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Hopf insulators are intriguing three-dimensional topological insulatorscharacterized by an integer topological invariant. They originate from themathematical theory of Hopf fibration and epitomize the deep connection betweenknot theory and topological phases of matter, which distinguishes them fromother classes of topological insulators. Here, we implement a model Hamiltonianfor Hopf insulators in a solid-state quantum simulator and report the firstexperimental observation of their topological properties, including fascinatingtopological links associated with the Hopf fibration and the integer-valuedtopological invariant obtained from a direct tomographic measurement. Ourobservation of topological links and Hopf fibration in a quantum simulatoropens the door to probe rich topological properties of Hopf insulators inexperiments. The quantum simulation and probing methods are also applicable tothe study of other intricate three-dimensional topological model Hamiltonians.

Observation of topological links associated with Hopf insulators in a solid-state quantum simulator