Double-slit Fraunhofer pattern as the signature of the Josephson effect between Berezinskii superconductors through the ferromagnetic vortex

I apply the recently developed formalism of generalized quasiclassical theory to show that using hybrid superconducting systems with non-collinear strong ferromagnets one can realize the Josephson junction between Berezinskii-type superconductors. The reported calculation reproduces main features observed in the recent experiment, namely the the slightly asymmetric double-slit Fraunhofer interference pattern of the Josephson current through the ferromagnetic vortex. The double-slit structure results from the spatially inhomogeneous Berezinskii state with the amplitude controlled by the local angle between magnetic moments in two ferromagnetic layers. The critical current asymmetry by the sign of magnetic field can signal the presence of spontaneous supercurrents generated by the non-coplanar magnetic texture near the core of the ferromagnetic vortex core. I demonstrate that ferromagnetic vortex can induce spontaneous vorticity in the odd-frequency order parameter manifesting the possibility of the emergent magnetic field to create topological defects.