Higher-Order Topological Insulators via Momentum-Space Nonsymmorphic Symmetries

TL;DR

This paper introduces a novel higher-order topological insulator model based on momentum-space nonsymmorphic symmetries, enabling the realization of quadrupole phases and $ ext{RP}^2$ topology with potential applications across various physical platforms.

Contribution

It constructs a new topological phase using momentum-space nonsymmorphic symmetries, expanding the understanding of high-order band topology beyond conventional assumptions.

Findings

Quantized bulk polarization and Wannier-sector polarization due to nonsymmorphic symmetries

Realization of $ ext{RP}^2$ quadrupole insulator in acoustic resonator arrays

Phase transitions driven by bulk energy gap closing

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.