A dynamical view on uncovering enigmatic topology
The theoretical proposal discloses a new form of topology by tracing its dynamics, providing a route for finally capturing this enigmatic phase of matter in experiments.
When the system has a nontrivial Euler invariant, the collective wave function starts tying knots and links dynamically. This corresponds to topological structures of netsted tori.
The past decades have witnessed a transformation in our understanding of matter with the mathematical field of topology, which quantifies global geometrical properties, being employed to characterize robust material properties. A broad quest to identify topologically distinct insulators and metals, and their distinguished physical signatures, was initiated. Very recently, a new topological phase of matter, characterized by so-called Euler invariants, has been proposed. This phase goes beyond any conventional characterization scheme and features a combination of properties of several distinct classes. The mathematical description involves parts of a spectrum (energy levels) that are insulating and parts that are metallic. In fact, this phase of matter can only be understood as a consequence of how these different parts have become entangled as electron wave functions are braided along each other. Given the mysterious nature of these Euler phases, it has been unclear how this invariant would manifest itself physically. In this Letter, this fundamental question is resolved by turning to a dynamical context. When an Euler phase is suddenly created, the system's collective wave function starts tying knots and links dynamically, see Figure. The resulting patterns can be understood by appealing to deep mathematical notions that describe how certain spaces can be topologically decomposed into consistent subspaces. The links and knots most importantly relate to distinct observables that can be measured in ultracold atomic setups, thereby opening the pursuit to capture and explore this new topology.Topological Euler class as a dynamical observable in optical lattices F. Nur Ünal, Adrien Bouhon, Robert-Jan Slager, Phys. Rev. Lett. 125 053601 (2020)