Josephson diode effects in twisted nodal superconductors

Recent Josephson tunneling experiments on twisted flakes of high-${\mathrm{T꜀}}^{}$ cuprate superconductor ${\mathrm{Bi}}_{\mathrm{₂}}{\mathrm{Sr₂}}^{}{\mathrm{CaCu}}_{\mathrm{₂}}{\mathrm{O₈₊ₓ}}^{}$ revealed a nonreciprocal behavior of the critical interlayer Josephson current, i.e., a Josephson diode effect. Motivated by these findings we study theoretically the emergence of the Josephson diode effect in twisted interfaces between nodal superconductors, and highlight a strong dependence on the twist angle $\theta$ and damping of the junction. In all cases, the theory predicts diode efficiency that vanishes exactly at $\theta ={45}^{\circ }$ and has a strong peak at a twist angle close to $\theta ={45}^{\circ }$, consistent with experimental observations. Near ${45}^{\circ }$, the junction breaks time-reversal symmetry $T$ spontaneously. We find that for underdamped junctions showing hysteretic behavior, this results in a dynamical Josephson diode effect in a part of the $T$-broken phase. The direction of the diode is trainable in this case by sweeping the external current bias. This effect provides a sensitive probe of spontaneous $T$ breaking. We then show that explicit $T$-breaking perturbations with the symmetry of a magnetic field perpendicular to the junction plane lead to a thermodynamic diode effect that survives even in the overdamped limit. We discuss an experimental protocol to probe the double-well structure in the Josephson free energy that underlies the tendency towards spontaneous $T$ breaking even if $T$ is broken explicitly. Finally, we show that in-plane magnetic fields can control the diode effect in the short junction limit, and predict the signatures of explicit $T$ breaking in Shapiro steps.