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) x Z(2) 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) x Z(2) broken state to a normal state due to a nontrivial interplay between U(1) vortices and Z(2) 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 Z(2) 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 Z(2) 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 Z(2) symmetry. Thus, there appears a new phase that is U(1) symmetric, but breaks Z(2) time-reversal symmetry, an anomalous dissipative (metallic) state.