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Gauge invariance, massless modes and topology of gauge fields in multi-band superconductors

Multi-phase physics is a new physics of multi-gap superconductors. Multi-band superconductors exhibit many interesting and novel properties. We investigate the dynamics of the phase-difference mode and show that this mode yields a new excitation mode. The phase-difference mode is represented as an abelian vector field. There are massless modes when the number of gaps is greater than three and the Josephson term is frustrated. The fluctuation of phase-difference modes with non-trivial topology leads to the existence of a fractional-quantum flux vortex in a magnetic field. A superconductor with a fractional-quantum flux vortex is regarded as a topological superconductor with the integer Chern number.