High-order time-reversal symmetry breaking normal state

Abstract

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₋ ≡ θ₁ − θ₂) exhibits an Ising-type Z₂ 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 ${\theta _ + } \equiv {{{\theta _1} + {\theta _2}} \over 2}$ 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 $\pm {\pi \over 2}$, 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-4e, superconductivity, which occurs above the low temperature Z₂-breaking charge-2e superconducting phase. Our results provide useful guidance for studying novel symmetry breaking phases in strongly correlated superconductors.