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I am trying to understand the physical meaning of phase in the order parameter of a superconductor. In particular,I was looking at this article that states the phase $e^{i\theta}$ in the BCS ground state

$$|\psi_G\rangle=\prod_k (u_k +v_k e^{i\theta}c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger)|\phi_0\rangle $$

is not a gauge invariance, but a global $U(1)$ phase rotation symmetry. However, the author (on page 6) then says that

The most significant difference between an antiferromagnet and a superfluid or superconductor with regard to the order parameter is that the broken rotational symmetry in the former case is much more evident to us, as all the macroscopic objects in our daily life experience violate rotational symmetry at one level or another...In the case a superconductor, we need a second superconductor to have a reference direction for the phase, and an interaction between the order parameter in both superconductors to detect a relative difference in the phases.

I can easily visualize this breaking of $U(1)$ symmetry in a antiferromagnet. However, I am still confused as to what, exactly, is the physical meaning of phase in the superconducting order parameter? In particular, how can I prescribe a physical meaning to the phase on the Cooper pairing term in the BCS ground state? As with my other post on a similar question I had, any explanations or resources at the level of Tinkham would be greatly appreciated.

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    $\begingroup$ Of course a phase is meaning less. What makes sense is a phase difference, hence the citation in your question. As in any situation with quantum mechanics, a phase difference is related to interference phenomena. The specific point with superconductivity is that the related particles are charged, and so interference is connected with a charged current (the one flowing in your computer, or your house). An important example of the role of phase difference in the phenomenology of superconductors is the Josephson effect. Also, the superconducting vortices can be explained using the phase. $\endgroup$
    – FraSchelle
    Apr 5, 2017 at 17:59
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    $\begingroup$ In quantum field theory, the phase (or the U(1) global rotation as it is named in your question) is related to the number of particles though the Noether theorem. There are enormous amonts of materials already on this website so I do not write an other answer. Check for Noether, Josephson, Superconductivity + Higgs, Ginzburg-Landau model, ... Note the phenomenology of the superconductivity can be understood as a Higgs mechanism for the U(1) gauge redundancy (when one makes the global U(1) symmetry/rotation a dynamical gauge field. i.e. give it the gauge structure of Maxwell's theory). $\endgroup$
    – FraSchelle
    Apr 5, 2017 at 18:05
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    $\begingroup$ This is a rather open-ended question... and physicsl meaning is a subjective notion - it typically means relating things to the already familiar cocnepts. $\endgroup$ Mar 29, 2021 at 2:18

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