Abstract
Moir´e superlattices form when van der Waals layers are stacked with a relative twist or lattice mismatch, allowing geometric control of electronic bands. Magic-angle twisted bilayer graphene exemplifies this platform, producing flat bands that drive correlated insulators and unconventional superconductivity. Yet the theoretical models used to describe these systems all have important shortcomings. The widely used Bistritzer–MacDonald (BM) model has limitations: its spectrum depends on which K-valley serves as the expansion anchor and it cannot properly describe the high-energy remote bands. The complementary single-layer Dirac–Harper model is valid only away from the degeneracy points between the layers. To overcome these deficiencies we first study uniaxially strained bilayer graphene, where enlarging the unit cell from two to four atoms exposes a block diagonal low/high-energy structure; this motivates a general full-spectrum continuum framework built directly from the microscopic tight-binding Hamiltonian via block-circulant diagonalization and full-range quasi-degenerate perturbation theory. The enlarged-cell model reproduces the critical parameters of the gap-closing transition. For one-dimensional incommensurate two-chain systems, the framework reduces to a matrix-valued almost-Mathieu operator with a fractal Cantor-set spectrum, keeping both chains so the description remains valid at the degeneracy points where the single-chain Aubry–Andr´e reduction fails. Applied to twisted bilayer graphene, the formalism reproduces the BM model in the low-energy first-shell limit and fixes the valley-ambiguity issue, while capturing the remote-band second shell. In the remote-band window, the second shell opens the degeneracies that the BM model leaves intact at the moir´e Γ point, producing a sharp van Hove singularity near 110–120 meV and an energy gap above it that the BM model lacks—a gap that offers the most direct experimental test of the theory. This unified framework applies to commensurate and quasiperiodic moir´e systems, providing a basis for future correlated-phase studies and a direct test of the predicted remote-band gap.