Quantum fluctuation-induced first-order breaking of time-reversal symmetry in unconventional superconductors

Abstract

Spontaneous time-reversal symmetry breaking in superconductors with competing non-degenerate pairing channels is an exotic quantum phase transition that could give rise to robust topological superconductivity and unusual magnetism. It is proposed mostly in two-dimensional systems and is signaled by a nonzero relative phase between the two superconducting order parameters, hence it should particularly be prone to order-parameter phase fluctuations. Nevertheless, the existing understanding of it is still at the mean-field level. Here, we illustrate the non-negligible effects of the phase fluctuations on such quantum phase transitions using the hole-doped square-lattice $t$-$J$ model as an example. We derive the phase fluctuation-corrected free energy and show that under the quantum phase fluctuations, the time-reversal asymmetric $s+id$ phase region splits off a dome featuring a first-order border with the $d$ phase, indicating the possibility of a phase separation into the time-reversal symmetric and asymmetric phases. The phase fluctuations also narrow the range of the $s+id$ phase considerably. We further discuss the implications of our findings for recent experiments on disorder-induced first-order quantum breakdown of superconductivity and promising high-temperature topological superconductivity in twisted cuprate Josephson junctions.