Compressibility and non-local charge response study of twisted bilayer graphene

Magic-angle twisted bilayer graphene (TBG) provides an exceptional platform for studying strongly correlated electrons. In this work, we apply the Gutzwiller approximation (GA) to an effective eight-band tight-binding model to explore its correlated behavior. We observe a cascade-like transition in the compressibility of the symmetric state upon doping, linked to the redistribution of electrons between localized and delocalized orbitals. Additionally, we develop a static linear response theory for GA to examine the charge response to inhomogeneous potentials within an efficient GA framework.