Acoustic Klein bottle insulator
Zhenhang Pu, Hailong He, Weiyin Deng, Xueqin Huang, Liping Ye, Jiuyang Lu, Manzhu Ke, Zhengyou Liu
Abstract
Internal and crystalline symmetries play critical roles in the classification of topological materials. Recently, the crystalline symmetries are found to allow projective representations by gauge fields and bring about new topological phases, such as M\"obius insulators in spinless systems. Here, we report an observation of a Klein bottle insulator (KBI) phase in phononic crystals under ${\mathbb{Z}}_{2}$ gauge fields. This intriguing insulator phase possesses, in momentum space rather than real space, a nonsymmorphic glide symmetry, under which the fundamental domain of the Brillouin zone is topologically equivalent to a Klein bottle. We exploit a bilayer structure that can be decomposed into a time-reversal-broken KBI and its time-reversal counterpart. In the acoustic KBI, a pair of edge states with a nonlocal twist are observed in experiment. All the theoretical and experimental results are in agreement and consistently validate the existence of the KBI for acoustic waves. Our work will promote the discovery of unexplored phases induced by gauge fields and offer possible applications in topological materials.