Spontaneous Time-Reversal Symmetry Breaking in the Normal Phase of Unconventional Superconductors

Time-reversal symmetry (TRS) plays a crucial role in the study of unconventional superconductivity in strongly correlated systems. When two superconducting order parameters with different pairing symmetries compete, TRS can be spontaneously broken due to a second order Josephson coupling between the competing order parameters. In this work, we show that TRS breaking transition can occur due to superconducting phase fluctuations before the onset of superconductivity. To illustrate this phenomenon, we employ the Ginzburg-Landau theory, and use an effective two-component XY-model to perform a renormalization group (RG) analysis to study superconducting phase fluctuations. In the TRS breaking normal state, neither of the pairing orders develops phase coherence, but their relative phase is pinned at $\pm \pi/2$. Monte Carlo simulations are also used to obtain the phase diagram for the coupled XY model at various coupling strengths, which agrees well with the RG calculation.