Realizing Hopf Insulators in Dipolar Spin Systems

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.

6 + 7 pages, 3 figures. See companion manuscript arxiv:2105.10504 for additional numerics and a detailed experimental proposal for realizing the Hopf insulator

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• Realization of a Hopf insulator in circuit systems
• Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond
• Teleportation of Berry curvature on the surface of a Hopf insulator
• Tune-out and magic wavelengths for ground-state ${}^{23}$Na${}^{40}$K molecules
• Floquet Engineering Ultracold Polar Molecules to Simulate Topological Insulators
• Massless multifold Hopf semimetals
• Extrinsic and Intrinsic Nonlinear Hall Effects across Berry-Dipole Transitions
• Real Hopf Insulator
• Quench dynamics of Hopf insulators
• Dipolar Weyl semimetals
• Non-Hermitian three-dimensional two-band Hopf insulator
• Topological phonons in arrays of ultracold dipolar particles
• Flat Bands in Three-dimensional Lattice Models with Non-trivial Hopf Index
• Representation-protected topology of spin-singlet $s$-wave superconductors
• Interaction effects of pseudospin-based magnetic monopoles and kinks in a doped dipolar superlattice gas
• PT-symmetric Non-Hermitian Hopf Metal
• Topological insulators and geometry of vector bundles
• Non-Hermitian Hopf insulators
• Twisted curve geometry underlying topological invariants
• Fragile topological insulators protected by rotation symmetry without spin-orbit coupling
• Correlated Hopf insulators