TY - JOUR

T1 - Twisted bilayer graphene. IV. Exact insulator ground states and phase diagram

AU - Lian, Biao

AU - Song, Zhi Da

AU - Regnault, Nicolas

AU - Efetov, Dmitri K.

AU - Yazdani, Ali

AU - Bernevig, B. Andrei

N1 - Funding Information:
We thank A. Cowsik and F. Xie for valuable discussions. We are also grateful to M. Zaletel for helpful comments and discussions on our results. B.A.B thanks O. Vafek for fruitful discussions, and for sharing their similar results on this problem before publication . This work was supported by the DOE Grant No. DE-SC0016239, the Schmidt Fund for Innovative Research, Simons Investigator Grant No. 404513, the Packard Foundation, the Gordon and Betty Moore Foundation through Grant No. GBMF8685 towards the Princeton theory program, and a Guggenheim Fellowship from the John Simon Guggenheim Memorial Foundation. Further support was provided by the NSF-EAGER Grant No. DMR 1643312, NSF-MRSEC Grants No. DMR-1420541 and No. DMR-2011750, ONR Grant No. N00014-20-1-2303, Gordon and Betty Moore Foundation through Grant No. GBMF8685 towards the Princeton theory program, BSF Israel US foundation Grant No. 2018226, and the Princeton Global Network Funds. B.L. acknowledges the support of Princeton Center for Theoretical Science at Princeton University during the early stage of this work. D.K.E. acknowledges support from the Ministry of Economy and Competitiveness of Spain through the Severo Ochoa program for Centres of Excellence in R&D (Grant No. SE5-0522), Fundació Privada Cellex, Fundació Privada Mir-Puig, the Generalitat de Catalunya through the CERCA program, funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 852927) and the La Caixa Foundation. A.Y. is supported by the Gordon and Betty Moore Foundation's EPiQS initiative Grant No. GBMF9469, DOE-BES Grant No. DE-FG02-07ER46419, NSF-MRSEC through the Princeton Center for Complex Materials Grants No. NSF-DMR-1420541 and No. NSF-DMR-1904442.
Publisher Copyright:
© 2021 American Physical Society.

PY - 2021/5/11

Y1 - 2021/5/11

N2 - We derive the exact insulator ground states of the projected Hamiltonian of magic-angle twisted bilayer graphene (TBG) flat bands with Coulomb interactions in various limits, and study the perturbations away from these limits. We define the (first) chiral limit where the AA stacking hopping is zero, and a flat limit with exactly flat bands. In the chiral-flat limit, the TBG Hamiltonian has a U(4)×U(4) symmetry, and we find that the exact ground states at integer filling -4≤ν≤4 relative to charge neutrality are Chern insulators of Chern numbers νC=4-|ν|,2-|ν|, »,|ν|-4, all of which are degenerate. This confirms recent experiments where Chern insulators are found to be competitive low-energy states of TBG. When the chiral-flat limit is reduced to the nonchiral-flat limit which has a U(4) symmetry, we find ν=0,±2 has exact ground states of Chern number 0, while ν=±1,±3 has perturbative ground states of Chern number νC=±1, which are U(4) ferromagnetic. In the chiral-nonflat limit with a different U(4) symmetry, different Chern number states are degenerate up to second-order perturbations. In the realistic nonchiral-nonflat case, we find that the perturbative insulator states with Chern number νC=0 (0<|νC|<4-|ν|) at integer fillings ν are fully (partially) intervalley coherent, while the insulator states with Chern number |νC|=4-|ν| are valley polarized. However, for 0<|νC|≤4-|ν|, the fully intervalley coherent states are highly competitive (0.005 meV/electron higher). At nonzero magnetic field |B|>0, a first-order phase transition for ν=±1,±2 from Chern number νC=sgn(νB)(2-|ν|) to νC=sgn(νB)(4-|ν|) is expected, which agrees with recent experimental observations. Lastly, the TBG Hamiltonian reduces into an extended Hubbard model in the stabilizer code limit.

AB - We derive the exact insulator ground states of the projected Hamiltonian of magic-angle twisted bilayer graphene (TBG) flat bands with Coulomb interactions in various limits, and study the perturbations away from these limits. We define the (first) chiral limit where the AA stacking hopping is zero, and a flat limit with exactly flat bands. In the chiral-flat limit, the TBG Hamiltonian has a U(4)×U(4) symmetry, and we find that the exact ground states at integer filling -4≤ν≤4 relative to charge neutrality are Chern insulators of Chern numbers νC=4-|ν|,2-|ν|, »,|ν|-4, all of which are degenerate. This confirms recent experiments where Chern insulators are found to be competitive low-energy states of TBG. When the chiral-flat limit is reduced to the nonchiral-flat limit which has a U(4) symmetry, we find ν=0,±2 has exact ground states of Chern number 0, while ν=±1,±3 has perturbative ground states of Chern number νC=±1, which are U(4) ferromagnetic. In the chiral-nonflat limit with a different U(4) symmetry, different Chern number states are degenerate up to second-order perturbations. In the realistic nonchiral-nonflat case, we find that the perturbative insulator states with Chern number νC=0 (0<|νC|<4-|ν|) at integer fillings ν are fully (partially) intervalley coherent, while the insulator states with Chern number |νC|=4-|ν| are valley polarized. However, for 0<|νC|≤4-|ν|, the fully intervalley coherent states are highly competitive (0.005 meV/electron higher). At nonzero magnetic field |B|>0, a first-order phase transition for ν=±1,±2 from Chern number νC=sgn(νB)(2-|ν|) to νC=sgn(νB)(4-|ν|) is expected, which agrees with recent experimental observations. Lastly, the TBG Hamiltonian reduces into an extended Hubbard model in the stabilizer code limit.

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U2 - 10.1103/PhysRevB.103.205414

DO - 10.1103/PhysRevB.103.205414

M3 - Article

AN - SCOPUS:85106351068

VL - 103

JO - Physical Review B

JF - Physical Review B

SN - 2469-9950

IS - 20

M1 - 205414

ER -