Multiscale Methods for Quantum Many-Body Systems

Quantum many-body system is principally a complicated problem with high complexity, where emergent phenomena are hidden behind strong correlations beyond the mean-field. Due to its exponential scaling, direct solution is beyond our ability and multiscale methods provide a natural framework for conquering strong correlations within the system with cost being reduced to an affordable level. In Ch. 1 I will briefly summarize the definition of quantum many-body systems and the ongoing efforts to the problems. After that, I will approach quantum many-body systems from three different aspects: two novel multiscale initialization methods inspired by multigrid method and mean-field theory respectively to solve quantum ground-state problems in Ch. 2, a fast Green's function representation applied to dynamical mean-field theory in Ch. 3, and a novel Green's function embedding framework called quantum defect embedding theory to approach strongly correlated electronic states in solids in Ch. 4. Finally, I will summarize what have been done and propose future steps in Ch. 5. The new methods developed allow a combination of low computational cost and good accuracy, helping people understand novel physical phenomena from their many-body nature.