A dark state of Chern bands: Designing flat bands with higher Chern number

Abstract

We introduce a scheme by which flat bands with higher Chern number $|C|>1$ can be designed in ultracold gases through a coherent manipulation of Bloch bands. Inspired by quantum-optics methods, our approach consists in creating a "dark Bloch band" by coupling a set of source bands through resonant processes. Considering a $\Lambda$ system of three bands, the Chern number of the dark band is found to follow a simple sum rule in terms of the Chern numbers of the source bands: $C_D=C_1+C_2-C_3$. Altogether, our dark-state scheme realizes a nearly flat Bloch band with predictable and tunable Chern number $C_D$. We illustrate our method based on a $\Lambda$ system, formed of the bands of the Harper-Hofstadter model, which leads to a nearly flat Chern band with $C_D=2$. We explore a realistic sequence to load atoms into the dark Chern band, as well as a probing scheme based on Hall drift measurements. Dark Chern bands offer a practical platform where exotic fractional quantum Hall states could be realized in ultracold gases.

25 pages, 7 figures, including Appendices. Substantial revisions, including a mathematical proof of the sum rule

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• Programmable Hamiltonian engineering with quadratic quantum Fourier transform
• Band engineered bilayer Haldane model: Evidence of multiple topological phase transitions
• Simulating a Chern Insulator with C = $\pm$2 on Synthetic Floquet Lattice