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

Abstract

The Hamiltonian of the magic-angle twisted symmetric trilayer graphene (TSTG) can be decomposed into a 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 $\nu =-3,-2,-1$ 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 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 $\nu$ 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 $w_0/w_1\gtrsim 0.7$, which possibly become 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 $\nu$ 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 $\nu =-2,-1,0$. Our results provide a simple picture for the electron interactions in TSTG systems, and reveal the connection between the TSTG and TBG ground states.

31 pages, 27 figures

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