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 ( θ − ≡ θ 1 − θ 2 ) exhibits an Ising-type Z 2 symmetry due to the second order Josephson coupling, where θ 1,2 are the phases of two gap functions. In contrast, the U (1) symmetry in the channel of θ _ + ≡θ _1 + θ _2 2 is intact. The phase locking, i.e., ordering of θ − , can take place in the phase fluctuation regime before the onset of superconductivity, i.e., when θ + is disordered. If θ − is pinned at ±π 2 , then time-reversal symmetry is broken in the normal state, otherwise, if θ − = 0, or, π , 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 θ + but with θ − disordered. This is the quartetting, or, charge-4 e , superconductivity, which occurs above the low temperature Z 2 -breaking charge-2 e superconducting phase. Our results provide useful guidance for studying novel symmetry breaking phases in strongly correlated superconductors.