Probe knots and Hopf insulators with ultracold atoms

TL;DR

This paper reveals that Hopf insulators host knotted and linked spin textures in momentum space, which can be experimentally probed using ultracold atoms and are robust to imperfections, opening new avenues in topological physics.

Contribution

It demonstrates the existence of knotted structures in Hopf insulators and proposes a method to detect them using ultracold atom experiments.

Findings

Knotted spin textures exist in Hopf insulators.

Knots and links can be detected via time-of-flight imaging.

Knotted structures are robust to experimental imperfections.

Abstract

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.