Effects of finite AA tunneling beyond the chiral limit

Characterize how the squared-Hamiltonian structures, zero-mode localization, magic-angle behavior, and related phenomena of chiral twisted bilayer graphene change when finite same-sublattice AA tunneling is restored, and determine why the first magic angle survives under this deformation.

Background

The analysis throughout the paper is primarily performed in the chiral limit, where the same-sublattice AA tunneling is set to zero and the Hamiltonian has an exact chiral symmetry. Real twisted bilayer graphene instead has reduced but nonzero AA tunneling because lattice relaxation does not eliminate it completely.

The paper explicitly identifies the extension to finite wAAw_{AA} as unresolved. In particular, it asks how the non-Abelian gauge-field description, zero-mode structures, magic-angle sequence, and the special role of the first magic angle are modified once chiral symmetry is no longer exact. The authors note that this issue is addressed in a separate work, but it remains an open problem in the present paper.

References

Several questions remain open. The squared-Hamiltonian approach should apply to other moiré systems with a chiral limit, such as twisted multilayer graphene, where flat bands with higher Chern numbers are known . It would be interesting to see whether a similar rescaling argument fixes their magic-angle sequences, and whether their gauge structure goes beyond $SU(2)$. A second direction is to use the coherent-state basis of Sec.~\ref{sec:localization}, labelled by the angular momentum $m$, as a starting point for projecting the Coulomb interaction, and to compare the resulting pairing estimates with the geometric superfluid weight of Eq.~eq:Dsconvgeom. Third, real samples are not chiral: the AA tunneling is reduced by relaxation but not removed. How the structures described here change for finite $w_{AA}$, and why the first magic angle is the one that survives, is the subject of a separate work .

— Theory of Magic Angles in Twisted Bilayer Graphene: from Non-Abelian Gauge Fields to Flat Bands  (2610.07734 - Navarro-Labastida et al., 6 Oct 2026) in Section Conclusions and Outlook