Real-Symmetric Hamiltonian Enables Near-Linear Scaling for Fast Million-Atom Electronic Structure Computations

The exploration of quantum phenomena in mesoscale materials, such as moire superlattices, is limited by the cubic scaling cost of conventional electronic structure methods. Here, we introduce a scalable tight binding framework that achieves near linear scaling, enabling mesoscopic quantum simulations. By transforming the complex Hermitian Bloch Hamiltonian into an equivalent real symmetric form, the method avoids dense diagonalization by combining sparse LDL decomposition with Sylvester's law of inertia for spectral slicing and global rank calibration. This formulation enables efficient band structure calculations for large scale systems, solving magic angle twisted bilayer graphene in minutes on a standard laptop and extending to 1.5 million atoms within days on a single workstation. Applying this framework to ultra low twist angle structures with atomistic strain relaxation, we find robust isolated low-energy band clusters over several finite ultra low angle windows down to 0.09 degree. Our framework provides an efficient computational platform for studying quantum materials at experimentally relevant length scales and supports data driven discovery in large scale moire systems.