Twisted bilayer graphene. III. Interacting Hamiltonian and exact symmetries

Bernevig, B. Andrei; Song, Zhi-Da; Regnault, Nicolas; Lian, Biao

doi:10.1103/physrevb.103.205413

Twisted bilayer graphene. III. Interacting Hamiltonian and exact symmetries

Journal Article · Tue May 11 00:00:00 EDT 2021 · Physical Review. B

DOI:https://doi.org/10.1103/physrevb.103.205413· OSTI ID:1852145

Bernevig, B. Andrei ^[1]; Song, Zhi-Da ^[2]; Regnault, Nicolas ^[3]; Lian, Biao ^[2]

Princeton Univ., NJ (United States); OSTI
Princeton Univ., NJ (United States)
Princeton Univ., NJ (United States); Univ. Paris-Saclay, Sorbonne (France); University Paris-Diderot (France); Sorbonne Univ., Paris (France)

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)]. 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 et al. [Phys. Rev. X 10, 031034 (2020)], for the eight lowest bands, we here prove that they are valid for projection into any 8 n max particle-hole symmetric TBG bands, with n_max > 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)]. Additionally, 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.

View Accepted Manuscript (DOE)

Research Organization:: Princeton University, NJ (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: SC0016239

OSTI ID:: 1852145

Journal Information:: Physical Review. B, Journal Name: Physical Review. B Journal Issue: 20 Vol. 103; ISSN 2469-9950

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

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Nematicity with a twist: Rotational symmetry breaking in a moiré superlattice Fernandes, Rafael M.; Venderbos, Jörn W. F. Science Advances, Vol. 6, Issue 32 https://doi.org/10.1126/sciadv.aba8834	journal	August 2020
Tuning superconductivity in twisted bilayer graphene Yankowitz, Matthew; Chen, Shaowen; Polshyn, Hryhoriy Science, Vol. 363, Issue 6431 https://doi.org/10.1126/science.aav1910	journal	January 2019
Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene Sharpe, Aaron L.; Fox, Eli J.; Barnard, Arthur W. Science, Vol. 365, Issue 6453 https://doi.org/10.1126/science.aaw3780	journal	July 2019
Ferromagnetism in magic-angle graphene Pixley, Jed H.; Andrei, Eva Y. Science, Vol. 365, Issue 6453 https://doi.org/10.1126/science.aay3409	journal	August 2019
Intrinsic quantized anomalous Hall effect in a moiré heterostructure Serlin, M.; Tschirhart, C. L.; Polshyn, H. Science, Vol. 367, Issue 6480 https://doi.org/10.1126/science.aay5533	journal	December 2019
Tuning electron correlation in magic-angle twisted bilayer graphene using Coulomb screening Liu, Xiaoxue; Wang, Zhi; Watanabe, K. Science, Vol. 371, Issue 6535 https://doi.org/10.1126/science.abb8754	journal	March 2021
DMRG study of strongly interacting $\mathbb{Z}_2$ flatbands: a toy model inspired by twisted bilayer graphene Eugenio, Paul; Dag, Ceren SciPost Physics Core, Vol. 3, Issue 2 https://doi.org/10.21468/SciPostPhysCore.3.2.015	journal	January 2020
Correlated states in twisted double bilayer graphene Shen, Cheng; Chu, Yanbang; Wu, QuanSheng Materials Cloud https://doi.org/10.24435/materialscloud:be-eq	dataset	January 2021

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Chiral approximation to twisted bilayer graphene: Exact intravalley inversion symmetry, nodal structure, and implications for higher magic angles Wang, Jie; Zheng, Yunqin; Millis, Andrew J. Physical Review Research, Vol. 3, Issue 2 https://doi.org/10.1103/physrevresearch.3.023155	journal	May 2021
Construction of low-energy symmetric Hamiltonians and Hubbard parameters for twisted multilayer systems using ab-initio input Davydov, Arkadiy; Choo, Kenny; Fischer, Mark H. arXiv https://doi.org/10.48550/arxiv.2012.12942	text	January 2020

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