Topological Skyrmion Phases of Matter

We introduce two-dimensional topological phases of matter defined by non-trivial homotopy groups into the literature, characterized either by a single Skyrmion number, known as the chiral topological Skymion insulator, or a pair of Skyrmion numbers, known as the helical topological Skyrmion insulator, which generalize and extend the concepts of the Chern insulator and quantum spin Hall insulator, respectively. We show each topological phase of matter is protected by a combination of a mirror symmetry and a generalized particle-hole symmetry equal to the product of particle-hole symmetry and spatial inversion symmetry. Despite these phases being protected in part by crystalline point group symmetries, the phases introduced here are very different from all others known: we characterize three kinds of phase transitions by which a Skyrmion number can change. One kind of topological phase transition occurs without the closing of energy gaps, which has important consequences for study of topologically non-trivial phases of matter. We find that each phase is realized in a tight-binding model relevant to Sr2RuO4 and discuss experimental realization given that the chiral topological Skyrmion insulator phase is realized for a parameter set used to characterize Sr2RuO4.