TY - JOUR KW - General Physics and Astronomy AU - Thomas Schuster AU - Felix Flicker AU - Ming Li AU - Svetlana Kotochigova AU - Joel Moore AU - Jun Ye AU - Norman Yao AB - 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 an accompanying paper [T. Schuster et al., Phys. Rev. A 103, AW11986 (2021)] we provide a specific experimental blueprint for implementing our proposal using ultracold polar molecules of 40 K 87 Rb. BT - Physical Review Letters DA - 2021-06 DO - 10.1103/physrevlett.127.015301 N2 - 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 an accompanying paper [T. Schuster et al., Phys. Rev. A 103, AW11986 (2021)] we provide a specific experimental blueprint for implementing our proposal using ultracold polar molecules of 40 K 87 Rb. PB - American Physical Society (APS) PY - 2021 EP - 015301 T2 - Physical Review Letters TI - Realizing Hopf Insulators in Dipolar Spin Systems VL - 127 SN - 0031-9007, 1079-7114 ER -