I still have some confusions over symmetry breaking in superconductivity.
To begin with it’s clear gauge symmetry can’t be spontaneously broken, since it’s not a symmetry to begin with. I want to understand :
(1) Can global, physical $U(1)$ symmetry be spontaneously broken? This seems to be clearly true, and is what happens in superconductivity. There is even no gauge invariance yet without coupling to gauge fields, so SSB works as usual.
The confusion is if one then couples the charged, SSB system to a classical, external gauge field, the system then has gauge invariance.
Is it true then: there are now degenerate groundstates which transform among each other under global $U(1)$, but each is invariant under gauge transformations?
As an example, in the paper byGreiter, he claims, for a BCS groundstate, \begin{equation} \lvert\psi_\phi\rangle=\prod_{k}u_k+v_ke^{i\phi}c^\dagger_{k\uparrow}c^\dagger_{-k\downarrow}\lvert 0\rangle \end{equation} under a $U(1)$ gauge transformation, both the operators and the state "label" must tranform simultaneously \begin{align} &c^\dagger_{k\sigma} \rightarrow e^{i\theta} c^\dagger_{k\sigma} \\ &\phi \rightarrow \phi-2\theta \end{align} and thus the state is an invariant singlet under "gauge" transformation.
However, under a physical transformation, only the phase $\phi$ transforms, and thus the BCS state is not invariant under a physical transformation, and hence a symmetry broken state.
(2) If the above picture is true, the local order parameter, the Cooper pair condensate, is gauge covariant, how can it be non-vanishing when there is a gauge field? Is it related to the fact the gauge field here is classical?
(3) Finally, what about with dynamical gauge fields, as in Higgs mechanism?
Is there any distinction between a global physical symmetry and a global gauge one? If so, the gauge symmetry can’t undergo SSB, but can the global symmetry?
If the global symmetry undergo SSB, there are degenerate goundatate(in thermodynamic limit ofc). However, In the notes by de Wit, he states in sec 12, there is no groundstate degeracy in a gauge theory, which is also a fact stated by t'Hooft that the groundstate must be unique. How does this reconcile with the picture of condensed matter physicists, that the "global" symmetry is spontaneously broken?
In QFT, however, the gauge fields are dynamical, and there is an integral over all gauge orbits, which then forces an operator like Cooper pair to vanish. Then is the magnitude of Cooper/Higgs field $\langle\lvert\phi\rvert\rangle$ a good order parameter?
