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In this question, I asked how the link can be made between the property of superconductivity and the BCS ground state. I would now like to extend this question to a general state. That is, if I have an unknown state $\left| \psi \right\rangle$ that has a non-vanishing order parameter (for example of s-wave type), $$ \left\langle \psi \right| c_{k, \uparrow} c_{-k, \downarrow} \left| \psi \right\rangle = \Delta $$ how can I know for sure that properties associated with superconductivity will be manifested in this state?

Note that I do not want to make assumptions regarding the state in question. Maybe we can say that it can approximate the ground state of some interacting hamiltonian, but nothing more.

Please do not hesitate to show or reference to explicit calculations.

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To a large extent, the phenomenology behind superconductivity is the phenomenology of a bosonic gas of charged particles. This was first described explicitly in

Schafroth, M. R. (1955). Superconductivity of a Charged Ideal Bose Gas. Physical Review, 100, 463–475.

and the idea was at the origin of London's arguments

London, F. (1961). Superfluids, volume I: Macroscopic theory of superconductivity (second edition). Dover Publications, Inc.

So as soon as you can prove you've got a gas of bosonic charged particles, you will get many aspects of superconductivity. In particular, the London's equations very well describe the electrodynamics of the superconducting condensates. This set of equations can be justified from the Schrödinger equation. The Schrödinger equation is in fact nothing more than the linear approximation of the Ginzburg-Landau functional, known to describe the onset of superconductivity quite well.

In short, what you really need is a macroscopic wave function of a charged condensate. As a matter of fact, superconductors behave like giant atoms as e.g. perfect diamagnetism is an illustration.

In addition, if you got two macroscopic wave functions and let them interact weakly, Feynmann showed by simple arguments that a current of probability density between the two wave functions exist, and that it is proportional to $\sin(\varphi)$, with $\varphi$ the phase difference between the two condensates. In addition, the phase difference evolves as $\dot{\varphi}\propto \Delta E $ (time derivative of the phase difference $\dot{\varphi}$ is proportional to the difference of energy $\Delta E$ between the two condensates). Adding charge to the wave functions, one recovers the phenomenology of the Josephson effect. In a way, Josephson phenomenology is the phenomenology of two weakly interacting macroscopic ground states, both representing charged particles.

Now the link with your order parameter may appear somewhat obscure. In fact it is not, since what you wrote is a technical writing for something easier to understand mathematically. The superconducting order parameter is first of all a complex order parameter. So it has an amplitude and a phase, and it has specific electrodynamics as described above (i.e. London's and Josephson's relations). You need nothing more to understand the phenomenology of superconductivity. But you need your exact writing to understand the microscopic origin of the superconducting state as the instability of the Fermi surface and the Cooper pairing.

In an other perspective, superconducting properties can be traced out to be an Anderson-Higgs mechanism applied to a gas of charged particle. There again the crucial thing you need is a complex order parameter. Consult

Weinberg, S. (1995). The Quantum Theory of Fields (Volume 2). Cambridge University Press.

Greiter, M. (2005). Is electromagnetic gauge invariance spontaneously violated in superconductors? Annals of Physics, 319, 18., also on arXiv:cond-mat/0503400.

for more details.

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