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The Hamiltonian is time-reversal invariant: $c_{k\uparrow}\rightarrow c_{-k,\downarrow}, c_{k\downarrow}\rightarrow -c_{-k,\uparrow}$. You can check that explicitly. The ground state is also invariant, because Cooper pairs are all spin singlet. One of the significant implications of time-reversal symmetry for s-wave superconductors is the Anderson's ...


If you don't have the gauge gauge invariance, it's not necessarily inconsistent, it's just describing more degrees of freedom, i.e. the 4 degrees of a massive spin 3/2 rather than the 2 degrees of freedom of a massless spin 3/2. To have gauge invariance for the spin 3/2 you can do it on on a background Einstein space with negative curvature, and you need to ...

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