Scalable Near-Linear Method for Fast Million-Atom Electronic Structure Computations

Abstract

The exploration of exotic quantum phenomena in mesoscale materials, such as moire superlattices, is fundamentally bottlenecked by the prohibitive cubic scaling barrier of conventional electronic structure methods. Here, we introduce a scalable tight binding framework that achieves a near linear empirical complexity, successfully bridging the scale gap between local first principles accuracy and macroscopic quantum simulations. By transforming the complex Hermitian Bloch Hamiltonian into an equivalent real symmetric form, our method bypasses dense diagonalization by coupling sparse LDL decomposition with Sylvester's law of inertia for direct spectral slicing and global rank calibration. This elegant mathematical reformulation enables unprecedentedly fast band structure calculations for large scale systems, solving magic angle twisted bilayer graphene in minutes on a standard laptop and scaling seamlessly to 1.5 million atoms within days on a single workstation. Applying this framework to the previously inaccessible ultra low twist angle regime incorporating atomistic strain relaxation, we unveil robust, isolated flat bands that persist down to 0.09 degree. Our framework establishes a versatile, high performance computing platform, opening a practical route toward the systematic, data driven discovery of quantum materials at experimental length scales.