High-order Time-Reversal Symmetry Breaking Normal State

Spontaneous time-reversal symmetry breaking plays an important role in studying strongly correlated unconventional superconductors. When two superconducting gap functions with different symmetries compete, the relative phase channel ($\theta_-\equiv \theta_1-\theta_2$) exhibits an Ising-type $Z_2$ symmetry due to the second order Josephson coupling, where $\theta_{1,2}$ are the phases of two gap functions respectively. In contrast, the $U(1)$ symmetry in the channel of $\theta_+\equiv \frac{\theta_1+\theta_2}{2}$ is intact. The phase locking, i.e., ordering of $\theta_-$, can take place in the phase fluctuation regime before the onset of superconductivity, i.e. when $\theta_+$ is disordered. If $\theta_-$ is pinned at $\pm\frac{\pi}{2}$, then time-reversal symmetry is broken in the normal state, otherwise, if $\theta_-=0$, or, $\pi$, rotational symmetry is broken, leading to a nematic normal state. In both cases, the order parameters possess a 4-fermion structure beyond the scope of mean-field theory, which can be viewed as a high order symmetry breaking. We employ an effective two-component $XY$-model assisted by a renormalization group analysis to address this problem. As a natural by-product, we also find the other interesting intermediate phase corresponds to ordering of $\theta_+$ but with $\theta_-$ disordered. This is the quartetting, or, charge-4e, superconductivity, which occurs above the low temperature $Z_2$-breaking charge-2e superconducting phase. Our results provide useful guidance for studying novel symmetry breaking phases in strongly correlated superconductors.