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  • Open Access

Robustness of Real-Space Topology in Moiré Systems

Kryštof Kolář1,2, Kang Yang1,*, Felix von Oppen1, and Christophe Mora3

  • *Present address: Department of Physics, Westlake University, Hangzhou 310030, Zhejiang, China.
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Phys. Rev. X 16, 031043 – Published 19 August, 2026

DOI: https://doi.org/10.1103/szqp-wp6h

Abstract

The appearance of fractional Chern insulators in moiré systems can be rationalized by the presence of a fictitious magnetic field associated with the spatial texture of layer-resolved electronic wave functions. Here, we present a systematic study of real-space topology and the associated fictitious magnetic fields in moiré systems. We first show that at the level of individual Bloch wave functions, the real-space Chern number, akin to a Pontryagin index, is a fragile marker. It generically vanishes except for specific limits where the Bloch functions exhibit fine-tuned zeros within the unit cell, such as the chiral limit of twisted bilayer graphene (TBG) or the adiabatic regime of twisted homobilayer transition metal dichalcogenides (TMDs). We then show that these limitations do not apply to textures associated with ensembles of Bloch wave functions, such as entire bands or the ensemble of states at a given energy. The Chern number of these textures defines a robust topological index protected by a spectral gap. We find that symmetries constrain it to be nonzero for both twisted TMDs and TBG across all twist angles and levels of corrugation, which can be verified in scanning tunneling microscopy measurements. By projection to the band ensemble texture, a single-component Hamiltonian under fictitious magnetic field emerges in broad regime, enabling a direct comparison of multicomponent bands to Landau-level wave functions. We also study real-space topology within the topological heavy fermion model of TBG, finding that the real-space topological features are supported only by the light c electrons.

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Popular Summary

Interpreting exotic electronic states in moiré materials remains challenging because intuitive real-space descriptions of emergent magnetic fields break down outside idealized theoretical limits. We addressed this limitation by systematically analyzing real-space electronic textures and their underlying topology beyond these idealized constraints. We established a robust topological index carried by the textures of entire band ensembles, allowing the associated band wave functions to be mapped directly onto states experiencing a texturally derived fictitious magnetic field. Our results show that this band-derived texture expands traditional characterizations of electronic states that rely purely on standard quantum geometry. This framework helps clarify the connection between spatial textures and nontrivial topology in real moiré systems. Our work provides a mathematical foundation that informs the search for new materials capable of supporting fractional Chern insulator phases without external magnetic fields.

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Phys. Rev. X 16, 031043– Published 19 August, 2026
  • Received 16 July 2025
  • Revised 17 May 2026
  • Accepted 14 July 2026
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