Spontaneous time-reversal symmetry breaking in superconductors with competing nondegenerate 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 $s+\mathrm{i}d$ 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 $s+\mathrm{i}d$ 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.

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• Phase transitions by order
• Quantum phase transitions
• Superconducting fluctuations
• Superconductivity
• T-symmetry