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 $\groupUZ$ $n$-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 $\groupUZ$ broken state to a normal state due to a nontrivial interplay between $\groupU$ vortices and $\groupZ$ 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 $\groupZ$ 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 $\groupZ$-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 $\groupZ$ symmetry. Thus, there appears a new phase which is $\groupU$-symmetric, but breaks $\groupZ$ time reversal symmetry, an anomalous dissipative (metallic) state.