Quantum Geometry of Moiré Flat Bands Beyond the Valley Paradigm

Abstract

Flat bands in moiré superlattices provide a fertile ground for correlated and topological phases, governed by their quantum geometric properties. While the valley-based paradigm captures key features in select materials, it breaks down in a growing class of systems lacking valley structure, where exotic phenomena such as twist-angle-tunable numbers of flat bands emerge. In this work, we develop and analyze tight-binding models for twisted heterobilayers of bipartite lattices, with a focus on the role of interlayer hybridization in generating flat-band quantum geometry. We demonstrate that sublattice-selective interlayer tunnelings in twisted dice lattice and graphene heterobilayers induce isolated flat bands at zero energy, whose number is tunable by the twist angle. Most importantly, these flat bands exhibit finite Berry curvature and a quantum metric of the Chern-insulator scale, generated through interlayer hybridization. This establishes a mechanism to induce quantum geometry in moiré flat bands beyond the valley paradigm. Our results chart a route to flat-band quantum geometry engineering in twisted bilayer bipartite lattices, with potential material realizations in oxide heterostructures, molecular lattices, and synthetic quantum matter.

Figures (12)

• Figure 1: (a) The lattice structure of the dice lattice, and (b) its schematic band structure. (c) The lattice graph of the dice lattice shows a bipartite structure, of which the partitions are separated by the black dashed line.
• Figure 2: (a) The lattice structure of the tb-D/G at $\theta_c\approx9.43^\circ$, of which the high-symmetrically stacked regions are marked by different colors. (b) Its lattice graph shows a bipartite structure, with the partitions separated by the black dashed line. The (blue, green, orange) hopping marks the interlayer tunnelings ($t_1$, $t_2$, $t_3$). (c) The number of zero-energy flat bands $N_{\text{flat}}$ in tb-D/G as a function of $\theta_c$.
• Figure 3: (a) Band structures of tb-D/G at a representative twist angle $\theta_c\approx3.15^\circ$ near the $K$ valley. (b) Band gaps between flat bands and high-energy bands as functions of $\theta_c$, ranging from $\theta_c\approx58.39^\circ$ to $\theta_c\approx1.61^\circ$.
• Figure 4: The modified quantum weight $\tilde{K}$ of tb-D/G with various $\phi_H$ for all types of interlayer tunnelings as functions of $\theta_c$, ranging from $\theta_c\approx58.39^\circ$ to $\theta_c\approx1.61^\circ$.
• Figure 5: Wave function compositions of isolated flat bands at the $K$ valley in tb-D/G with (011)-, (110)-, and (111)-types of interlayer tunnelings, as functions of $\theta_c$ from $\theta_c\approx58.39^\circ$ to $\theta_c\approx1.61^\circ$. $\rho_{i}^{(l)}$ denotes the partial charge on sublattice $i$ of layer $l$ ($l = D, G$ for dice lattice and graphene, respectively).