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

The lowest order Josephson coupling, $J_1(\theta )\cos (\phi )$, between two d-wave superconductors with phase-difference $\phi$ across the junction vanishes when their relative orientation is rotated by $\theta =\pi \mathrm{/}4$. However, in the presence of inhomogeneity, $J_1(\mathbf{r})$ is non-zero locally, with a sign that fluctuates in space. We show that such a random $J_1$ generates a global second-harmonic Josephson coupling, $J_2\cos (2\phi )$, with a sign that favors $\phi =\pm \pi \mathrm{/}2$, i.e., spontaneous breaking of time reversal symmetry. The magnitude of $J_2$ is substantially enhanced if the spatial correlations of $J_1(\mathbf{r})$ 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.