  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 prove the Fermi velocity
vanishes at $\alpha \approx .586$, and put rigorous minimum and maximum bounds
on the location of this zero, 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 assumptions which can be checked numerically: a
bound below on the smallest eigenvalue of a positive semi-definite, Hermitian
$81 \times 81$ matrix which is essentially the square of the projected
Hamiltonian, and an assumption on the validity of the negative value of a real
18th order polynomial approximating the numerator of the Fermi velocity when
evaluated at a specific value of $\alpha$. Since these assumptions can be
verified up to high precision using standard numerical methods, together with
TKV's work our result proves existence of at least one perfectly flat band of
the chiral model.
