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It was recently shown that a chiral topological phase emerges from thecoupling of two twisted monolayers of superconductingBi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ for certain twist angles. In this work, wereveal the behavior of such twisted superconducting bilayers with $d_{x^2-y^2}$pairing symmetry in presence of applied magnetic field. Specifically, we showthat the emergent vortex matter can serve as smoking gun for detection oftopological superconductivity in such bilayers. Moreover, we report twodistinct skyrmionic states that characterize the chiral topological phase, andprovide full account of their experimental signatures and their evolution withthe twist angle.

Vortical versus skyrmionic states in the topological phase of a twisted bilayer with $d$-wave superconducting pairing