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I am trying to make an analogy between the spontaneous symmetry breaking occurring in magnetism and that of superconducvity, but I am struggling to make it precise.

In magnetism, we know that interactions are aligning the directions of all spins in the system, thus the system has a single magnetic orientation. The hamiltonian we can use to describe such a mechanism is of the form

$$ H = - J \sum_{i,j} \vec{S_i} \cdot \vec{S_j} $$ In superconductivity, the phase of the system is said to be uniform. Can we think of the superconductivity interaction as a "phase aligning" mechanism? If so, can this be understood within BCS theory? And as a side question, what do we mean exactly when we say that the "phase" of the electrons is uniform?

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