Collective excitations in multi-band superconductivity: Leggett modes and time-reversal symmetry

We study collective excitations in three- and four-band superconductors with inter-band frustration, which may lead to a violation of time-reversal symmetry. Using a low-energy spin-Hamiltonian, we find that a massless Leggett mode occurs in a wide parameter region of this four-band system. This massless mode is related to a degeneracy between a time-reversal-symmetry-breaking state and a time-reversal-symmetric state. Therefore, the mass of this collective mode characterizes well the time-reversal symmetry in frustrated multi-band superconductors.