Magic distances in twisted bilayer graphene

Twisted bilayer graphene exhibits isolated, relatively flat electronic bands near charge neutrality when the interlayer rotation is tuned to specific magic angles. These small misalignments, typically below 1.1°, result in long-period moiré patterns with anomalous electronic properties, posing severe challenges for accurate atomistic simulations due to the large supercell sizes required. Here, we introduce a framework to map arbitrarily stacked graphene bilayers, characterized by specific rotation angles corresponding to precise interplanar distances, onto an equivalence class represented by magic-angle twisted bilayer graphene. Using a continuum model, we derive the equivalence relation defining this class and extend its implementation to tight-binding approaches. We further explore the applicability of this mapping within density functional theory, demonstrating that the magic-angle physics can be efficiently studied using twisted bilayer graphene configurations with larger stacking angles and computationally manageable supercell sizes. This approach offers a pathway for ab initio investigations into unconventional topological phases and emergent excitations in the low-energy quasi-flat bands of twisted bilayer materials.