[Paper Review] Chess-board Acoustic Crystals with Momentum-space Nonsymmorphic Symmetries

Yanqiu Wang, Chen Zhang+4 more|arXiv (Cornell University)|May 11, 2023

Quantum, superfluid, helium dynamics4 citations

TL;DR

This paper introduces a 2D acoustic crystal with a chess-board pattern of π and 0 fluxes that realizes momentum-space nonsymmorphic symmetries—specifically glide reflections—due to projective representations induced by the flux structure. Experimentally demonstrated, these symmetries reduce the Brillouin zone topology from a torus to a Klein bottle, enabling new topological phases with distinct invariants, thus extending topological classification beyond conventional frameworks.

ABSTRACT

Spatial symmetries appearing in both real and momentum space are of fundamental significance to crystals. However, in the conventional framework, every space group in real space, either symmorphic or nonsymmorphic, corresponds to a symmorphic dual in momentum space. Our experiment breaks the framework by showing that in a 2D acoustic crystal with chess-board pattern of $π$ and 0 fluxes, mirror reflections are manifested nonsymmorphically as glide reflections in momentum space. These momentum-space nonsymmorphic symmetries stem from projective, rather than ordinary, representations of the real-space symmetries due to the peculiar flux pattern. Moreover, our experiment demonstrates that the glide reflection can reduce the topological type of the Brillouin zone from the torus to the Klein bottle, resulting in novel topological phases with new topological invariants. Since crystalline topologies are based on momentum-space symmetries, our work paves the way for utilizing engineerable gauge fluxes over artificial crystals to extend the current topological classifications into the broader regime of momentum-space nonsymmorphic symmetries.

Motivation & Objective

• To explore the emergence of momentum-space nonsymmorphic symmetries in artificial crystals with engineered gauge fluxes.
• To challenge the conventional duality between real-space and momentum-space symmetries in crystallography.
• To demonstrate that projective representations from flux patterns can induce glide reflections in momentum space.
• To show that such symmetries alter the topology of the Brillouin zone, transforming it from a torus to a Klein bottle.
• To extend topological classification in condensed matter systems by incorporating momentum-space nonsymmorphic symmetries.

Proposed method

• Designing a 2D acoustic crystal with a chess-board pattern of π and 0 magnetic fluxes to break conventional real-to-momentum space symmetry duality.
• Using projective representations of real-space symmetries to induce nonsymmorphic behavior in momentum space, specifically glide reflections.
• Engineering the system to realize mirror symmetries in real space that manifest as glide reflections in momentum space.
• Measuring the band structure and symmetry properties to confirm the presence of momentum-space nonsymmorphic symmetries.
• Analyzing the topological invariants under the modified Brillouin zone topology, identifying the transition from torus to Klein bottle.

Experimental results

Research questions

• RQ1 Can momentum-space nonsymmorphic symmetries emerge in artificial crystals through engineered gauge fluxes?
• RQ2 How do projective representations of real-space symmetries influence momentum-space symmetry operations?
• RQ3 What topological consequences arise when the Brillouin zone topology changes from a torus to a Klein bottle?
• RQ4 Can such systems host new topological phases not accessible under conventional symmetry classifications?
• RQ5 To what extent can gauge flux engineering extend the classification of topological crystalline phases?

Key findings

• The system realizes momentum-space glide reflections due to projective representations induced by the π/0 flux pattern, breaking the conventional real-to-momentum symmetry duality.
• Mirror symmetries in real space manifest as nonsymmorphic glide reflections in momentum space, a phenomenon not captured by standard space group theory.
• The Brillouin zone topology is topologically reduced from a torus to a Klein bottle due to the momentum-space nonsymmorphic symmetry.
• This topological change enables the emergence of novel topological phases with distinct topological invariants.
• The results demonstrate a pathway to engineer and classify topological phases using gauge fluxes in artificial crystals beyond conventional symmetry constraints.