Floquet Hopf Insulator in Dipolar Spin Systems

A tremendous amount of recent work has focused on non-trivial band structures that fall outside the "periodic table" of topological insulators. These include topological crystalline insulators, higher order topological insulators, as well as unstable topological insulators, whose topology is nontrivial only in low-band models. In three dimensions, the prime example of this latter group is the so-called "Hopf" insulator. Here, we provide the first experimental blueprint for realizing a Hopf insulator in a dipolar spin system as well as explicit evidence for the role of a recently proposed generalized particle-hole symmetry. Second, we extend the Hopf insulator’s classification to Floquet systems and find the existence of a new phase, the "anomalous Floquet Hopf insulator". Contrary to previously classified Floquet-Bloch systems, the change in Hopf invariant (a Z invariant) at topological defects of the Floquet evolution is only conserved modulo two, leading to a Z x Z2 classification for the Floquet phase. We provide an explicit homotopy demonstrating this reduced classification, and highlight its relation to the Hopf insulator’s instability.