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

## TL;DR

This paper reports the first experimental observation of topological links and the Hopf fibration in a solid-state quantum simulator, demonstrating the topological properties of Hopf insulators through direct measurement.

## Contribution

It implements a model Hamiltonian for Hopf insulators in a quantum simulator and directly observes their topological invariants and links, bridging knot theory and topological phases.

## Key findings

- Observation of topological links in a quantum simulator
- Direct measurement of the Hopf topological invariant
- Implementation of a Hopf insulator model in solid-state system

## Abstract

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 fascinating 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.