Abstract
This thesis develops variational Gutzwiller theory for realistic multi-orbital strongly correlated systems and applies it to twisted bilayer graphene (TBG), where correlated insulating, metallic, and superconducting phases appear in close proximity. We first establish an efficient and robust implementation of the multi-orbital Gutzwiller method, including the original-basis formulation, stable variational equations, analytical Jacobians and gradients, practical density-matrix parametrizations, and the use of gauge freedom to reduce computational cost. Benchmark calculations demonstrate that this implementation captures correlation effects beyond Hartree-Fock and reproduces known strong-coupling behavior. We then apply the method to an eight-band model of TBG with correlated f orbitals and itinerant c orbitals. The resulting phase diagram reveals a Fermi-liquid dome that separates a weakly correlated BCS-like superconducting regime from a strongly correlated superconducting regime. In the latter, pairing is tied to local spin-valley-orbital structure within a moiré unit cell and is better understood in terms of preformed local pairs that gain itinerancy through charge fluctuations and hybridization. Finally, we develop a linear-response framework within the variational Gutzwiller method and derive an analytical expression for the static compressibility. Applied to symmetric TBG, this formalism naturally explains the cascade-like compressibility minima as a consequence of strong correlations in the localized orbitals. Together, these results establish variational Gutzwiller theory as a powerful framework for studying correlated normal and superconducting phases in moiré materials.