Projected Hartree-Fock Study in Twisted Symmetric Trilayer Graphene

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 (2021)] at integer fillings \nu = −3, −2, −1, and 0 measured from charge neutrality. We study the phase diagram with w0/w1, 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 ν are in the same phases as the tensor products of a Dirac semimetal with the filling \nu TBG insulator ground states, which are spin- valley polarized at \nu = −3, and fully (partially) intervalley coherent at \nu = −2, 0 (\nu = −1) in the flat bands. An exception is \nu = −3 with w0/w1 > 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 ν 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 \nu = −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.