Probing the Josephson effect in twisted nodal superconductors

Motivated by the recent proposals for unconventional emergent physics in twisted bilayers of nodal superconductors, we study the Josephson effect at the twisted interface between d-wave superconductors. We demonstrate that the critical current can exhibit a nonmonotonic temperature dependence with a maximum at a nonzero temperature as well as a complex dependence on the twist angle at low temperatures. Effects of interface inhomogeneity are also discussed and demonstrated that they can drive topological to trivial superconducting transitions. Close to 45 degree twist we find that the critical current does not vanish due to Cooper pair cotunneling, which leads to a strong second harmonic in the current-phase relation and a putative transition to a time-reversal breaking topological superconducting phase. We show how the behavior of critical current in a magnetic field and under microwave drive can yield unambiguous evidence of Cooper pair cotunneling.