Skyrmionic chains and lattices in s +i d superconductors

We report characteristic vortex configurations in $s+id$ superconductors with time-reversal symmetry breaking, exposed to magnetic field. A vortex in the $s+id$ state tends to have an opposite phase winding between $s$- and $d\ensuremath{-}\mathrm{wave}$ condensates. We find that this peculiar feature together with the competition between $s$- and $d\ensuremath{-}\mathrm{wave}$ symmetry results in three distinct classes of vortical configurations. When either $s$ or $d$ condensate absolutely dominates, vortices form a conventional lattice. However, when one condensate is relatively dominant, vortices organize in chains that exhibit skyrmionic character, separating the chiral components of the $s\ifmmode\pm\else\textpm\fi{}id$ order parameter into domains within and outside the chain. Such skyrmionic chains are found stable even at high magnetic field. When $s$ and $d$ condensates have comparable strength, vortices split cores in two chiral components to form full-fledged skyrmions, i.e., coreless topological structures with an integer topological charge, organized in a lattice. We provide characteristic magnetic field distributions of all states, enabling their identification in, e.g., scanning Hall probe and scanning SQUID experiments. These unique vortex states are relevant for high-${T}_{c}$ cuprate and iron-based superconductors, where the relative strength of competing pairing symmetries is expected to be tuned by temperature and/or doping level, and can help distinguish $s+is$ and $s+id$ superconducting phases.