Three-dimensional $\mathcal{P}\mathcal{T}$-symmetric topological phases with Pontryagin index

Abstract

We report on a certain class of three-dimensional topological insulators and semimetals protected by spinless $\mathcal{P}\mathcal{T}$ symmetry, hosting an integer-valued bulk invariant. We show using homotopy arguments that these phases host multi-gap topology, providing a realization of a single $\mathbb{Z}$ invariant in three spatial dimensions that is distinct from the Hopf index. We identify this invariant with the Pontryagin index, which describes BPST instantons in particle physics contexts and corresponds to a 3-sphere winding number. We study naturally arising multi-gap linked nodal rings, topologically characterized by split-biquaternion charges, which can be removed by non-Abelian braiding of nodal rings, even without closing a gap. We additionally connect the describing winding number in terms of gauge-invariant combinations of non-Abelian Berry connection elements, indicating relations to Pontryagin characteristic class in four dimensions. These topological configurations are furthermore related to fully non-degenerate multi-gap phases that are characterized by a pair of winding numbers relating to two isoclinic rotations in the case of four bands and can be generalized to an arbitrary number of bands. From a physical perspective, we also analyze the edge states corresponding to this Pontryagin index as well as their dissolution subject to the gap-closing disorder. Finally, we elaborate on the realization of these novel non-Abelian phases, their edge states and linked nodal structures in acoustic metamaterials and trapped-ion experiments.

16+6 pages, 7+2 figures

References in corpus (60)

• Topological Insulators
• Topological insulators and superconductors
• Weyl and Dirac Semimetals in Three Dimensional Solids
• Topological Field Theory of Time-Reversal Invariant Insulators
• Classification of topological quantum matter with symmetries
• Topological Crystalline Insulators
• Periodic table for topological insulators and superconductors
• Topological quantum chemistry
• Prediction and observation of the first antiferromagnetic topological insulator
• Robust axion insulator and Chern insulator phases in a two-dimensional antiferromagnetic topological insulator
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Antiferromagnetic topological insulators
• The space group classification of topological band insulators
• Topological classification of crystalline insulators through band structure combinatorics
• Topology of crystalline insulators and superconductors
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• 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
• Quantitative mappings between symmetry and topology in solids
• On the $Z_2$ classification of Quantum Spin Hall Models
• Disclination Loops, Hedgehogs, and All That
• Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
• Non-Abelian band topology in noninteracting metals
• Impurity Bound States and Greens Function Zeroes as Local Signatures of Topology
• Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems
• Non-Abelian reciprocal braiding of Weyl points and its manifestation in $\text{ZrTe}$
• Topological surface states in three-dimensional magnetic insulators
• Topological Euler class as a dynamical observable in optical lattices
• Surfaces of Axion Insulators
• Experimental observation of non-Abelian topological charges and bulk-edge correspondence
• Observation of non-Abelian topological semimetals and their phase transitions
• Dual gauge field theory of quantum liquid crystals in two dimensions
• Geometric approach to fragile topology beyond symmetry indicators
• Topological Insulators from Group Cohomology
• Unified bulk-boundary correspondence for band insulators
• Intrinsic axion insulating behavior in antiferromagnetic MnBi${}_6$Te${}_{10}$
• Topological correspondence between magnetic space group representations
• Hopf Insulators and Their Topologically Protected Surface States
• The translational side of topological band insulators
• Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates
• Berry-phase description of Topological Crystalline Insulators
• Stiefel-Whitney classes and topological phases in band theory
• Symmetry representation approach to topological invariants in $C_{2z}T$-symmetric systems
• Non-abelian topology of nodal-line rings in $\mathcal{P}\mathcal{T}$-symmetric systems
• Symmetry protected Spin Quantum Hall phases in 2-Dimensions
• Hopf characterization of two-dimensional Floquet topological insulators
• Surface theorem for the Chern-Simons axion coupling
• Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase
• Four-band non-Abelian topological insulator and its experimental realization
• Multi-gap topology and non-Abelian braiding of phonons from first principles
• Subdimensional topologies, indicators and higher order phases
• Floquet Hopf Insulators
• Observation of topological Euler insulators with a trapped-ion quantum simulator
• Manipulation and braiding of Weyl nodes using symmetry-constrained phase transitions
• Teleportation of Berry curvature on the surface of a Hopf insulator
• Zero-energy states bound to a magnetic pi-flux vortex in a two-dimensional topological insulator
• Phase transitions of non-Abelian charged nodal links in a spring-mass system
• Measuring Hopf links and Hopf invariants in a quenched topological Raman lattice
• Chirality flip of Weyl nodes and its manifestation in strained MoTe${}_2$
• Real Hopf Insulator

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• From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory
• Anomalous multi-gap topological phases in periodically driven quantum rotors
• Loop unitary and phase band topological invariant in generic multi-band Chern insulators
• Euler band topology in superfluids and superconductors
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• Hinge modes of three-dimensional Euler insulators
• Gapped spinful phases obtained via Gutzwiller projections of Euler states
• Three-dimensional spinless Euler insulators with rotational symmetry