Robustness of real-space topology in moir\'e systems

TL;DR

This paper investigates the robustness of real-space topological properties in moiré systems, revealing that while individual Bloch wavefunctions' topological markers are fragile, ensemble-based textures exhibit robust, symmetry-protected topological indices observable in experiments.

Contribution

It demonstrates that ensemble-based real-space topological textures in moiré systems are robust and symmetry-protected, unlike individual Bloch wavefunction markers, with implications for experimental detection.

Findings

Individual Bloch wavefunctions' Chern number is fragile and vanishes in general.

Ensemble textures have a robust, symmetry-protected Chern number.

Symmetries ensure nonzero topological indices across twist angles and corrugations.

Abstract

The appearance of fractional Chern insulators in moir\'e systems can be rationalized by the presence of a fictitious magnetic field associated with the spatial texture of layer-resolved electronic wavefunctions. Here, we present a systematic study of real-space topology and the associated fictitious magnetic fields in moir\'e systems. We first show that at the level of individual Bloch wavefunctions, 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 zeroes within the unit cell, such as the chiral limit of twisted bilayer graphene (TBG) or the adiabatic regime of twisted homobilayer transition metal dichalcogenides (TMD). We then show that these limitations do not apply to textures associated with ensembles of Bloch wavefunctions, 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, implying experimental signatures in scanning tunneling microscopy measurements. 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.