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Quantum simulation for topological Euler insulators
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DOI：10.1038/s42005-022-01001-2.png)
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Although recent studies have established a powerful framework to search for and classify topological phases based on symmetry indicators, there exists a large class of fragile topology beyond the description. The Euler class characterizing the topology of two-dimensional real wave functions is an archetypal fragile topology underlying some important properties. However, as a minimum model of fragile topology, the two-dimensional topological Euler insulator consisting of three bands remains a significant challenge to be implemented in experiments. Here, we experimentally realize a three-band Hamiltonian to simulate a topological Euler insulator with a trapped-ion quantum simulator. Through quantum state tomography, we successfully evaluate the Euler class, Wilson loop flow, entanglement spectra and Berry phases to show the topological properties of the Hamiltonian. The flexibility of the trapped-ion quantum simulator further allows us to probe dynamical topological features including skyrmion-antiskyrmion pairs and Hopf links in momentum-time space from quench dynamics. A new class of topological systems has been recently discovered, it has been dubbed the Euler insulator and can manifest in two-dimensional systems. The authors present an experimental implementation for realizing the Euler insulator with 171Yb+ ion trapped by an electrode-surface trap where the topological indicators such as the Euler class, Wilson loop flow and entanglement spectrum for the theoretical and experimental models clearly show the presence of a fragile phase
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