[Paper Review] Higher-order Klein bottle topological insulator in three-dimensional acoustic crystals

Yu-Liang Tao, Mou Yan+6 more|arXiv (Cornell University)|May 16, 2023

Topological Materials and Phenomena47 references4 citations

TL;DR

This paper proposes a novel three-dimensional higher-order topological insulator in acoustic crystals protected by momentum-space glide reflection symmetries, which deform the fundamental topological manifold from a torus to a Klein bottle. The authors introduce two topological invariants based on the quadrupole moment and Wannier Hamiltonians on the Klein bottle, and experimentally verify gapless hinge modes in a 3D-printed acoustic crystal, demonstrating a new class of topological phases rooted in momentum-space nonsymmorphic symmetries.

ABSTRACT

Topological phases of matter are classified based on symmetries, with nonsymmorphic symmetries like glide reflections and screw rotations being of particular importance in the classification. In contrast to extensively studied glide reflections in real space, introducing space-dependent gauge transformations can lead to momentum-space glide reflection symmetries, which may even change the fundamental domain for topological classifications, e.g., from a torus to a Klein bottle. Here we discover a new class of three-dimensional (3D) higher-order topological insulators, protected by a pair of momentum-space glide reflections. It supports gapless hinge modes, as dictated by the quadrupole moment and Wannier Hamiltonians defined on a Klein bottle manifold, and we introduce two topological invariants to characterize this phase. Our predicted topological hinge modes are experimentally verified in a 3D-printed acoustic crystal, providing direct evidence for 3D higher-order Klein bottle topological insulators. Our results not only showcase the remarkable role of momentum-space glide reflections in topological classifications, but also pave the way for experimentally exploring physical effects arising from momentum-space nonsymmorphic symmetries.

Motivation & Objective

• To identify and classify a new class of three-dimensional higher-order topological insulators protected by momentum-space glide reflection symmetries.
• To explore how momentum-space nonsymmorphic symmetries, such as glide reflections, alter the fundamental topological manifold from a torus to a Klein bottle.
• To develop topological invariants—based on the quadrupole moment and Wannier Hamiltonians—specifically adapted to the Klein bottle geometry.
• To provide experimental realization and verification of the predicted topological hinge modes in a 3D-printed acoustic crystal.

Proposed method

• Theoretical construction of a 3D acoustic crystal model with engineered momentum-space glide symmetries.
• Derivation of topological invariants using the quadrupole moment and Wannier Hamiltonians defined on a Klein bottle manifold.
• Use of space-dependent gauge transformations to induce momentum-space glide symmetries, altering the topological classification framework.
• Numerical computation of the Wannier band structure and topological invariants to confirm the nontrivial phase.
• Design and fabrication of a 3D-printed acoustic crystal to realize the predicted topological phase.
• Experimental measurement of acoustic modes to confirm the presence of gapless hinge modes.

Experimental results

Research questions

• RQ1 How do momentum-space glide reflection symmetries influence the topological classification of 3D systems?
• RQ2 Can a higher-order topological insulator be realized in a 3D acoustic crystal with a Klein bottle fundamental domain?
• RQ3 What new topological invariants are required to characterize a topological phase on a non-orientable manifold like the Klein bottle?
• RQ4 How do space-dependent gauge transformations lead to momentum-space symmetries that alter the topological invariant framework?
• RQ5 Can the predicted topological hinge modes be experimentally observed in a 3D acoustic crystal?

Key findings

• A new class of 3D higher-order topological insulators is identified, protected by a pair of momentum-space glide reflection symmetries.
• The fundamental topological manifold is shown to be a Klein bottle rather than a torus due to the momentum-space glide symmetries, altering the classification framework.
• Two distinct topological invariants are introduced—one based on the quadrupole moment and another on the Wannier Hamiltonian on the Klein bottle—enabling characterization of the phase.
• Numerical simulations confirm the existence of robust gapless hinge modes in the bulk gap, consistent with the predicted topological invariants.
• Experimental measurements on a 3D-printed acoustic crystal directly observe the predicted topological hinge modes, validating the theoretical predictions.
• The results demonstrate that momentum-space nonsymmorphic symmetries can give rise to nontrivial topological phases with unique geometric and topological properties.