Realization of a Hopf insulator in circuit systems

Abstract

Three-dimensional (3D) two-band Hopf insulators are a paradigmatic example of topological phases beyond the topological classifications based on powerful methods like $K$-theory and symmetry indicators.Since this class of topological insulating phases was theoretically proposed in 2008, they have attracted significant interest owing to their conceptual novelty, connection to knot theory, and many fascinating physical properties. However, because their realization requires special forms of long-range spin-orbit coupling (SOC), they have not been achieved in any 3D system yet. Here we report the first experimental realization of the long-sought-after Hopf insulator in a 3D circuit system. To implement the Hopf insulator, we construct basic pseudo-spin modules and connection modules that can realize $2\times 2$-matrix elements and then design the circuit network according to a tight-binding Hopf insulator Hamiltonian constructed by the Hopf map. By simulating the band structure of the designed circuit network and calculating the Hopf invariant, we find that the circuit realizes a Hopf insulator with Hopf invariant equaling $4$. Experimentally, we measure the band structure of a printed circuit board and find the observed properties of the bulk bands and topological surface states (TSS) are in good agreement with the theoretical predictions, verifying the bulk-boundary correspondence of the Hopf insulator. Our scheme brings the experimental study of Hopf insulators to reality and opens the door to the implementation of more unexplored topological phases beyond the known topological classifications.

References in corpus (38)

• Classification of topological insulators and superconductors in three spatial dimensions
• Classification of topological quantum matter with symmetries
• Topological quantum chemistry
• The (High Quality) Topological Materials In The World
• Catalogue of Topological Electronic Materials
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Towards ideal topological materials: Comprehensive database searches using symmetry indicators
• Topolectrical circuits
• Time Reversal Invariant Topologically Insulating Circuits
• Topological properties of linear circuit lattices
• Nodal-link semimetals
• Experimental Observation of Higher-Order Topological Anderson Insulators
• Measuring topology by dynamics: Chern number from linking number
• Nodal-knot semimetals
• Topological surface states in three-dimensional magnetic insulators
• Topological Euler class as a dynamical observable in optical lattices
• Circuit Implementation of a Four-Dimensional Topological Insulator
• Measuring Topological Number of a Chern-Insulator from Quench Dynamics
• Probing the Berry Curvature and Fermi Arcs of a Weyl Circuit
• Observation of novel topological states in hyperbolic lattices
• 4D spinless topological insulator in a periodic electric circuit
• Hopf Insulators and Their Topologically Protected Surface States
• Topological nodal states in circuit lattice
• Non-abelian gauge fields in circuit systems
• Topolectrical-circuit realization of 4D hexadecapole insulator
• Noncommutative geometry for three-dimensional topological insulators
• Hopf characterization of two-dimensional Floquet topological insulators
• Symmetry Protected Topological Hopf Insulator and its Generalizations
• Topological Hopf-Chern Insulators and the Hopf Superconductor
• Electric circuit simulations of $n$th-Chern insulators in $2n$-dimensional space and their non-Hermitian generalizations for arbitral $n$
• Observation of topological Euler insulators with a trapped-ion quantum simulator
• Teleportation of Berry curvature on the surface of a Hopf insulator
• Realizing Hopf Insulators in Dipolar Spin Systems
• Majorana Zero Modes Protected by Hopf Invariant in Topologically Trivial Superconductors
• Observation of topological links associated with Hopf insulators in a solid-state quantum simulator
• What's knot to like? Observation of a linked loop quantum state
• $N$-band Hopf insulator
• Floquet Engineering Ultracold Polar Molecules to Simulate Topological Insulators

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• Non-Abelian inverse Anderson transitions
• General Theory of Momentum-Space Nonsymmorphic Symmetry
• Observation of Large-Number Corner Modes in $\mathbb{Z}$-class Higher-Order Topolectrical Circuits
• Extrinsic and Intrinsic Nonlinear Hall Effects across Berry-Dipole Transitions
• Engineering topological states and quantum-inspired information processing using classical circuits
• Dipolar Weyl semimetals
• Berry-dipole Semimetals
• Topological Insulators with Hybrid-order Boundary States
• Berry curvature inside parity-time-symmetry protected exceptional surface
• Hierarchical Topological States in Thermal Diffusive Networks
• From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory
• Brillouin Platycosms and Topological Phases
• Flat Bands in Three-dimensional Lattice Models with Non-trivial Hopf Index
• Floquet-Engineering Weyl Points and Linked Fermi Arcs from Straight Nodal Lines
• PT-symmetric Non-Hermitian Hopf Metal
• Hopf Semimetals
• Boundary Flat Bands with Topological Spin Textures Protected by Sub-chiral Symmetry
• Non-Hermitian Hopf insulators
• Electric circuit analog of Landau-Zener tunneling using time-varying elements
• Emergence of Nodal-Knot Transitions by Disorder
• Metal-Insulator Coexistence and Gap-Crossing Domain-Wall Modes in an Aubry-André Model with Nonlocal Hopping
• From orthogonal link to phase vortex in generalized dynamical Hopf insulators
• Correlated Hopf insulators