Nambu-Goldstone-Leggett modes in multi-condensate superconductors

Multi-gap superconductors exhibit interesting properties. In an $N$-gap superconductor, we have in general $U(1)^N$ phase invariance. This multiple-phase invariance is partially or totally spontaneously broken in a superconductor. The Nambu-Goldstone modes, as well as Higgs modes, are important and will play an important role in multi-condensate superconductors. The additional phase invariance leads to a new quantum phase, with help of frustrated Josephson effects, such as the time-reversal symmetry breaking, the emergence of massless modes and fractionally quantized-flux vortices. There is a possibility that half-flux vortices exist in two-component superconductors in a magnetic field. The half-quantum flux vortex can be interpreted as a monopole, and two half-flux vortices form a bound state connected by a domain wall. There is an interesting analogy between quarks and fractionally quantized-flux vortices in superconductors.