Large-scale numerical methods based on wave propagation

Common computational methods in condensed matter physics typically
rely on the stationary Schrödinger equation, which involves diagonalizing the
Hamiltonian and poses challenges for large systems. This talk primarily focuses on
the transformation of solved problems from the stationary Schrödinger equation to the time-dependent Schrödinger equation, thereby circumventing the need for
diagonalization and enabling large-scale simulations and calculations of complex
systems. The main topics include: (1) The linear TBPM method, based on the tight-
binding approximation, allows for non-diagonalized calculations of electronic,
optical, transport, plasmonic and magnetic properties, achieved through the wave
propagation. This method improves the scale by 5-6 orders of magnitude compared to traditional methods, enabling the study of complex quantum systems comprising
billions of atoms or even larger. Typical examples of low-dimensional systems,
heterostructures, fractals, and quasicrystals will be presented. (2) The linear DFPM
method, based on density functional theory (DFT), enables non-diagonalized self-
consistent calculations of charge density and Hamiltonian through wave propagation
in real space. This approach extends DFT to systems consisting of millions of atoms
and can be applied to large-scale first-principle calculations of ground state properties (PM), molecular dynamics simulations (MD), excited states (TDDFT), quasiparticles (GW), excitons (GW-BSE), and magnetic properties (MFT). (3) Finally, a brief overview of recent advancements in methods based on the wave propagation in quantum spin systems, quantum computer simulations, Hubbard models, and phonon calculations is provided.