Phase-fluctuation Induced Time-Reversal Symmetry Breaking Normal State.

Spontaneous time-reversal symmetry (TRS) breaking plays an important role in studying strongly correlated unconventional superconductors. When the superconducting gap functions with different pairing symmetries compete, an Ising ($Z_2$) type symmetry breaking occurs due to the locking of the relative phase $\Delta\theta_{12}$ via a second order Josephson coupling. The phase locking can take place even in the normal state in the phase fluctuation regime before the onset of superconductivity. If $\Delta\theta_{12}$ is locked at $\pm\frac{\pi}{2}$, then TRS is broken, otherwise, if $\Delta\theta=0$, or, $\pi$, rotational symmetry is broken leading to a nematic state. In both cases, the order parameters possess a 4-fermion structure, which is beyond the scope of mean-field theory. We employ an effective two-component $XY$-model assisted by a renormalization group (RG) analysis to address this problem. In addition, a quartetting, or, charge-"4e", superconductivity can occur above $T_c$. Monte-Carlo simulations are also performed whose results are in a good agreement with the RG analysis. Our results provide useful guidance for studying novel symmetry breakings in strongly correlated superconductors.