Josephson diode effects in twisted nodal superconductors

Recent Josephson tunneling experiments on twisted flakes of high-${T}_{c}$ cuprate superconductor ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+x}$ 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 $\ensuremath{\theta}$ and damping of the junction. In all cases, the theory predicts diode efficiency that vanishes exactly at $\ensuremath{\theta}={45}^{\ensuremath{\circ}}$ and has a strong peak at a twist angle close to $\ensuremath{\theta}={45}^{\ensuremath{\circ}}$, consistent with experimental observations. Near ${45}^{\ensuremath{\circ}}$, the junction breaks time-reversal symmetry $\mathcal{T}$ spontaneously. We find that for underdamped junctions showing hysteretic behavior, this results in a dynamical Josephson diode effect in a part of the $\mathcal{T}$-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 $\mathcal{T}$ breaking. We then show that explicit $\mathcal{T}$-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 $\mathcal{T}$ breaking even if $\mathcal{T}$ 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 $\mathcal{T}$ breaking in Shapiro steps.