Twisted symmetric trilayer graphene. II. Projected Hartree-Fock study

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 Călugăru et al. [Phys. Rev. B103, 195411 (2021)2469-995010.1103/PhysRevB.103.195411] at integer fillings , and 0 measured from charge neutrality. We study the phase diagram with , the ratio of and interlayer hoppings, and the displacement field, which introduces an interlayer potential and hybridizes the TBG-like bands with the Dirac bands. At small , we find the ground states at all fillings are in the same phases as the tensor products of a Dirac semimetal with the filling TBG insulator ground states, which are spin-valley polarized at , and fully (partially) intervalley coherent at () in the flat bands. An exception is with , which possibly becomes a metal with competing orders at small due to charge transfers between the Dirac and flat bands. At strong where the bandwidths exceed interactions, all the fillings enter a metal phase with small or zero valley polarization and intervalley coherence. Lastly, at intermediate , semimetal or insulator phases with zero intervalley coherence may arise for . Our results provide a simple picture for the electron interactions in TSTG systems, and reveal the connection between the TSTG and TBG ground states.