This note extends a striking mathematical identity about “magic angles” in the chiral model of twisted bilayer graphene. Earlier work by Becker et al. showed that, for the exact Bistritzer–MacDonald potential, the sum of the fourth powers of the generalized magic angles equals 8π/√3. The author proves a version of that identity for a larger family of potentials that share the lattice and rotational symmetries of the model.

The paper works with the chiral continuum model. In that model one studies a differential operator that depends on a complex potential U(z) and a parameter α that is essentially the inverse of the physical twist angle. Magic angles are special values of α at which the operator develops flat electronic bands. Mathematically, these magic values appear as inverse eigenvalues of a compact operator built from U(z) and the derivative operators. The author assumes the potential respects the model symmetries and can be written with finitely many Fourier modes of a specific form.

Under these assumptions the main result is an explicit trace formula for the fourth power sum of the inverse magic angles. Writing the potential in the chosen Fourier basis with real coefficients c_n and a lattice parameter K, the paper shows

sum_{α in A} α^{-4} = 8 sqrt(3) π K^4 \sum_{n,m} (c_n^2 c_m^2 + 2 c_n^2 c_m c_{-n-m})/(n^2 + n m + m^2).

Here the left sum counts each magic value with its algebraic multiplicity. The c_n are the Fourier coefficients that describe the allowed potential, and the integer K comes from the dual lattice used in the model. For the exact Bistritzer–MacDonald potential, choosing the corresponding coefficients recovers the earlier 8π/√3 identity reported by Becker et al.