Twisted bilayer graphene.VI. An Exact Diagonalization Study of Twisted Bilayer Graphene at Non-Zero Integer Fillings

Fang Xie; Aditya Cowsik; Zhi-Da Song; Biao Lian; B. Andrei Bernevig; Nicolas Regnault

doi:10.1103/PhysRevB.103.205416

paper

Twisted bilayer graphene.VI. An Exact Diagonalization Study of Twisted Bilayer Graphene at Non-Zero Integer Fillings

arXiv:2010.00588 · doi:10.1103/PhysRevB.103.205416

Abstract

Using exact diagonalization, we study the projected Hamiltonian with Coulomb interaction in the 8 flat bands of first magic angle twisted bilayer graphene. Employing the U(4) (U(4) $\times$ U(4)) symmetries in the nonchiral (chiral) flat band limit, we reduced the Hilbert space to an extent which allows for study around $ν = \pm 3, \pm 2, \pm 1$ fillings. In the first chiral limit $w_{0} / w_{1} = 0$ where $w_{0}$ ( $w_{1}$ ) is the $AA$ ( $A B$ ) stacking hopping, we find that the ground-states at these fillings are extremely well-described by Slater determinants in a so-called Chern basis, and the exactly solvable charge $\pm 1$ excitations found in [arXiv:2009.14200] are the lowest charge excitations up to system sizes $8 \times 8$ (for restricted Hilbert space) in the chiral-flat limit. We also find that the Flat Metric Condition (FMC) used in [arXiv:2009.11301,2009.11872,2009.12376,2009.13530,2009.14200] for obtaining a series of exact ground-states and excitations holds in a large parameter space. For $ν = - 3$ , the ground state is the spin and valley polarized Chern insulator with $ν_{C} = \pm 1$ at $w_{0} / w_{1} ≲ 0.9$ (0.3) with (without) FMC. At $ν = - 2$ , we can only numerically access the valley polarized sector, and we find a spin ferromagnetic phase when $w_{0} / w_{1} ≳ 0.5 t$ where $t \in [0, 1]$ is the factor of rescaling of the actual TBG bandwidth, and a spin singlet phase otherwise, confirming the perturbative calculation [arXiv:2009.13530]. The analytic FMC ground state is, however, predicted in the intervalley coherent sector which we cannot access [arXiv:2009.13530]. For $ν = - 3$ with/without FMC, when $w_{0} / w_{1}$ is large, the finite-size gap $Δ$ to the neutral excitations vanishes, leading to phase transitions. Further analysis of the ground state momentum sectors at $ν = - 3$ suggests a competition among (nematic) metal, momentum $M_{M}$ ( $π$ ) stripe and $K_{M}$ -CDW orders at large $w_{0} / w_{1}$ .

arXiv comment22+26 pages, 15+20 figures. Added discussion about spin polarized simulation and intervalley coherent states at $ν = - 2$ filling factor. Updated references

Twisted bilayer graphene.VI. An Exact Diagonalization Study of Twisted Bilayer Graphene at Non-Zero Integer Fillings

Abstract

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