my question is: why is phonon number not conserved in the aforementioned context?
My bet is the following: the phonon field is constructed by taking the harmonic approximation on the ionic many-body potential energy operator, diagonalizing the dynamical matrix (be it in real space for all lattice cells, or in k space for each wave vector k), expanding in terms of the resulting normal modes that uncouple the oscillators and defining the Bose ladder operators from these. It is then shown that a general property of the phonon spectrum of a crystal is the linearity of the dispersion relation in the acoustic branches near the $\Gamma$ point, $\omega(\boldsymbol{k}) \propto |\boldsymbol{k}|$. Therefore, this means that there are phonon modes with very low energy, namely those with k nearest to $\Gamma$. It is standard to consider a quasi-continuous wave vector, owing to the dimensions of the crystalline sample. In that case, it is true that one may excite phonon modes with an arbitrarily small energy contribution, and hence would understand that the phonon number is not well-defined. Is that the ultimate reason why we take $\mu=0$ from the start? In that case, this should not strictly hold true if $\boldsymbol{k}$ is noticeably quantized, right?
Thank you very much in advance.
