Brillouin Klein bottle from artificial gauge fields

Abstract

A Brillouin zone is the unit for the momentum space of a crystal. It is topologically a torus, and distinguishing whether a set of wave functions over the Brillouin torus can be smoothly deformed to another leads to the classification of various topological states of matter. Here, we show that under ${\mathbb{Z}}_2$ gauge fields, i.e., hopping amplitudes with phases ±1, the fundamental domain of momentum space can assume the topology of a Klein bottle. This drastic change of the Brillouin zone theory is due to the projective symmetry algebra enforced by the gauge field. Remarkably, the non-orientability of the Brillouin Klein bottle corresponds to the topological classification by a ${\mathbb{Z}}_2$ invariant, in contrast to the Chern number valued in $\mathbb{Z}$ for the usual Brillouin torus. The result is a novel Klein bottle insulator featuring topological modes at two edges related by a nonlocal twist, radically distinct from all previous topological insulators. Our prediction can be readily achieved in various artificial crystals, and the discovery opens a new direction to explore topological physics by gauge-field-modified fundamental structures of physics.

Topological states are exploited based on crystalline symmetry, but under artificial gauge fields, symmetries may satisfy projective algebras, which remains less studied. Here, the authors reveal that projective symmetry algebra leads to momentum-space nonsymmorphic symmetry, resulting in new topological states over a momentum-space Klein bottle.