Brillouin Klein Bottle From Artificial Gauge Fields

TL;DR

This paper reveals that in systems with $\

Contribution

It introduces a new topological phase where the Brillouin zone takes the form of a Klein bottle due to $\

Findings

Brillouin zone can assume Klein bottle topology under $\

Topological classification is given by a $\

Discovery of a Klein-bottle insulator with unique edge modes

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 $\pm 1$, 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.