From Bellissard's gap labeling to atomistic quasiperiodic bilayers
Abstract
Bellissard's framework uses covariant observables to describe aperiodic solids. We examine how butterfly-like spectra in twisted bilayers can be related to bulk gap labels and topological response. A bulk gap permits a spectral projector and an integrated density of states. A topological response requires an additional pairing of that projector. In an illustrative finite honeycomb bilayer, an exact second-moment identity isolates the interlayer contribution to spectral variance. Angle-resolved spectra, finite state counts, an uncoupled reference, and size and broadening checks show how coupling redistributes spectral weight without assigning a bulk gap label. We then examine the requirements for atomistic electronic-structure calculations, including nonorthogonal orbitals, projected densities of states, and basis functions that move during relative sliding. The resulting framework specifies the steps needed to connect atomistic spectra with bulk state counting and topological response.
Related
- On the types of quasiperiodic potentials and level surfaces of quasiperiodic functions
- Accurately computing quasiperiodic parabolic equations within finite-size domains via modeling quasiperiodic boundary conditions
- Arithmetic localization for quasiperiodic long-range operators with monotone potentials
- Smooth perturbation theory and generic Cantor spectrum for multi-dimensional monotone quasiperiodic operators
- Dynamical Signatures of Selective Gap Closing in Multi-Dimensional Monotone Quasiperiodic Operators
BibTeX
@misc{kuhne2026from,
title = {{From Bellissard's gap labeling to atomistic quasiperiodic bilayers}},
author = {Thomas D. Kühne and Vladislav Efremkin and Emil Prodan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36981},
eprint = {2609.36981},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36981},
}
RIS
TY - UNPB
TI - From Bellissard's gap labeling to atomistic quasiperiodic bilayers
AU - Thomas D. Kühne
AU - Vladislav Efremkin
AU - Emil Prodan
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36981
UR - https://arxiv.org/abs/2609.36981
AB - Bellissard's framework uses covariant observables to describe aperiodic solids. We examine how butterfly-like spectra in twisted bilayers can be related to bulk gap labels and topological response. A bulk gap permits a spectral projector and an integrated density of states. A topological response requires an additional pairing of that projector. In an illustrative finite honeycomb bilayer, an exact second-moment identity isolates the interlayer contribution to spectral variance. Angle-resolved spectra, finite state counts, an uncoupled reference, and size and broadening checks show how coupling redistributes spectral weight without assigning a bulk gap label. We then examine the requirements for atomistic electronic-structure calculations, including nonorthogonal orbitals, projected densities of states, and basis functions that move during relative sliding. The resulting framework specifies the steps needed to connect atomistic spectra with bulk state counting and topological response.
ER -