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

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• Classical statistical mechanics
• Josephson effect
• Josephson junctions
• Phase transitions
• Superconducting gap
• Superconducting order parameter
• XY model
• d-wave
• Cuprates
• Superconducting devices