Frustrated edge currents in bilayers formed of s- and d-wave superconductors

We explore edge currents in heterostructures formed of a high-$T_c$ cuprate and a conventional $s$-wave superconductors. The resulting $d\pm is$ superconductor spontaneously breaks time reversal symmetry and, remarkably, exhibits large edge currents along certain edge directions in spite of being topologically trivial. In addition we find that the edge currents are frustrated such that they appear to emerge from or flow into sample corners, seemingly violating charge conservation. Careful self-consistent solutions that guarantee charge conservation are required to understand how this frustration is resolved in physical systems. Calculations within the Ginzburg-Landau theory framework and fully self-consistent microscopic lattice models reveal intriguing patterns of current reversals depending on edge orientation, accompanied by spontaneous formation of magnetic flux patterns which can be used to detect these phenomena experimentally. Our study illuminates the interplay between time-reversal symmetry breaking and unconventional superconductivity in high-$T_c$ superconducting heterostructures, and shows that sizable edge currents are possible even in the absence of non-trivial bulk topology.