TY - JOUR
T1 - Twisted bilayer graphene. II. Stable symmetry anomaly
AU - Song, Zhi Da
AU - Lian, Biao
AU - Regnault, Nicolas
AU - Bernevig, B. Andrei
N1 - Funding Information:
We thank Aditya Cowsik and Fang Xie for valuable discussions. This work was supported by US Department of Energy 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 NSF-EAGER Grant No. DMR 1643312, NSF-MRSEC Grants No. DMR-1420541 and No. DMR-2011750, ONR Grant No. N00014-20-1-2303, the 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.
Publisher Copyright:
© 2021 American Physical Society.
PY - 2021/5/11
Y1 - 2021/5/11
N2 - We show that the entire continuous model of twisted bilayer graphene (TBG) (and not just the two active bands) with particle-hole symmetry is anomalous and hence incompatible with lattice models. Previous works [e.g., Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, Phys. Rev. Lett. 123, 036401 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.036401; J. Ahn, S. Park, and B.-J. Yang, Phys. Rev. X 9, 021013 (2019)2160-330810.1103/PhysRevX.9.021013; H. C. Po, L. Zou, T. Senthil, and A. Vishwanath, Phys. Rev. B 99, 195455 (2019)2469-995010.1103/PhysRevB.99.195455; J. Kang and O. Vafek, Phys. Rev. X 8, 031088 (2018)2160-330810.1103/PhysRevX.8.031088; M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi, K. Kuroki, and L. Fu, Phys. Rev. X 8, 031087 (2018)2160-330810.1103/PhysRevX.8.031087; J. Liu, J. Liu, and X. Dai, Phys. Rev. B 99, 155415 (2019)2469-995010.1103/PhysRevB.99.155415; L. Zou, H. C. Po, A. Vishwanath, and T. Senthil, Phys. Rev. B 98, 085435 (2018)2469-995010.1103/PhysRevB.98.085435] found that the two flatbands in TBG possess a fragile topology protected by the C2zT symmetry. Song et al. [Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, Phys. Rev. Lett. 123, 036401 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.036401] also pointed out an approximate particle-hole symmetry (P) in the continuous model of TBG. In this paper, we numerically confirm that P is indeed a good approximation for TBG and show that the fragile topology of the two flatbands is enhanced to a P-protected stable topology. This stable topology implies 4l+2 (l N) Dirac points between the middle two bands. The P-protected stable topology is robust against arbitrary gap closings between the middle two bands and the other bands. We further show that, remarkably, this P-protected stable topology, as well as the corresponding 4l+2 Dirac points, cannot be realized in lattice models that preserve both C2zT and P symmetries. In other words, the continuous model of TBG is anomalous and cannot be realized on lattices. Two other topology related topics, with consequences for the interacting TBG problem, i.e., the choice of Chern band basis in the two flatbands and the perfect metal phase of TBG in the so-called second chiral limit, are also discussed.
AB - We show that the entire continuous model of twisted bilayer graphene (TBG) (and not just the two active bands) with particle-hole symmetry is anomalous and hence incompatible with lattice models. Previous works [e.g., Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, Phys. Rev. Lett. 123, 036401 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.036401; J. Ahn, S. Park, and B.-J. Yang, Phys. Rev. X 9, 021013 (2019)2160-330810.1103/PhysRevX.9.021013; H. C. Po, L. Zou, T. Senthil, and A. Vishwanath, Phys. Rev. B 99, 195455 (2019)2469-995010.1103/PhysRevB.99.195455; J. Kang and O. Vafek, Phys. Rev. X 8, 031088 (2018)2160-330810.1103/PhysRevX.8.031088; M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi, K. Kuroki, and L. Fu, Phys. Rev. X 8, 031087 (2018)2160-330810.1103/PhysRevX.8.031087; J. Liu, J. Liu, and X. Dai, Phys. Rev. B 99, 155415 (2019)2469-995010.1103/PhysRevB.99.155415; L. Zou, H. C. Po, A. Vishwanath, and T. Senthil, Phys. Rev. B 98, 085435 (2018)2469-995010.1103/PhysRevB.98.085435] found that the two flatbands in TBG possess a fragile topology protected by the C2zT symmetry. Song et al. [Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, Phys. Rev. Lett. 123, 036401 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.036401] also pointed out an approximate particle-hole symmetry (P) in the continuous model of TBG. In this paper, we numerically confirm that P is indeed a good approximation for TBG and show that the fragile topology of the two flatbands is enhanced to a P-protected stable topology. This stable topology implies 4l+2 (l N) Dirac points between the middle two bands. The P-protected stable topology is robust against arbitrary gap closings between the middle two bands and the other bands. We further show that, remarkably, this P-protected stable topology, as well as the corresponding 4l+2 Dirac points, cannot be realized in lattice models that preserve both C2zT and P symmetries. In other words, the continuous model of TBG is anomalous and cannot be realized on lattices. Two other topology related topics, with consequences for the interacting TBG problem, i.e., the choice of Chern band basis in the two flatbands and the perfect metal phase of TBG in the so-called second chiral limit, are also discussed.
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U2 - 10.1103/PhysRevB.103.205412
DO - 10.1103/PhysRevB.103.205412
M3 - Article
AN - SCOPUS:85106351986
SN - 2469-9950
VL - 103
JO - Physical Review B
JF - Physical Review B
IS - 20
M1 - 205412
ER -