Probe Knots and Hopf Insulators with Ultracold Atoms

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.More than a century ago, Lord Kelvin propounded his celebrated "vortex atom" theory, which asserted that atoms are made of vortex knots in the aether [1]. Knot theory has since become a central subject in topology and began to undertake important roles in diverse fields of sciences. Biologists discovered that molecular knots and links in DNA are crucial in vital processes of replication, transcription and recombination [2,3]. In supramolecular chemistry, complex knotted structures have been demonstrated in the laboratory [4][5][6] and were shown to be essential for the crystallization and rheological properties of polymers [7]. In physics, knot has appeared in many subfields, such as classical field theory [8,9] In quantum condensed matter physics, topology is crucial in understanding exotic phases and phase transitions. Notable examples include the discoveries of the Berezinskii-Kosterlitz-Thouless transition [26,27] and the quantum Hall effect [28]. More recently, another new class of materials, dubbed topological insulators, was theoretically predicted and experimentally observed [29][30][31]. In general, they are topological phases protected by system symmetries, which cannot be smoothly connected to the trivial phase if the respective symmetries are preserved [32]. For these materials, the nontriviality originates from the interplay between symmetry and topology. Many peculiar physical phenomena are predicted or observed in experiments, including, for instance, quan- [35,36]. However, despite these striking progresses, the relation between topological insulators and knot theory remains largely unexplored. It is unclear whether topological insulators carry nontrivial knot structures and how we can visualize these structures.In this paper, we reveal intriguing knot and link structures hidden in a particular class of topological insulators. Specifically, we find various nontrivial knot and link structures encoded in the spin textures of Hopf insulators. Hopf insulators are special three-dimensional (3D) to...

Nodal-link semimetals

“…This fact suggests that Eq. (15) describes a (Floquet) Hopf insulators [98][99][100][101][102][103], or more precisely, a Floquet Hopf-Chern insulator [102], because the Chern number C(k z ) = −2 for arbitrary k z , as found in our numerical calculation. In the definition of Hopf invariant [95,98], a nonsingular global Berry potential is needed, which is impossible in the presence of nonzero Chern number, nevertheless, we can study topological surface states.…”

“…A purpose of this paper is to construct models with mutually linked nodal rings.Instead of taking trial-and-error approaches, we put forward a general method based on Hopf maps [95,96]. They play special roles in quantum spin systems [95,97], topological Hopf insulators [98][99][100][101][102][103], liquid-crystal solitons[104], quench dynamics of Chern insulators [105], and minimal models for topologically trivial superconductor-based Majorana zero modes [106]. Mathematically, a Hopf map is a nontrivial mapping from a 3-sphere S 3 to a 2-sphere S 2 , which possesses a nonzero Hopf invariant [95,96,98].…”

“…This is a quite general approach to construct nodal-link semimetals. As an example, we take [98][99][100][101][102][103]…”

Nodal-knot semimetals

“…The most common choices of a 1 and a 3 , say a 1 = cos k x +cos k y +cos k z −m 0 , a 3 = sin k z , yield ordinary nodal lines resembling Fig.1(a). Taking advantage of the Hopf mappings [75][76][77][78][79][80][81][82][83], nodal links such as the one shown in Fig.1(c) can be constructed [66]. This method is sufficiently general to generate nodal links with any integer linking numbers (including the simplest Hopf link), however, it is unable to generate a nodal knot, which contains only one nodal line.…”

Majorana Zero Modes Protected by a Hopf Invariant in Topologically Trivial Superconductors

“…Mappings from a 3D torus T 3 to S 2 inherit the nontrivial topology from the mappings S 3 → S 2 . The Hopf invariant has found interesting applications in nonlinear σ models and spin systems [82,84], Hopf insulators [85][86][87][88][89][90], liquid crystals [91], and quench dynamics of Chern insulators [92,93].…”