Fluctuation modes in multi-gap superonductivity and theory of spontaneous symmetry breaing

We discuss the Nambu-Goldstone boson and the WardTakahashi identity in a system with spontaneous symmetry breaking. We give a formula for the Nambu-Goldstone boson for a general transformation group G. The Ward-Takahashi identity is modified in the presence of the Nambu-Goldstone boson. We apply this theory to examine low-excitation modes in superconductors. In a multi-gap superconductor, the multiplephase invariance is partially or totally broken. The NambuGoldstoneLeggett modes, as well as Higgs modes, are important and will play an important role in superconductors. The properties of these modes are dependent upon inter-band couplings (Josephson couplings). Frustrated Josephson couplings induce many interesting phenomena in a multi-gap superconductor [1-4]. For example, new excitation modes appear in a phase with time-reversal symmetry breaking. The sine-Gordon model for the Leggett mode can be generalized to a nonabelian case [5]. [1] T. Yanagisawa et al., J. Phys. Soc. Jpn. 79, 114706 (2010). [2] Y. Tanaka et al., Solid State Commun. 150, 1980 (2010). [3] T. Yanagisawa et al., J. Phys. Soc. Jpn. 81, 024712 (2012). [4] T. Yanagisawa et al., J. Phys. Soc. Jpn. 82, 124704 (2013). [5] T. Yanagisawa, EPL 113, 41001 (2016).