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 C\ifmmode \u{a}\else \u{a}\fi{}lug\ifmmode \u{a}\else \u{a}\fi{}ru et al. [Phys. Rev. B 103, 195411 (2021)] at integer fillings $\ensuremath{\nu}=\ensuremath{-}3,\phantom{\rule{0.222222em}{0ex}}\ensuremath{-}2,\phantom{\rule{0.222222em}{0ex}}\ensuremath{-}1$, 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 $\ensuremath{\nu}$ are in the same phases as the tensor products of a Dirac semimetal with the filling $\ensuremath{\nu}$ TBG insulator ground states, which are spin-valley polarized at $\ensuremath{\nu}=\ensuremath{-}3$, and fully (partially) intervalley coherent at $\ensuremath{\nu}=\ensuremath{-}2,0$ ($\ensuremath{\nu}=\ensuremath{-}1$) in the flat bands. An exception is $\ensuremath{\nu}=\ensuremath{-}3$ with ${w}_{0}/{w}_{1}\ensuremath{\gtrsim}0.7$, 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 $\ensuremath{\nu}$ 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 $\ensuremath{\nu}=\ensuremath{-}2,\phantom{\rule{0.222222em}{0ex}}\ensuremath{-}1,\phantom{\rule{0.222222em}{0ex}}0$. Our results provide a simple picture for the electron interactions in TSTG systems, and reveal the connection between the TSTG and TBG ground states.