Inhomogeneity-Induced Time-Reversal Symmetry Breaking in Cuprate Twist-Junctions*

The lowest order Josephson coupling, J1(θ)cos(ϕ), between two d-wave superconductors with phase-difference ϕ across the junction vanishes when their relative orientation is rotated by θ=π/4. However, in the presence of inhomogeneity, J1(r) is non-zero locally, with a sign that fluctuates in space. We show that such a random J1 generates a global second-harmonic Josephson coupling, J2cos(2ϕ), with a sign that favors ϕ=±π/2, i.e., spontaneous breaking of time reversal symmetry. The magnitude of J2 is substantially enhanced if the spatial correlations of J1(r) extend over large distances, such as would be expected in the presence of large amplitude twist-angle angle disorder or significant local electronic nematicity. We argue that this effect likely accounts for the recent observations in twisted Josephson junctions between high temperature superconductors.