Collective modes and short-time dynamics of the time-reversal symmety broken s+is superconductor

Motivated by the recent observation of the time-reversal symmetry broken
state in K-doped BaFe_2As_2 systems, we theoretically study the
collective modes and the short time dynamics of this state using an
effective four-band model with two hole and two electron pockets.
We consider the phase transition from s^+- to s⁺⁺ between the hole gaps
via an intermediate s+is state induced by hole doping. The transition
occurs, when the electron bands are incipient. As a result the
amplitude and phase modes are coupled giving rise to a variety of collective modes.
In particular, in the s^+- state, we find the Higgs mode at frequencies similar to a two-band model with an
absent Leggett mode, while in the s+is and s⁺⁺ state, we uncover a new coupled collective soft mode.
This mode becomes gapless at the boundaries between the s+is state and
the s^+- and the s⁺⁺ state, respectively.