A Numerical Perspective on Moir\'e Superlattices: From Single-Particle Properties to Many-Body Physics

Moir\'e superlattices in two-dimensional materials provide a versatile platform to explore strongly correlated and topological phases. This work presents a practical theoretical workflow for studying the correlated and topological states in moir\'e systems, combining continuum modeling, Hartree-Fock mean-field approximations, many-body perturbation theory, and exact diagonalizations. We focus on the numerical implementation of these methods, emphasizing subtleties such as remote band effects, inhomogeneous and dynamical screening, double counting problem, etc., which are often swept under the rug in theoretical works. The workflow enables a systematic investigation of symmetry-breaking ground state properties, quasiparticle excitation properties and fractional Chern insulator phases emerging from moir\'e superlattices, providing insights that are directly relevant to experimental observations. By bridging technical details and physical interpretations, this work aims to guide both theorists and experimentalists in understanding and predicting correlated phenomena in moir\'e materials.