Non-Abelian Braiding of Topological Edge Bands

TL;DR

This paper introduces a scheme to realize non-Abelian braiding of multiple topological edge bands using gauge-enriched symmetry, demonstrated experimentally in acoustic crystals with three and four braided edge bands, advancing topological physics.

Contribution

It develops a method to achieve non-Abelian braiding of topological edge states in physical models, moving beyond simple Abelian braids and providing experimental validation.

Findings

Demonstrated non-Abelian braiding in acoustic crystals with three and four edge bands

Linked braiding structures to topological winding in eigenvalue space

Realized non-Abelian braiding topology on a physical crystal platform

Abstract

Braiding is a geometric concept that manifests itself in a variety of scientific contexts from biology to physics, and has been employed to classify bulk band topology in topological materials. Topological edge states can also form braiding structures, as demonstrated recently in a type of topological insulators known as M\"obius insulators, whose topological edge states form two braided bands exhibiting a M\"obius twist. While the formation of M\"obius twist is inspiring, it belongs to the simple Abelian braid group $\mathbb{B}_2$. The most fascinating features about topological braids rely on the non-Abelianness in the higher-order braid group $\mathbb{B}_N$ ($N \geq 3$), which necessitates multiple edge bands, but so far it has not been discussed. Here, based on the gauge enriched symmetry, we develop a scheme to realize non-Abelian braiding of multiple topological edge bands. We propose tight-binding models of topological insulators that are able to generate topological edge states forming non-Abelian braiding structures. Experimental demonstrations are conducted in two acoustic crystals, which carry three and four braided acoustic edge bands, respectively. The observed braiding structure can correspond to the topological winding in the complex eigenvalue space of projective translation operator, akin to the previously established point-gap winding topology in the bulk of the Hatano-Nelson model. Our work also constitutes the realization of non-Abelian braiding topology on an actual crystal platform, but not based on the "virtual" synthetic dimensions.