Higher-Order Topological Insulators via Momentum-Space Nonsymmorphic Symmetries

Abstract

The topology of the Brillouin zone, foundational in topological physics, is always assumed to be a torus. We theoretically report the construction of Brillouin real projective plane (${\mathrm{RP}}^2$) and the appearance of quadrupole insulating phase, which are enabled by momentum-space nonsymmorphic symmetries stemming from ${\mathbb{Z}}_2$ synthetic gauge fields. We show that the momentum-space nonsymmorphic symmetries quantize bulk polarization and Wannier-sector polarization nonlocally across different momenta, resulting in quantized corner charges and an isotropic binary bulk quadrupole phase diagram, where the phase transition is triggered by a bulk energy gap closing. Under open boundary conditions, the nontrivial bulk quadrupole phase manifests either trivial or nontrivial edge polarization, resulting from the violation of momentum-space nonsymmorphic symmetries under lattice termination. We present a concrete design for the ${\mathrm{RP}}^2$ quadrupole insulator based on acoustic resonator arrays and discuss its feasibility in optics, mechanics, and electrical circuits. Our results show that deforming the Brillouin manifold creates opportunities for realizing high-order band topology.

5 pages,5 figures

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Cited by in corpus (8)

• Weyl points on non-orientable manifolds
• Quantized crystalline-electromagnetic responses in insulators
• Brillouin Platycosms and Topological Phases
• Exceptional Non-Hermitian Topology Associated with Non-Toroidal Brillouin Zones
• Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry
• Exceptional topology on nonorientable manifolds
• Asymmetric real topology of conduction and valence bands
• Breakdown of the symmetry constraint in a Floquet topological insulator