Probe Knots and Hopf Insulators with Ultracold Atoms
Dong-Ling Deng1,2,3**, Sheng-Tao Wang1,4,3, Kai Sun1, L.-M. Duan1,3
1Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA 2Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, MD 20742-4111, USA 3Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084 4Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
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.
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See the supplemental material for details of the stereographic coordinates, a sketch of the spin texture, adding random perturbations, the $\epsilon$-neighborhood method and an experimental protocol to realize Hopf insulators with ultracold atoms in optical lattices
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2018, Vol. 35 
