  Motivated by recent experiments on twisted junctions of cuprate
superconductors (SC), it was proposed [1] that at zero temperature, a random
first order Josephson coupling $J_1(\textbf{r}) \cos \phi$ generates an
"effective" global second order coupling, $J_2\cos(2\phi)$, with a sign that
favors $\phi = \pm \pi/2$, 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 $\bar{J}_1=0$, 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 $\bar{J}_1\ne 0$, we show that the two
transitions split linearly for small $\bar{J}_1 \ll \kappa$ (where $\kappa$ is
the in-plane SC stiffness), and that $T_{\mathrm{TRSB}}$ vanishes for $\bar
J_1> J_c$ where $ J_c= \overline{J^2_1}/\kappa$ in the weak disorder limit.
  [1] A. C. Yuan, Y. Vituri, E. Berg, B. Spivak, and S. A. Kivelson,
Inhomogeneity-induced time-reversal symmetry breaking in cuprate
twist-junctions, arXiv preprint arXiv:2305.15472 (2023)
