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 (${\theta }_{-}\equiv {\theta }_1-{\theta }_2$) 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.

arXiv comment7 pages + appendix, 2 figures. References are added

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Cited by in corpus (5)

• Nematic Superconductivity and Its Critical Vestigial Phases in the Quasi-crystal
• Charge 4$e$ superconductor: a wavefunction approach
• Frustrated superconductivity and sextetting order
• Local condensation of charge-$4e$ superconductivity at a nematic domain wall
• Fate of Berezinskii-Kosterlitz-Thouless Paired Phase in Coupled $XY$ Models