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, and Vishwanath (TKV). TKV proved that for inverse twist angles α 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 α ≈ 0.586. In this work, we give a proof that the Fermi velocity vanishes for at least one α between 0.57 and 0.61 by rigorously justifying TKV’s formal expansion of the Fermi velocity over a sufficiently large interval of α 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 computation. The propositions give a bound below on the spectral gap of the projected Hamiltonian, an Hermitian 80 × 80 matrix whose spectrum is symmetric about 0, and verify that two real 18-th order polynomials, which approximate the numerator of the Fermi velocity, take values with a definite sign when evaluated at specific values of α. Together with TKV’s work, our result proves the existence of at least one perfectly flat band of the chiral model.

A simple derivation of moiré-scale continuous models for twisted bilayer graphene

“…The asymptotic equality (30) leads us to introduce the following effective model for the propagation of low-energy wave-packets in TBG:…”

“…A rigorous mathematical derivation of the BM model from a tight-binding Hamiltonian whose parameters satisfy suitable scaling laws, was recently proposed in [29]. A simplified chiral BM model, obtained by setting w AA = 0, was notably used in [27,2,30] to prove the existence of perfectly flat bands at the Fermi level for a sequence of so-called "magic" angles. From a physical point of view, the existence of partially occupied almost flat bands in the single-particle picture may reflect the presence of localized strongly correlated electrons, and provide a possible explanation of the experimentally observed superconducting behavior [25].…”

Bistritzer-MacDonald dynamics in twisted bilayer graphene

“…Becker-Ge-Wittsten have also studied the transport and spectral properties of this model in [5]. Two of the authors of the present work gave a computer-assisted proof of existence of the first magic angle of the "chiral" approximation to the Bistritzer-MacDonald model in [44]. Numerical methods for computing electronic properties of twisted multilayer materials which exploit similar approximations to those made in the Bistritzer-MacDonald model have been developed [11,15,17,34,36,37].…”

“…The translation and rotation symmetries of H BM are also discussed in the supplementary material of [44].…”

Discrete honeycombs, rational edges and edge states

“…[2][3][4]24] or models of multilayer structures, such as twisted bilayer graphene, e.g. [5,7,27,29]. Parallel questions for quantum graph models [6,20] would also be of interest.…”