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- equivalent to theta 1 - theta 2) exhibits an Ising-type Z2 symmetry due to the second order Josephson coupling, where theta 1,2 are the phases of two gap functions. In contrast, the U(1) symmetry in the channel of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta _ + } \equiv {{{\theta _1} + {\theta _2}} \over 2}$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \pm {\pi \over 2}$$\end{document}, 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 Z2-breaking charge-2e superconducting phase. Our results provide useful guidance for studying novel symmetry breaking phases in strongly correlated superconductors.