Spontaneous Time-Reversal Symmetry Breaking in unconventional Superconductors

Time-reversal symmetry plays a crucial role in the study of unconventional superconductivity in strongly correlated systems. When two superconducting order parameters with different pairing symmetries compete, time-reversal symmetry can be spontaneously broken due to a 2^nd order Josephson coupling between the competing order parameters. In this work, we show that time-reversal symmetry 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 analysis to study superconducting phase fluctuations. In the time-reversal symmetry breaking normal state, neither of the pairing orders develops phase coherence, but their relative phase is pinned at ±π/2.