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The Hamiltonian of the magic-angle twisted symmetric trilayer graphene (TSTG)can be decomposed into a TBG-like flat band Hamiltonian and a high-velocityDirac fermion Hamiltonian. We use Hartree-Fock mean field approach to study theprojected Coulomb interacting Hamiltonian of TSTG developed in C\u{a}lug\u{a}ruet al. [Phys. Rev. B 103, 195411 (2021)] at integer fillings $\nu=-3, -2, -1$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 displacementfield, which introduces an interlayer potential $U$ and hybridizes the TBG-likebands with the Dirac bands. At small $U$, we find the ground states at allfillings $\nu$ are in the same phases as the tensor products of a Diracsemimetal with the filling $\nu$ TBG insulator ground states, which arespin-valley polarized at $\nu=-3$, and fully (partially) intervalley coherentat $\nu=-2,0$ ($\nu=-1$) in the flat bands. An exception is $\nu=-3$ with$w_0/w_1 \gtrsim 0.7$, which possibly become a metal with competing orders atsmall $U$ due to charge transfers between the Dirac and flat bands. At strong$U$ where the bandwidths exceed interactions, all the fillings $\nu$ enter ametal phase with small or zero valley polarization and intervalley coherence.Lastly, at intermediate $U$, semimetal or insulator phases with zerointervalley coherence may arise for $\nu=-2,-1,0$. Our results provide a simplepicture for the electron interactions in TSTG systems, and reveal theconnection between the TSTG and TBG ground states.

TSTG II: Projected Hartree-Fock Study of Twisted Symmetric Trilayer Graphene