Massless modes and abelian gauge fields in multi-band superconductors

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In N-band superconductors, the U(1)ⁿ phase invariance is spontaneously broken. We propose a model for N-band superconductors where the phase differences between gaps are represented by abelian gauge fields. This model corresponds to an SU(N) gauge theory with the abelian projection. We show that there are massless modes as well as massive modes when the number of gaps N is greater than 3. There are N-3 massless modes and two massive modes near a minimum of the Josephson potential when N bands are equivalent and Josephson couplings are frustrated. The global symmetry U(1)ⁿ⁻¹ is broken down by the Josephson term to U(1)ⁿ⁻³. A non-trivial configuration of the gauge field, that is, a monopole singularity of the gauge field results in a fractional quantum-flux vortex. The fractional quantum-flux vortex corresponds to a monopole in superconductors.