Topological skyrmion phases of matter

We introduce topological phases of matter defined by skyrmions in the ground state spin---or pseudospin---expectation value textures in the Brillouin zone, the chiral and helical topological skyrmion phases of matter. These phases are protected by a symmetry present in all centrosymmetric superconductors. We consider a tight-binding model for spin-triplet superconductivity in transition metal oxides and find it realizes each of these topological skyrmion phases. The chiral phase is furthermore realized for a parameter set characterizing Sr$_2$RuO$_4$ with spin-triplet superconductivity. We also find two types of topological phase transitions by which the skyrmion number can change. The second type occurs without the closing of energy gaps in a system described by a quadratic Hamiltonian without breaking the protecting symmetries when spin-orbit coupling is non-negligible. This contradicts the "flat band" limit assumption of the entanglement spectrum, Wilson loops, and the ten-fold way classification scheme of topological phases of matter.