Second Euler number in four dimensional synthetic matter

Abstract

Two-dimensional Euler insulators are novel kind of systems that host multi-gap topological phases, quantified by a quantised first Euler number in their bulk. Recently, these phases have been experimentally realised in suitable two-dimensional synthetic matter setups. Here we introduce the second Euler invariant, a familiar invariant in both differential topology (Chern-Gauss-Bonnet theorem) and in four-dimensional Euclidean gravity, whose existence has not been explored in condensed matter systems. Specifically, we firstly define two specific novel models in four dimensions that support a non-zero second Euler number in the bulk together with peculiar gapless boundary states. Secondly, we discuss its robustness in general spacetime-inversion invariant phases and its role in the classification of topological degenerate real bands through real Grassmannians. In particular, we derive from homotopy arguments the minimal Bloch Hamiltonian form from which the tight-binding models of any second Euler phase can be generated. Considering more concretely the gapped Euler phase associated with the tangent bundle of the four-sphere, we show that the bulk band structure of the nontrivial 4D Euler phase necessarily exhibits triplets of linked nodal surfaces (where the three types of nodal surfaces are formed by the crossing of the three successive pairs of bands within one four-band subspace). Finally, we show how to engineer these new topological phases in a four-dimensional ultracold atom setup. Our results naturally generalize the second Chern and spin Chern numbers to the case of four-dimensional phases that are characterised by real Hamiltonians and open doors for implementing such unexplored higher-dimensional phases in artificial engineered systems, ranging from ultracold atoms to photonics and electric circuits.

arXiv comment15 pages, 5 figures

References in corpus (86)

• Topological Photonics
• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Topological Crystalline Insulators
• Quantized Electric Multipole Insulators
• Periodic table for topological insulators and superconductors
• Higher-Order Topological Insulators
• Topological quantum chemistry
• Topological characterization of periodically-driven quantum systems
• Topological Bands for Ultracold Atoms
• Zoo of quantum-topological phases of matter
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Topological Quantum Matter with Ultracold Gases in Optical Lattices
• The space group classification of topological band insulators
• Topological classification of crystalline insulators through band structure combinatorics
• Synthetic gauge fields in synthetic dimensions
• Symmetry Protected Topological phases of Quantum Matter
• A Four Dimensional Generalization of the Quantum Hall Effect
• Photonic topological pumping through the edges of a dynamical four-dimensional quantum Hall system
• Reading between the lines of four-dimensional gauge theories
• Topology of crystalline insulators and superconductors
• Exploring 4D Quantum Hall Physics with a 2D Topological Charge Pump
• Fermion Path Integrals And Topological Phases
• Topological quantum matter in synthetic dimensions
• Fragile Topology and Wannier Obstructions
• Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle
• Synthetic dimensions in integrated photonics: From optical isolation to 4D quantum Hall physics
• Higher-Order Topology, Monopole Nodal Lines, and the Origin of Large Fermi Arcs in Transition Metal Dichalcogenides XTe${}_2$ (X=Mo,W)
• Topological quantum matter with cold atoms
• Topological classification with additional symmetries from Clifford algebras
• Band Topology and Linking Structure of Nodal Line Semimetals with Z2 Monopole Charges
• Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
• Four-Dimensional Quantum Hall Effect in a Two-Dimensional Quasicrystal
• Four-Dimensional Quantum Hall Effect with Ultracold Atoms
• Impurity Bound States and Greens Function Zeroes as Local Signatures of Topology
• Quantum Hall Effect in Higher Dimensions
• Non-Abelian reciprocal braiding of Weyl points and its manifestation in $\text{ZrTe}$
• Topological Euler class as a dynamical observable in optical lattices
• Circuit Implementation of a Four-Dimensional Topological Insulator
• Observation of a non-Abelian Yang Monopole: From New Chern Numbers to a Topological Transition
• Experimental observation of non-Abelian topological charges and bulk-edge correspondence
• Observation of non-Abelian topological semimetals and their phase transitions
• Unified theory of PT and CP invariant topological metals and nodal superconductors
• Unsupervised machine learning and band topology
• $PT$ Symmetric Real Dirac Fermions and Semimetals
• Geometric approach to fragile topology beyond symmetry indicators
• Disconnected Elementary Band Representations, Fragile Topology, and Wilson Loops as Topological Indices: An Example on the Triangular Lattice
• Interplay between electronic topology and crystal symmetry: Dislocation-line modes in topological band-insulators
• Topological one-way fiber of second Chern number
• 4D spinless topological insulator in a periodic electric circuit
• Topological correspondence between magnetic space group representations
• Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates
• Phase-Space Berry Phases in Chiral Magnets: Dzyaloshinskii-Moriya Interaction and the Charge of Skyrmions
• The translational side of topological band insulators
• Symmetry representation approach to topological invariants in $C_{2z}T$-symmetric systems
• Revealing tensor monopoles through quantum-metric measurements
• Experimental Observation of Tensor Monopoles with a Superconducting Qudit
• Four-Dimensional Higher-Order Chern Insulator and Its Acoustic Realization
• Hopf characterization of two-dimensional Floquet topological insulators
• Weyl nodal surfaces
• Second Chern Number and Non-Abelian Berry Phase in Topological Superconducting Systems
• Topological acoustic triple point
• Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase
• Multi-gap topology and non-Abelian braiding of phonons from first principles
• Electric circuit simulations of $n$th-Chern insulators in $2n$-dimensional space and their non-Hermitian generalizations for arbitral $n$
• Subdimensional topologies, indicators and higher order phases
• Topological energy conversion through bulk or boundary of driven systems
• Tensor Berry connections and their topological invariants
• Observation of topological Euler insulators with a trapped-ion quantum simulator
• Manipulation and braiding of Weyl nodes using symmetry-constrained phase transitions
• Non-Abelian Majorana modes protected by an emergent second Chern number
• Floquet-engineering of nodal rings and nodal spheres and their characterization using the quantum metric
• Topology and entanglement in quench dynamics
• Yang-Mills Instantons from Gravitational Instantons
• Four-dimensional semimetals with tensor monopoles: From surface states to topological responses
• Realization of an atomic quantum Hall system in four dimensions
• Non-Abelian tensor Berry connections in multi-band topological systems
• Thomas Precession, Berry potential and the Meron
• Topological quantum quench dynamics carrying arbitrary Hopf and second-Chern numbers
• Chirality flip of Weyl nodes and its manifestation in strained MoTe${}_2$
• Revealing the boundary Weyl physics of the four-dimensional Hall effect via phason engineering in metamaterials
• Tensor monopoles and negative magnetoresistance effect in optical lattices
• Topological Electromagnetic Effects and Higher Second Chern Numbers in Four-Dimensional Gapped Phases
• Linking topological features of the Hofstadter model to optical diffraction figures
• $\pi$-fluxes, semi-metals and flat bands in artificial materials

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