Time reversal symmetry breakdown in normal and superconducting states in frustrated three-band systems

We discuss the phase diagram and phase transitions in $U(1)\times \text{groupZ}2$ three-band superconductors with broken time reversal symmetry. We find that beyond mean field approximation and for sufficiently strong frustration of interband interactions there appears an unusual metallic state precursory to a superconducting phase transition. In that state, the system is not superconducting. Nonetheless, it features a spontaneously broken $\text{groupZ}2$ time reversal symmetry. By contrast, for weak frustration of interband coupling the energy of a domain wall between different $\text{groupZ}2$ states is low and thus fluctuations restore broken time reversal symmetry in the superconducting state at low temperatures.