I have been taught that in Ginzburg-Landau model for superconductors the transition to the superconductive states lead to the spontaneous symmetry breaking of global $U(1)$ symmetry.
While I have understood the significance of this from a mathematical point of view I have not yet a clear idea of what is the physical significance of this.
If I consider the ground state, clearly I have a collection of configuration with the same energy but different global phase. However, I don't see how this is affecting the physical observable of the system.
The condensate density $|\psi|$ is clearly unaffected. The current $\vec j_s \propto \nabla \theta + \vec A$ depends only on the gradient of the phase and the magnetic field $\vec B = \nabla \times \vec A$ is independent.
Are these states different only in the mathematical representation but, in fact, physically indistinguishable? In this case why do we even care about this symmetry breaking?