PreprintCC BY 4.0arXiv
Dirac operators on periodic planar graphs: the right definition advocated by flat bands
Abstract
Massive discrete Dirac-type operators on periodic planar graphs are investigated. Using Floquet--Bloch decomposition, we first show that neglecting the faces leads to spurious flat bands that are unrelated to the graph's geometry. Then, for the correct Dirac--Hodge operator Dm, we establish a spectral reduction expressing σ(Dm) through the graph Laplacian on vertices −ΔV and the Laplacian on faces −ΔF. As a consequence, Dm carries a flat band precisely when one of these scalar Laplacians does. We finally apply this reduction to the study of flat bands of the operator Dm on Archimedean lattices and contrast the outcome with that on the Lieb lattice.
Related
- Sufficiently wide strips uniquely minimize the planar periodic fractional perimeter
- Back to the Definition: Estimating Step-Level Advantages via Trajectory Graphs for Agentic Reinforcement Learning
- Band edges of periodic Schrödinger operators are generically isolated and nondegenerate
- Irreducibility of the Bloch variety for periodic Schrödinger operators in arbitrary dimension
- Bounds for B-coloring planar and outerplanar graphs
BibTeX
@misc{kato2026dirac,
title = {{Dirac operators on periodic planar graphs: the right definition advocated by flat bands}},
author = {Koichi Kato and Daniel Parra and Serge Richard},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10964},
eprint = {2610.10964},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10964},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dirac operators on periodic planar graphs: the right definition advocated by flat bands
AU - Koichi Kato
AU - Daniel Parra
AU - Serge Richard
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10964
UR - https://arxiv.org/abs/2610.10964
AB - Massive discrete Dirac-type operators on periodic planar graphs are investigated. Using Floquet--Bloch decomposition, we first show that neglecting the faces leads to spurious flat bands that are unrelated to the graph's geometry. Then, for the correct Dirac--Hodge operator D_m, we establish a spectral reduction expressing σ(D_m) through the graph Laplacian on vertices −Δ_V and the Laplacian on faces −Δ_F. As a consequence, D_m carries a flat band precisely when one of these scalar Laplacians does. We finally apply this reduction to the study of flat bands of the operator D_m on Archimedean lattices and contrast the outcome with that on the Lieb lattice.
ER -