Observation of topological Euler insulators with a trapped-ion quantum simulator

Abstract

Symmetries play a crucial role in the classification of topological phases of matter. 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, such as non-Abelian braiding of crossing nodes and higher-order topology. 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 and entanglement spectra to show the topological properties of the Hamiltonian. We also measure the Berry phases of the lowest energy band, illustrating the existence of four crossing points protected by the Euler class. 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. Our results show the advantage of quantum simulation technologies for studying exotic topological phases and open a new avenue for investigating fragile topological phases in experiments.

19 pages including appendixes, 12 figures

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• Probing Complex-energy Topology via Non-Hermitian Absorption Spectroscopy in a Trapped Ion Simulator
• Realization of a Hopf insulator in circuit systems
• Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase
• Quantized Integrated Shift Effect in Multigap Topological Phases
• Optical manifestations and bounds of topological Euler class
• Non-Abelian Hopf-Euler insulators
• Quantum Valley Hall effect without Berry curvature
• Three-dimensional $\mathcal{P}\mathcal{T}$-symmetric topological phases with Pontryagin index
• Interferometry of non-Abelian band singularities and Euler class topology
• Disorder-induced topological quantum phase transitions in multi-gap Euler semimetals
• Bulk-edge correspondence of Stiefel-Whitney and Euler insulators through the entanglement spectrum and cutting procedure
• Non-Abelian quantum geometric tensor in degenerate topological semimetals
• Realizing and detecting Stiefel-Whitney insulators in an optical Raman lattice
• Anomalous geometric transport signatures of topological Euler class
• Exact projected entangled pair ground states with topological Euler invariant
• Optical signatures of Euler superconductors
• Fractional corner charges in threefold-symmetric two-dimensional materials with fragile topology
• Floquet non-Abelian topological charges and edge states
• Coexisting Euler and Stiefel-Whitney Topological Phases in Elastic Metamaterials
• Individually tunable tunnelling coefficients in optical lattices using local periodic driving
• Second Euler number in four dimensional synthetic matter
• Bulk-entanglement spectrum correspondence in $PT$- and $PC$-symmetric topological insulators and superconductors
• Anomalous multi-gap topological phases in periodically driven quantum rotors
• High-chirality and multiquaternion Weyl nodes in hexagonal ReO${}_3$
• Coupled-wire construction of non-Abelian higher-order topological phases
• Gapped spinful phases obtained via Gutzwiller projections of Euler states
• Three-dimensional spinless Euler insulators with rotational symmetry
• Non-abelian Geometric Quantum Energy Pump