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

TL;DR

This study uses Hartree-Fock calculations to explore the phase diagram of twisted symmetric trilayer graphene, revealing how interlayer coupling and displacement fields influence electronic phases and connect to twisted bilayer graphene states.

Contribution

It presents a comprehensive Hartree-Fock analysis of TSTG, elucidating the interplay of interlayer coupling, displacement field, and electron interactions across various fillings.

Findings

Ground states at small displacement field resemble tensor products of Dirac semimetal and TBG insulators.

At strong displacement fields, all fillings become metallic with minimal valley polarization.

Intermediate displacement fields can induce semimetal or insulator phases with zero intervalley coherence.

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 C\u{a}lug\u{a}ru 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.