Phase transitions and anomalous normal state in superconductors with broken time-reversal symmetry

Using Monte Carlo simulations, we explore the phase diagram and the phase transitions in U(1)×Z₂ n-band superconductors with spontaneously broken time-reversal symmetry (also termed $s+is$ superconductors), focusing on the three-band case. In the limit of infinite penetration length, the system under consideration can, for a certain parameter regime, have a single first-order phase transition from a U(1)×Z₂ broken state to a normal state due to a nontrivial interplay between U(1) vortices and Z₂ domain walls. This regime may also apply to multicomponent superfluids. For other parameters, when the free energy of the domain walls is low, the system undergoes a restoration of broken Z₂ time-reversal symmetry at temperatures lower than the temperature of the superconducting phase transition. We show that inclusion of fluctuations can strongly suppress the temperature of the Z₂ transition when frustration is weak. The main result of our paper is that for relatively short magnetic field penetration lengths, the system has a superconducting phase transition at a temperature lower than the temperature of the restoration of the broken Z₂ symmetry. Thus, there appears a new phase that is U(1) symmetric, but breaks Z₂ time-reversal symmetry, an anomalous dissipative (metallic) state.