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 θ1,2 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.
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We thank Fan Yang, Yu-Bo Liu, and Jing Zhou for helpful discussions. Meng Zeng, Hong-Ye Hu, and Yi-Zhuang You are supported by a startup funding of UCSD and the National Science Foundation (Grant No. DMR-2238360). Congjun Wu is supported by the National Natural Science Foundation of China (Grant Nos. 12234016, and 12174317). This work has been supported by the New Cornerstone Science Foundation.
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Zeng, M., Hu, LH., Hu, HY. et al. High-order time-reversal symmetry breaking normal state. Sci. China Phys. Mech. Astron. 67, 237411 (2024). https://doi.org/10.1007/s11433-023-2287-8
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DOI: https://doi.org/10.1007/s11433-023-2287-8