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

TL;DR

This paper rigorously proves the existence of a specific twist angle in the chiral model of bilayer graphene where the Fermi velocity vanishes, confirming the presence of a perfectly flat band as previously suggested by formal expansions.

Contribution

The authors provide a rigorous proof of a twist angle where the Fermi velocity vanishes in the chiral model, validating prior formal predictions through computer-assisted analysis.

Findings

Existence of at least one twist angle with zero Fermi velocity between 0.57 and 0.61.

Verification of the spectral gap and sign of polynomials related to the Fermi velocity.

Confirmation of a perfectly flat band in 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 moir\'e $K$ 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.