  We study superconducting three-band systems within strong-coupling Eliashberg
theory. In particular, we search for phase-frustrated superconducting systems
with spontaneous time-reversal symmetry breaking (TRSB) states. The emergence
of TRSB states in multiband systems has so far been studied using microscopic
weak-coupling BCS theory or more phenomenological effective field theories such
as multi-component Ginzburg-Landau theories. For systems with three disjoint
Fermi surfaces whose electrons experience interactions mediated by phonons, we
present a microscopic analysis showing that TRSB states also exist within a
strong-coupling microscopic theory. The systems we consider have sizable
electron-phonon couplings, putting them into the strong-coupling regime. They
are thus a fitting description for strong-coupling materials such as some of
the iron pnictides. Moreover, as the TRSB states are challenging to find
numerically, we calculate the free energy of multiband systems within
strong-coupling theory and make explicit use of it to pin down the TRSB states'
elusive nature. Since Eliashberg theory is well incorporated with
first-principles calculations, our strong-coupling approach might help
facilitate a more efficient search for candidate materials that can exhibit
TRSB.
