Topological Hopf insulating phases of magnetic materials

Hopf insulators are examples of three-dimensional, time-reversal symmetry breaking states, whose topological properties cannot be addressed by the Altland-Zirnbauer scheme of symmetry classification. Our current understanding of these exotic insulators is based on the third homotopy classification of two-band, Bloch Hamiltonians. Can any realistic magnetic material support Hopf insulator phase? To answer this question affirmatively, we have developed third homotopy classification scheme for n-band, Bloch Hamiltonians, belonging to the unitary class, with n≥2. We also describe the nature of surface states and topological phase transitions for Hopf insulators. Our results lay the foundations for the theoretical and experimental search for new types of topological states in spin-orbit coupled, magnetic materials.