The Bistritzer–MacDonald (BM) model, introduced by Bistritzer and MacDonald [Proc. Natl. Acad. Sci. U. S. A. 108, 12233–12237 (2011); arXiv:1009.4203], attempts to capture electronic properties of twisted bilayer graphene (TBG), even at incommensurate twist angles, by using an effective periodic model over the bilayer moiré pattern. Starting from a tight-binding model, we identify a regime where the BM model emerges as the effective dynamics for electrons modeled as wave-packets spectrally concentrated at monolayer Dirac points up to error that can be rigorously estimated. Using measured values of relevant physical constants, we argue that this regime is realized in TBG at the first “magic” angle.
Equivalently, the scale of variation of the wave-packet envelope in real space is proportional to γ−1.
That this θ-dependence is small was already noted in Ref. 1, and neglecting it does not affect the prediction of the first magic angle.
Note that we use tildes here to reserve for the Fourier transform of a continuous function f.
Here, we are intentionally vague about the Taylor-expansion remainder for the sake of readability. We will make the error term precise when necessary for our proofs.
Here, we refer to the translation symmetry of the individual layers, which is generally broken by the interlayer hopping. For specific “rational” twist angles, H will retain exact translation symmetry with respect to “supercell” lattice vectors, which are distinct from the monolayer lattice vectors. Note that we do not assume the rationality of the twist angle anywhere in the present work. BM models are periodic with respect to the moiré lattice, which is well-defined for generic twist angles.
Note that, with this definition, has units of energy, but has units of energy times area. Since |Γ| has units of area, the previous equation has units of energy as expected.
The rotations by are, equivalently, the momentum differences between the Dirac points measured with respect to the equivalent monolayer Dirac points obtained by rotating the Dirac points (20) by .
Note that the lower bound already follows from D1 ≥ D2, so the upper bound is the non-trivial assumption here.
In the independent electron approximation, electrons in materials occupy states defined by eigenfunctions of an effective single-particle Hamiltonian. The highest energy attained by such electrons in their ground state is known as the Fermi level, and the collection of all such electrons is known as the Fermi sea.
Note that here we use the notation Ψ to distinguish between the wave-packets noted by ψ.
Note that normally in the ansatz, there would be a time-dependent phase part, but in this case, since (by convention) the monolayer Bloch bands equal zero at the Dirac points, this phase is zero for all time.
We denote by A† the conjugate transpose of a matrix A.