Exactly solvable model of randomly coupled twisted superconducting bilayers

Motivated by recent experiments on twisted junctions of cuprate superconductors (SCs), it was proposed [Yuan et al., Inhomogeneity-induced time-reversal symmetry breaking in cuprate twist junctions, Phys. Rev. B 108, L100505 (2023)] that at zero temperature, a random first order Josephson coupling ${J}_{1}(\mathrm{r})cos\ensuremath{\phi}$ generates an effective global second-order coupling, ${J}_{2}cos(2\ensuremath{\phi})$, with a sign that favors $\ensuremath{\phi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$, i.e., spontaneous breaking of time reversal symmetry (TRS). To obtain a more controlled understanding of the suggested disorder-induced-order mechanism, we construct an exactly solvable lattice mean-field model and prove that when the disorder-average ${\overline{J}}_{1}=0$, the model exhibits a TRS breaking phase for all temperatures below the SC transition, i.e., ${T}_{c}={T}_{\mathrm{TRSB}}$, regardless of the specific form of disorder. In the presence of nonzero ${\overline{J}}_{1}\ensuremath{\ne}0$, we show that the two transitions split linearly for small ${\overline{J}}_{1}\ensuremath{\ll}\ensuremath{\kappa}$ (where $\ensuremath{\kappa}$ is the in-plane SC stiffness), and that ${T}_{\mathrm{TRSB}}$ vanishes for ${\overline{J}}_{1}>{J}_{c}$ where ${J}_{c}=\overline{{J}_{1}^{2}}/\ensuremath{\kappa}$ in the weak disorder limit.