# Realizing Hopf Insulators in Dipolar Spin Systems

## TL;DR

This paper proposes a method to realize Hopf insulators in dipolar spin systems, leveraging long-range interactions and spin orientations to achieve topologically protected edge states in a three-dimensional lattice.

## Contribution

It introduces a novel approach to realize Hopf insulators in dipolar spin systems, detailing how dipole interactions and spin configurations enable topological phases.

## Key findings

- Robust gapless edge states at smooth and sharp edges.
- Feasibility of implementation with ultracold polar molecules.
- Long-range dipolar interactions enable precise topological structure.

## Abstract

The Hopf insulator is a weak topological insulator characterized by an insulating bulk with conducting edge states protected by an integer-valued linking number invariant. The state exists in three-dimensional two-band models. We demonstrate that the Hopf insulator can be naturally realized in lattices of dipolar-interacting spins, where spin exchange plays the role of particle hopping. The long-ranged, anisotropic nature of the dipole-dipole interactions allows for the precise detail required in the momentum-space structure, while different spin orientations ensure the necessary structure of the complex phases of the hoppings. Our model features robust gapless edge states at both smooth edges, as well as sharp edges obeying a certain crystalline symmetry, despite the breakdown of the two-band picture at the latter. In a companion manuscript [2105.10504], we provide a specific experimental blueprint for implementing our proposal using ultracold polar molecules of $^{40}$K$^{87}$Rb.