TY - JOUR

T1 - Twisted bilayer graphene. III. Interacting Hamiltonian and exact symmetries

AU - Bernevig, B. Andrei

AU - Song, Zhi Da

AU - Regnault, Nicolas

AU - Lian, Biao

N1 - Funding Information:
We thank Aditya Cowsik and Fang Xie for valuable discussions. B.A.B. thanks Oskar Vafek for fruitful discussions and for sharing similar results on this problem before publication . This work was supported by 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 toward the Princeton theory program, and a Guggenheim Fellowship from the John Simon Guggenheim Memorial Foundation. Further support was provided by NSF-EAGER Grant No. DMR 1643312, NSF-MRSEC Grants No. DMR-1420541 and No. DMR-2011750, ONR Grant No. N00014-20-1-2303, BSF Israel-US Foundation Grant No. 2018226, and the Princeton Global Network Funds. B.L. acknowledges the support of the Princeton Center for Theoretical Science at Princeton University during the early stage of this work.
Publisher Copyright:
© 2021 American Physical Society.

PY - 2021/5/11

Y1 - 2021/5/11

N2 - We derive the explicit Hamiltonian of twisted bilayer graphene (TBG) with Coulomb interaction projected into the flat bands and study the symmetries of the Hamiltonian. First, we show that all projected TBG Hamiltonians can be written as positive semidefinite Hamiltonians, an example of which was found in work by Kang and Vafek [Phys. Rev. Lett. 122, 246401 (2019)PRLTAO0031-900710.1103/PhysRevLett.122.246401]. We then prove that the interacting TBG Hamiltonian exhibits an exact U(4) symmetry in the exactly flat band (nonchiral-flat) limit. We further define, besides a first chiral limit where the AA stacking hopping is zero, a second chiral limit where the AB/BA stacking hopping is zero. In the first chiral-flat limit (or second chiral-flat limit) with exactly flat bands, the TBG is enhanced to have an exact U(4)×U(4) symmetry, whose generators are different between the two chiral limits. While in the first chiral limit and in the nonchiral case these symmetries have been found in work by Bultinck [Phys. Rev. X 10, 031034 (2020)2160-330810.1103/PhysRevX.10.031034], for the eight lowest bands, we here prove that they are valid for projection into any 8nmax particle-hole symmetric TBG bands, with nmax>1 being the practical case for small twist angles <1. Furthermore, in the first or second chiral-nonflat limit without flat bands, an exact U(4) symmetry still remains. We also elucidate the link between the U(4) symmetry presented here and the similar but different U(4) of Kang and Vafek [Phys. Rev. Lett. 122, 246401 (2019)PRLTAO0031-900710.1103/PhysRevLett.122.246401]. Furthermore, we show that our projected Hamiltonian can be viewed as the normal-ordered Coulomb interaction plus a Hartree-Fock term from passive bands, and exhibits a many-body particle-hole symmetry which renders the physics symmetric around charge neutrality. We also provide an efficient parametrization of the interacting Hamiltonian. The existence of two chiral limits with an enlarged symmetry suggests a possible duality of the model yet undiscovered.

AB - We derive the explicit Hamiltonian of twisted bilayer graphene (TBG) with Coulomb interaction projected into the flat bands and study the symmetries of the Hamiltonian. First, we show that all projected TBG Hamiltonians can be written as positive semidefinite Hamiltonians, an example of which was found in work by Kang and Vafek [Phys. Rev. Lett. 122, 246401 (2019)PRLTAO0031-900710.1103/PhysRevLett.122.246401]. We then prove that the interacting TBG Hamiltonian exhibits an exact U(4) symmetry in the exactly flat band (nonchiral-flat) limit. We further define, besides a first chiral limit where the AA stacking hopping is zero, a second chiral limit where the AB/BA stacking hopping is zero. In the first chiral-flat limit (or second chiral-flat limit) with exactly flat bands, the TBG is enhanced to have an exact U(4)×U(4) symmetry, whose generators are different between the two chiral limits. While in the first chiral limit and in the nonchiral case these symmetries have been found in work by Bultinck [Phys. Rev. X 10, 031034 (2020)2160-330810.1103/PhysRevX.10.031034], for the eight lowest bands, we here prove that they are valid for projection into any 8nmax particle-hole symmetric TBG bands, with nmax>1 being the practical case for small twist angles <1. Furthermore, in the first or second chiral-nonflat limit without flat bands, an exact U(4) symmetry still remains. We also elucidate the link between the U(4) symmetry presented here and the similar but different U(4) of Kang and Vafek [Phys. Rev. Lett. 122, 246401 (2019)PRLTAO0031-900710.1103/PhysRevLett.122.246401]. Furthermore, we show that our projected Hamiltonian can be viewed as the normal-ordered Coulomb interaction plus a Hartree-Fock term from passive bands, and exhibits a many-body particle-hole symmetry which renders the physics symmetric around charge neutrality. We also provide an efficient parametrization of the interacting Hamiltonian. The existence of two chiral limits with an enlarged symmetry suggests a possible duality of the model yet undiscovered.

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U2 - 10.1103/PhysRevB.103.205413

DO - 10.1103/PhysRevB.103.205413

M3 - Article

AN - SCOPUS:85106347506

SN - 2469-9950

VL - 103

JO - Physical Review B

JF - Physical Review B

IS - 20

M1 - 205413

ER -