[Paper Review] Observation of topological Euler insulators with a trapped-ion quantum simulator

W. -D. Zhao, Yusheng Yang+10 more|arXiv (Cornell University)|Jan 23, 2022

Quantum many-body systems4 citations

TL;DR

This study experimentally realizes a three-band topological Euler insulator using a trapped-ion quantum simulator, demonstrating its fragile topology via quantum state tomography. The team measures the Euler class ξ = 2, Wilson loop flow, entanglement spectra, and Berry phases, confirming four protected nodal points. Dynamical probing reveals skyrmion-antiskyrmion pairs and Hopf links in momentum-time space, establishing a direct link between Euler topology and nontrivial quench dynamics.

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.

Motivation & Objective

• To experimentally realize a minimal model of fragile topology—two-dimensional three-band topological Euler insulator—beyond conventional symmetry-indicator classifications.
• To demonstrate the existence of Euler-class-protected nodal points and nontrivial topology in a controlled quantum platform.
• To probe dynamical topological features such as skyrmion-antiskyrmion pairs and Hopf links in momentum-time space via quench dynamics.
• To validate theoretical predictions of fragile topology in a platform with high control and measurement fidelity.

Proposed method

• A single 171Yb+ ion is trapped in a surface-electrode trap to simulate a three-band Hamiltonian in momentum space.
• Quantum state tomography is performed on momentum-resolved eigenstates to reconstruct the full wavefunction and extract topological invariants.
• The Euler class ξ is evaluated from the Wilson loop flow and entanglement spectra, confirming fragile topology.
• Berry phases of the lowest band are measured to identify four nodal points protected by the Euler class.
• Quench dynamics are implemented by initializing the system in the |(0,0,1)⟩ state and evolving under the Euler Hamiltonian H_E(k) = 2n(k)n(k)^T - I.
• The time-evolving state is mapped to a Hopf map in momentum-time space, revealing nontrivial Hopf links and skyrmion-antiskyrmion structures via the Hopf invariant χ = ±1.

Experimental results

Research questions

• RQ1 Can a minimal three-band model of fragile topology, the topological Euler insulator, be experimentally realized in a quantum simulator?
• RQ2 What are the measurable topological invariants—such as the Euler class, Wilson loop flow, and entanglement spectra—that confirm the fragile topology of the Euler insulator?
• RQ3 How do quench dynamics in the Euler Hamiltonian reveal nontrivial topological structures like skyrmion-antiskyrmion pairs and Hopf links in momentum-time space?
• RQ4 To what extent can trapped-ion systems probe dynamical topological features beyond equilibrium topology?

Key findings

• The Euler class ξ is experimentally measured as ξ = 2, confirming the fragile topological nature of the three-band Hamiltonian.
• Four nodal points in momentum space are identified via Berry phase measurements, protected by the Euler class and robust under trivial band addition.
• Wilson loop flow and entanglement spectra exhibit nontrivial winding, providing direct evidence of the Euler insulator phase.
• Quench dynamics reveal a pair of skyrmion-antiskyrmion structures in momentum space, corresponding to regions where the vector a(k) winds over S² with opposite Chern numbers.
• The Hopf invariant χ = ±1 is numerically confirmed in two separate momentum-time regions, indicating the presence of Hopf links and antilinks with opposite linking numbers.
• The dynamical behavior of the Euler Hamiltonian is shown to be isomorphic to that of a Chern Hamiltonian via a Hopf map construction, linking Euler and Chern topologies through quench dynamics.