Multiband strong-coupling superconductors with spontaneously broken time-reversal symmetry*

States that spontaneously break time-reversal symmetry in superconducting multiband systems have been intensely researched following the experimental discovery of Fe-based superconductors (FeSCs). So far, these states have been mainly studied by means of microscopic weak-coupling BCS theory or more phenomenological effective field theories such as multi-component Ginzburg-Landau theories. However, since some FeSCs are in the strong-coupling regime, we consider if and how the spontaneous breaking of time-reversal symmetry occurs in strong-coupling theories. To elucidate these questions, we employ Eliashberg theory to search for time-reversal symmetry-breaking states in multiband systems. Deriving the free energy of multiband systems, we use it to pin down the elusive nature of such states. From our results, we discuss the relevant microscopic mechanisms at play in breaking time-reversal symmetry, with special emphasis on the FeSCs.