Existence of the first magic angle for the chiral model of bilayer graphene

Abstract

We consider the chiral model of twisted bilayer graphene introduced by Tarnopolsky-Kruchkov-Vishwanath (TKV). TKV have proved that for inverse twist angles $\alpha$ such that the effective Fermi velocity at the moiré $K$ point vanishes, the chiral model has a perfectly flat band at zero energy over the whole Brillouin zone. By a formal expansion, TKV found that the Fermi velocity vanishes at $\alpha \approx .586$. In this work, we give a proof that the Fermi velocity vanishes for at least one $\alpha$ between $.57$ and $.61$ by rigorously justifying TKV's formal expansion of the Fermi velocity over a sufficiently large interval of $\alpha$ values. The idea of the proof is to project the TKV Hamiltonian onto a finite dimensional subspace, and then expand the Fermi velocity in terms of explicitly computable linear combinations of modes in the subspace, while controlling the error. The proof relies on two propositions whose proofs are computer-assisted, i.e., numerical computation together with worst-case estimates on the accumulation of round-off error which show that round-off error cannot possibly change the conclusion of the computation. The propositions give a bound below on the spectral gap of the projected Hamiltonian, an Hermitian $80\times 80$ matrix whose spectrum is symmetric about $0,$ and verify that two real 18th order polynomials, which approximate the numerator of the Fermi velocity, take values with definite sign when evaluated at specific values of $\alpha$. Together with TKV's work our result proves existence of at least one perfectly flat band of the chiral model.

61 pages (last 7 pages are supplementary material), 11 figures. Improved after review comments: (1) proved that the Fermi velocity perturbation series converges for $|\alpha |<1/3$ (2) added proof that round-off error cannot affect the conclusions of our numerical computations (3) more extensive discussion of related work

References in corpus (1)

• Spectral characterization of magic angles in twisted bilayer graphene

Cited by in corpus (11)

• Mathematics of magic angles in a model of twisted bilayer graphene
• Chiral limit and origin of topological flat bands in twisted transition metal dichalcogenide homobilayers
• Bistritzer-MacDonald dynamics in twisted bilayer graphene
• Interacting models for twisted bilayer graphene: a quantum chemistry approach
• A simple derivation of moiré-scale continuous models for twisted bilayer graphene
• Neutral magic-angle bilayer graphene: Condon instability and chiral resonances
• Flat Bands and High Chern Numbers in Twisted Multilayer Graphene
• Exact ground state of interacting electrons in magic angle graphene
• Optical response of alternating twisted trilayer graphene
• Review of the tight-binding method applicable to the properties of moiré superlattices
• Classically forbidden regions in the chiral model of twisted bilayer graphene. With an appendix by Zhongkai Tao and Maciej Zworski