Model of collective modes in three-band superconductors with repulsive interband interactions

I consider a simple model of a three-band superconductor with repulsive interband interactions. The frustration, associated with the odd number of bands, leads to the possible existence of an intrinsically complex time reversal symmetry breaking order parameter. In such state the fluctuations of the different gaps are strongly coupled, and this leads to the development of novel collective modes, which mix phase and amplitude oscillations. I study these fluctuations using a simple microscopic model and derive the dispersion for two physically distinct modes, which are gapped by energy less than $2\Delta$, and apparently present for all values of interband couplings.

Figure 1

The possible superconducting states in a three-band system. The angles between the arrows represent the relative phase between the different gaps. On the left is the effective two-gap $s_{\pm }$ state, in the middle the three-gap $s_{\pm }$ state, and on the right the TRSB three-gap state (see Ref. 14 for details). The complex state is possible if one or three of the interband pairing terms are repulsive.

Figure 2

The two possible phase-amplitude modes around the TRSB state. The red/blue/purple arrows represent phase oscillations coupled with increase/decrease/no change of the amplitude of a given gap.

Figure 3

The energy gap of the mixed modes as a function of $A$, as derived from Eq. (7) (blue curve) and Eq. (8) (red dashed curve). Apparently the modes can exist both for small and large $J$ (negative and positive $A$).