# Realizing Hopf Insulators in Dipolar Spin Systems.

Abstract

The Hopf insulator represents a topological state of matter that exists outside the conventional ten-fold way classification of topological insulators. Its topology is protected by a linking number invariant, which arises from the unique topology of knots in three dimensions. We predict that three-dimensional arrays of driven, dipolar-interacting spins are a natural platform to experimentally realize the Hopf insulator. In particular, we demonstrate that certain terms within the dipolar interaction elegantly generate the requisite non-trivial topology, and that Floquet engineering can be used to optimize dipolar Hopf insulators with large gaps. Moreover, we show that the Hopf insulator's unconventional topology gives rise to a rich spectrum of edge mode behaviors, which can be directly probed in experiments. Finally, we present a detailed blueprint for realizing the Hopf insulator in lattice-trapped ultracold dipolar molecules; focusing on the example of {}^{40}^{87}b, we provide quantitative evidence for near-term experimental feasibility.