Observation of Topological Links Associated with Hopf Insulators in a Solid-State Quantum Simulator

Hopf insulators are intriguing three-dimensional topological insulators characterized by an integer topological invariant. They originate from the mathematical theory of Hopf fibration and epitomize the deep connection between knot theory and topological phases of matter, which distinguishes them from other classes of topological insulators. Here, we implement a model Hamiltonian for Hopf insulators in a solid-state quantum simulator and report the first experimental observation of their topological properties, including nontrivial topological links associated with the Hopf fibration and the integer-valued topological invariant obtained from a direct tomographic measurement. Our observation of topological links and Hopf fibration in a quantum simulator opens the door to probe rich topological properties of Hopf insulators in experiments. The quantum simulation and probing methods are also applicable to the study of other intricate three-dimensional topological model Hamiltonians.

Probe Knots and Hopf Insulators with Ultracold Atoms

Knots and links are fascinating and intricate topological objects.Their influence spans from DNA and molecular chemistry to vortices in superfluid helium,defects in liquid crystals and cosmic strings in the early universe.Here we find that knotted structures also exist in a peculiar class of three-dimensional topological insulators—the Hopf insulators.In particular,we demonstrate that the momentum-space spin textures of Hopf insulators are twisted in a nontrivial way,which implies the presence of various knot and link structures.We further illustrate that the knots and nontrivial spin textures can be probed via standard time-of-flight images in cold atoms as preimage contours of spin orientations in stereographic coordinates.The extracted Hopf invariants,knots,and links are validated to be robust to typical experimental imperfections.Our work establishes the existence of knotted structures in Hopf insulators,which may have potential applications in spintronics and quantum information processing.

Quantum anomalous Hall effect in two-dimensional magnetic insulator heterojunctions

Recent years have witnessed tremendous success in the discovery of topological states of matter.Particularly,sophisticated theoretical methods in time-reversal-invariant topological phases have been developed,leading to the comprehensive search of crystal database and the prediction of thousands of topological materials.In contrast,the discovery of magnetic topological phases that break time reversal is still limited to several exemplary materials because the coexistence of magnetism and topological electronic band structure is rare in a single compound.To overcome this challenge,we propose an alternative approach to realize the quantum anomalous Hall(QAH)effect,a typical example of magnetic topological phase,via engineering two-dimensional(2D)magnetic van der Waals heterojunctions.