Non-Abelian Hopf-Euler insulators

Abstract

We discuss a class of three-band non-Abelian topological insulators in three dimensions that carry a single bulk Hopf index protected by spatiotemporal ($\mathcal{P}\mathcal{T}$) inversion symmetry. These phases may also host subdimensional topological invariants given by the Euler characteristic class, resulting in real Hopf-Euler insulators. Such systems naturally realize helical nodal structures in the three-dimensional Brillouin zone, providing a physical manifestation of the linking number described by the Hopf invariant. We show that, by opening a gap between the valence bands of these systems, one finds a fully-gapped ``flag'' phase, which displays a three-band multi-gap Pontryagin invariant. Unlike the previously reported $\mathcal{P}\mathcal{T}$-symmetric four-band real Hopf insulator, which hosts a $\mathbb{Z}\oplus \mathbb{Z}$ invariant, these phases are not unitarily equivalent to two copies of a complex two-band Hopf insulator. We show that such uncharted phases can be obtained through dimensional extension of two-dimensional Euler insulators, and that they support (i) an optical bulk integrated circular shift effect quantized by the Hopf invariant, (ii) quantum-geometric breathing in the real space Wannier functions, and (iii) surface Euler topology on boundaries. Consequently, our findings pave the way for novel experimental realizations of real-space quantum-geometry, as these systems may be directly simulated by utilizing synthetic dimensions in metamaterials or ultracold atoms.

16+11 pages, 5+6 figures

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Cited by in corpus (20)

• Quantized Integrated Shift Effect in Multigap Topological Phases
• Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals
• The multi-state geometry of shift current and polarization
• Anomalous geometric transport signatures of topological Euler class
• Quantum geometric bounds for observables: Linear responses, Drude weight, and orbital magnetization
• Diverging shift current responses in the gapless limit of two-dimensional systems
• Optical signatures of Euler superconductors
• Quantum Geometric Origin of the Intrinsic Nonlinear Hall Effect
• Quantum geometric bounds in spinful systems with trivial band topology
• Quantization of spin circular photogalvanic effect in altermagnetic Weyl semimetals
• From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory
• Representation-protected topology of spin-singlet $s$-wave superconductors
• Non-Hermitian Hopf insulators
• Superfluid stiffness of superconductors with delicate topology
• Quantum geometric tensors from sub-bundle geometry
• Anomalous multi-gap topological phases in periodically driven quantum rotors
• Disorder-induced delocalization and reentrance in a Chern-Hopf insulator
• Orbital magnetization reveals multiband topology
• Gapped spinful phases obtained via Gutzwiller projections of Euler states
• Hinge modes of three-dimensional Euler insulators