Brillouin Klein bottle from artificial gauge fields

Z. Y. Chen, Shengyuan A. Yang and Y. X. Zhao ()
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Z. Y. Chen: Nanjing University
Shengyuan A. Yang: Singapore University of Technology and Design
Y. X. Zhao: Nanjing University

Nature Communications, 2022, vol. 13, issue 1, 1-5

Abstract: 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 $${{\mathbb{Z}}}_{2}$$ 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 $${{\mathbb{Z}}}_{2}$$ Z 2 invariant, in contrast to the Chern number valued in $${\mathbb{Z}}$$ 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.

Date: 2022
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DOI: 10.1038/s41467-022-29953-7

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