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Efficient Prediction of Superlattice and Anomalous Miniband Topology from Quantum Geometry

Valentin Crépel1 and Jennifer Cano2,1

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Phys. Rev. X 15, 011004 – Published 13 January, 2025

DOI: https://doi.org/10.1103/PhysRevX.15.011004

Abstract

Two-dimensional materials subject to long-wavelength modulations have emerged as novel platforms to study topological and correlated quantum phases. In this article, we develop a versatile and computationally inexpensive method to predict the topological properties of materials subjected to a superlattice potential by combining degenerate perturbation theory with the method of symmetry indicators. In the absence of electronic interactions, our analysis provides a systematic rule to find the Chern number of the superlattice-induced miniband starting from the harmonics of the applied potential and a few material-specific coefficients. Our method also applies to anomalous (interaction-generated) bands, for which we derive an efficient algorithm to determine all Chern numbers compatible with a self-consistent solution to the Hartree-Fock equations. Our approach gives a microscopic understanding of the quantum anomalous Hall insulators recently observed in rhombohedral graphene multilayers.

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Popular Summary

Stacking atomically thin sheets at small relative angles creates what is known as a moiré heterostructure. These synthetic structures—and, more generally, superlattice materials—can host a plethora of exotic electronic states. But predicting which structures are best suited for observing a particular state of interest is extremely difficult. Full-fledged calculations rely on computing the full superlattice energy-band structure with thousands of atoms per unit cell—a process often computationally intensive and analytically opaque—which is then used to derive effective models that are finally studied using costly state-of-the-art numerical methods. In this work, we introduce a fast and versatile method to bypass these computational hurdles.

By combining degenerate perturbation theory with the symmetry-indicator method, we predict the topological properties of superlattice-induced and moiré minibands using only the quantum geometry of the base material and the real-space geometry of the long-wavelength additional potential. Interaction-driven contributions can also be incorporated, extending the method to anomalous minibands. We demonstrate the predictive power of this method on a type of graphene heterostructure, revealing the origin of its integer anomalous Hall effect. Our method also agrees with predictions of the continuum model for twisted bilayer transition metal dichalcogenides.

The negligible computational cost of our proposed method and its extreme predictive power offer a clear path to a high-throughput search of topological moiré materials.

Article Text

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Phys. Rev. X 15, 011004– Published 13 January, 2025
  • Received 9 July 2024
  • Revised 17 October 2024
  • Accepted 25 November 2024
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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