Twisted bilayer graphene. VI. An exact diagonalization study at nonzero integer filling

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

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Abstract

Using exact diagonalization, we study the projected Hamiltonian with the Coulomb interaction in the eight flat bands of first magic angle twisted bilayer graphene. Employing the U(4) [U(4)×U(4)] symmetries in the nonchiral (chiral) flat band limit, we reduced the Hilbert space to an extent that allows for study around ν=±3,±2,±1 fillings. In the first chiral limit w0/w1=0, where w0 (w1) is the AA (AB) 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 ±1 excitations found in Bernevig et al. [Phys. Rev. B 103, 205415 (2021)10.1103/PhysRevB.103.205415] are the lowest charge excitations up to system sizes 8×8 (for restricted Hilbert space) in the chiral-flat limit. We also find that the flat metric condition (FMC) used by Bernevig et al. [Phys. Rev. B 103, 205411 (2021)10.1103/PhysRevB.103.205411], Song et al. [Phys. Rev. B 103, 205412 (2021)10.1103/PhysRevB.103.205412], Bernevig et al. [Phys. Rev. B 103, 205413 (2021)10.1103/PhysRevB.103.205413], Lian et al. [Phys. Rev. B 103, 205414 (2021)10.1103/PhysRevB.103.205414], and Bernevig et al. [Phys. Rev. B 103, 205415 (2021)10.1103/PhysRevB.103.205415] 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 νC=±1 at w0/w1 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 w0/w1 0.5t where t [0,1] is the factor of rescaling of the actual TBG bandwidth, and a spin singlet phase otherwise, confirming the perturbative calculation [Lian. et al., Phys. Rev. B 103, 205414 (2021)10.1103/PhysRevB.103.205414, Bultinck et al., Phys. Rev. X 10, 031034 (2020)2160-330810.1103/PhysRevX.10.031034]. The analytic FMC ground state is, however, predicted in the intervalley coherent sector which we cannot access [Lian et al., Phys. Rev. B 103, 205414 (2021)10.1103/PhysRevB.103.205414, Bultinck et al., Phys. Rev. X 10, 031034 (2020)2160-330810.1103/PhysRevX.10.031034]. For ν=-3 with/without FMC, when w0/w1 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 MM (π) stripe and KM-CDW orders at large w0/w1.

Original languageEnglish (US)
Article number205416
JournalPhysical Review B
Volume103
Issue number20
DOIs
StatePublished - May 11 2021

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics

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