Exactly Solvable Model of Randomly Coupled Twisted Superconducting Bilayers*

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 $J_1(r) cos phi$ generates an ``effective" global second order coupling, $J_2cos(2phi)$, 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 $ar{J}_1=0$, 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 $ar{J}_1 e 0$, we show that the two transitions split linearly for small $ar{J}_1 ll kappa$ (where $kappa$ is the in-plane SC stiffness), and that $T_{mathrm{TRSB}}$ vanishes for $ar J_1>  J_c$ 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, Phys. Rev. B 108, L100505 (2023)