Origin of band flatness and constraints of higher Chern numbers

Abstract

Flat bands provide a natural platform for emergent electronic states beyond Landau paradigm. Among those of particular importance are flat Chern bands, including bands of higher Chern numbers ($C$$>$$1$). We introduce a new framework for band flatness through wave functions, and classify the existing isolated flat bands in a "periodic table" according to tight binding features and wave function properties. Our flat band categorization encompasses seemingly different classes of flat bands ranging from atomic insulators to perfectly flat Chern bands and Landau Levels. The perfectly flat Chern bands satisfy Berry curvature condition $F_{xy}=\text{Tr}{\mathcal{G}}_{ij}$ which on the tight-binding level is fulfilled only for infinite-range models. Most of the natural Chern bands fall into category of $C=1$; the complexity of creating higher-$C$ flat bands is beyond the current technology. This is due to the breakdown of the microscopic stability for higher-$C$ flatness, seen atomistically e.g. in the increase of the hopping range bound as $\propto$$\sqrt{C}a$. Within our new formalism, we indicate strategies for bypassing higher-$C$ constraints and thus dramatically decreasing the implementation complexity.

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• Drude weight of an interacting flat-band metal
• Quantum Geometry and the Hidden Scales in Materials
• Fubini-Study metric and topological properties of flat band electronic states: the case of an atomic chain with $s-p$ orbitals
• Quantum geometric tensor and the topological characterization of the extended Su-Schrieffer-Heeger model
• Simple realization of a fragile topological lattice with quasi flat-bands in a microcavity array
• Quantum metric driven transition between superfluid and incoherent fluid
• Fractional Wannier Orbitals and Tight-Binding Gauge Fields for Kitaev Honeycomb Superlattices with Flat Majorana Bands
• Topologically nontrivial flat bands and quantum Hall crossovers in square-octagon lattice materials
• Exceptional flat bands in bipartite non-Hermitian lattices
• Modulation of quantum geometry and its coupling to pseudo-electric field by dynamic strain