Atomistic theory of the moiré Hofstadter butterfly in magic-angle graphene
Bibliographic record
Abstract
We present here a Hofstadter's butterfly spectrum for the magic-angle twisted bilayer graphene obtained using an ab initio-based multimillion-atom tight-binding model. We incorporate a hexagonal boron nitride substrate and out-of-plane atomic relaxation. The effects of a magnetic field are introduced via the Peierls modification of the long-range tight-binding matrix elements and the Zeeman spin splitting effects. A nanoribbon geometry is studied, and the quantum size effects for the sample widths up to 1 $\textmu{}\mathrm{m}$ are analyzed both for a large energy window and for the flat band around the Fermi level. For sufficiently wide ribbons, where the role of the finite geometry is minimized, we obtain and plot the Hofstadter spectrum and identify the in-gap Chern numbers by counting the total number of chiral edge states crossing these gaps. Subsequently, we examine the Wannier diagrams to identify the insulating states at charge neutrality. We establish the presence of three types of electronic states: moir\'e, mixed, and conventional. These states describe both the bulk Landau levels and the edge states crossing gaps in the spectrum. The evolution of the bulk moir\'e flat band wave functions in the magnetic field is investigated, predicting a decay of the electronic density from the moir\'e centers as the magnetic flux increases. Furthermore, the spatial properties of the three types of edge states are studied, illustrating the evolution of their localization as a function of the nanoribbon momentum.
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How this classification was reachedexpand
Direct model labels (unvalidated)
Per-model category and study-design labels from the labeling rounds. They are machine output, unvalidated, and the disagreement between models ships as data. No study design here is MEDLINE-validated yet.
| Model arm | Categories | Study design | Confidence |
|---|---|---|---|
| gemma | no category Domain: not available · Genre: Empirical About the Canadian research system: no · About a Canadian topic: no | Theoretical or conceptual | low |
| gpt | no category Domain: not available · Genre: Methods About the Canadian research system: no · About a Canadian topic: no | Theoretical or conceptual | medium |
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedLabeled directly by 2 models reading the full record.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".