Projectively enriched symmetry and topology in acoustic crystals

TL;DR

This paper introduces the concept of projectively enriched symmetries in acoustic crystals, revealing new topological phases resulting from gauge symmetry effects that extend current topological classification frameworks.

Contribution

It demonstrates, through theory, simulation, and experiment, how gauge symmetry enriches crystal symmetries, leading to novel topological states in acoustic lattices.

Findings

Discovery of a $Z_2$ gauge field inducing projective translation symmetries

Observation of a M"{o}bius topological insulator phase

Identification of graphene-like semimetal phases

Abstract

Symmetry plays a key role in modern physics, as manifested in the revolutionary topological classification of matter in the past decade. So far, we seem to have a complete theory of topological phases from internal symmetries as well as crystallographic symmetry groups. However, an intrinsic element, i.e., the gauge symmetry in physical systems, has been overlooked in the current framework. Here, we show that the algebraic structure of crystal symmetries can be projectively enriched due to the gauge symmetry, which subsequently gives rise to new topological physics never witnessed under ordinary symmetries. We demonstrate the idea by theoretical analysis, numerical simulation, and experimental realization of a topological acoustic lattice with projective translation symmetries under a $Z_2$ gauge field, which exhibits unique features of rich topologies, including a single Dirac point, M\"{o}bius topological insulator and graphene-like semimetal phases on a rectangular lattice. Our work reveals the impact when gauge and crystal symmetries meet together with topology, and opens the door to a vast unexplored land of topological states by projective symmetries.