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

Abstract

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. B 103, 195411 (2021)] at integer fillings 𝜈=−3,−2,−1, and 0 measured from charge neutrality. We study the phase diagram with 𝑤0/𝑤1, the ratio of 𝐴⁢𝐴 and 𝐴⁢𝐵 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 𝜈 TBG insulator ground states, which are spin-valley polarized at 𝜈=−3, and fully (partially) intervalley coherent at 𝜈=−2,0 (𝜈=−1) in the flat bands. An exception is 𝜈=−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 𝜈=−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.