▎ 摘　　要

The Hamiltonian of the magic-angle twisted symmetric trilayer graphene (TSTG) can be decomposed into a twisted-bilayer-graphene- (TBG-) like flat band Hamiltonian and a high-velocity Dirac fermion Hamiltonian. We use Hartree-Fock mean field approach to study the projected Coulomb interacting Hamiltonian of TSTG developed in Calugaru et al. [Phys. Rev. B 103, 195411 '20:.)] at integer fillings v = -3, -2, -1, and 0 measured from charge neutrality. We study the phase diagram with w(0)/w(1), the ratio of AA and AB interlayer hoppings, and the displacement field, which introduces an interlayer potential U and hybridizes the TBG-like bands with the Dirac bands. At small U, we find the ground states at all fillings v are in the same phases as the tensor products of a Dirac semimetal with the filling v TBG insulator ground states, which are spin-valley polarized at v = -3, and fully (partially) intervalley coherent at v = -2, 0 (v = -1) in the flat bands. An exception is v = -3 with w(0)/w(1) greater than or similar to 0.7, which possibly becomes a metal with competing orders at small U due to charge transfers between the Dirac and flat bands. At strong U where the bandwidths exceed interactions, all the fillings v enter a metal phase with small or zero valley polarization and intervalley coherence. Lastly, at intermediate U, semimetal or insulator phases with zero intervalley coherence may arise for v = -2, -1, 0. Our results provide a simple picture for the electron interactions in TSTG systems, and reveal the connection between the TSTG and TBG ground states.