Twisted bilayer graphene. I. Matrix elements, approximations, perturbation theory, and a k · p two-band model
Abstract
We investigate the twisted bilayer graphene (TBG) model of Bistritzer and MacDonald (BM) [Bistritzer and MacDonald, Proc. Natl. Acad. Sci. 108, 12233 (2011)] to obtain an analytic understanding of its energetics and wave functions needed for many-body calculations. We provide an approximation scheme for the wave functions of the BM model, which first elucidates why the BM KM-point centered original calculation containing only four plane waves provides a good analytical value for the first magic angle (θM ≈ 1°). The approximation scheme also elucidates why most of the many-body matrix elements in the Coulomb Hamiltonian projected to the active bands can be neglected. By applying our approximation scheme at the first magic angle to a ΓM-point centered model of six plane waves, we analytically understand the reason for the small ΓM-point gap between the active and passive bands in the isotropic limit w0 = w1. Furthermore, we analytically calculate the group velocities of the passive bands in the isotropic limit, and show that they are almost doubly degenerate, even away from the ΓM point, where no symmetry forces them to be. Furthermore, moving away from the ΓM and KM points, we provide an explicit analytical perturbative understanding as to why the TBG bands are flat at the first magic angle, despite the first magic angle is defined by only requiring a vanishing KM-point Dirac velocity. We derive analytically a connected “magic manifold” w1 = $$2\sqrt{1 + w^{2}_{0}}$$ $$-\sqrt{2 + 3w^2_0}$$, on which the bands remain extremely flat as w 0 is tuned between the isotropic (w0 = w1) and chiral (w0 = 0) limits. We analytically show why going away from the isotropic limit by making w0 less (but not larger) than w1 increases the ΓM-point gap between the active and the passive bands. Finally, by perturbation theory, we provide an analytic ΓM point k ∙ p two-band model that reproduces the TBG band structure and eigenstates within a certain w0, w1 parameter range. Further refinement of this model are discussed, which suggest a possible faithful representation of the TBG bands by a two-band ΓM point k ∙ p model in the full w0, w1 parameter range.
- Authors:
-
- Princeton Univ., NJ (United States)
- Princeton Univ., NJ (United States); Sorbonne Univ., Paris (France); University Paris-Diderot (France)
- Publication Date:
- Research Org.:
- Princeton Univ., NJ (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC)
- OSTI Identifier:
- 1852143
- Grant/Contract Number:
- SC0016239
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review. B
- Additional Journal Information:
- Journal Volume: 103; Journal Issue: 20; Journal ID: ISSN 2469-9950
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Materials Science; Physics; Bilayer films; Graphene; Two-dimensional electron system; Perturbation theory; k dot p method
Citation Formats
Bernevig, B. Andrei, Song, Zhi-Da, Regnault, Nicolas, and Lian, Biao. Twisted bilayer graphene. I. Matrix elements, approximations, perturbation theory, and a k·p two-band model. United States: N. p., 2021.
Web. doi:10.1103/physrevb.103.205411.
Bernevig, B. Andrei, Song, Zhi-Da, Regnault, Nicolas, & Lian, Biao. Twisted bilayer graphene. I. Matrix elements, approximations, perturbation theory, and a k·p two-band model. United States. https://doi.org/10.1103/physrevb.103.205411
Bernevig, B. Andrei, Song, Zhi-Da, Regnault, Nicolas, and Lian, Biao. Tue .
"Twisted bilayer graphene. I. Matrix elements, approximations, perturbation theory, and a k·p two-band model". United States. https://doi.org/10.1103/physrevb.103.205411. https://www.osti.gov/servlets/purl/1852143.
@article{osti_1852143,
title = {Twisted bilayer graphene. I. Matrix elements, approximations, perturbation theory, and a k·p two-band model},
author = {Bernevig, B. Andrei and Song, Zhi-Da and Regnault, Nicolas and Lian, Biao},
abstractNote = {We investigate the twisted bilayer graphene (TBG) model of Bistritzer and MacDonald (BM) [Bistritzer and MacDonald, Proc. Natl. Acad. Sci. 108, 12233 (2011)] to obtain an analytic understanding of its energetics and wave functions needed for many-body calculations. We provide an approximation scheme for the wave functions of the BM model, which first elucidates why the BM KM-point centered original calculation containing only four plane waves provides a good analytical value for the first magic angle (θM ≈ 1°). The approximation scheme also elucidates why most of the many-body matrix elements in the Coulomb Hamiltonian projected to the active bands can be neglected. By applying our approximation scheme at the first magic angle to a ΓM-point centered model of six plane waves, we analytically understand the reason for the small ΓM-point gap between the active and passive bands in the isotropic limit w0 = w1. Furthermore, we analytically calculate the group velocities of the passive bands in the isotropic limit, and show that they are almost doubly degenerate, even away from the ΓM point, where no symmetry forces them to be. Furthermore, moving away from the ΓM and KM points, we provide an explicit analytical perturbative understanding as to why the TBG bands are flat at the first magic angle, despite the first magic angle is defined by only requiring a vanishing KM-point Dirac velocity. We derive analytically a connected “magic manifold” w1 = $2\sqrt{1 + w^{2}_{0}}$ $-\sqrt{2 + 3w^2_0}$, on which the bands remain extremely flat as w 0 is tuned between the isotropic (w0 = w1) and chiral (w0 = 0) limits. We analytically show why going away from the isotropic limit by making w0 less (but not larger) than w1 increases the ΓM-point gap between the active and the passive bands. Finally, by perturbation theory, we provide an analytic ΓM point k ∙ p two-band model that reproduces the TBG band structure and eigenstates within a certain w0, w1 parameter range. Further refinement of this model are discussed, which suggest a possible faithful representation of the TBG bands by a two-band ΓM point k ∙ p model in the full w0, w1 parameter range.},
doi = {10.1103/physrevb.103.205411},
journal = {Physical Review. B},
number = 20,
volume = 103,
place = {United States},
year = {Tue May 11 00:00:00 EDT 2021},
month = {Tue May 11 00:00:00 EDT 2021}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Topology-Bounded Superfluid Weight in Twisted Bilayer Graphene
journal, April 2020
- Xie, Fang; Song, Zhida; Lian, Biao
- Physical Review Letters, Vol. 124, Issue 16
Origin of Magic Angles in Twisted Bilayer Graphene
journal, March 2019
- Tarnopolsky, Grigory; Kruchkov, Alex Jura; Vishwanath, Ashvin
- Physical Review Letters, Vol. 122, Issue 10
Ground State and Hidden Symmetry of Magic-Angle Graphene at Even Integer Filling
journal, August 2020
- Bultinck, Nick; Khalaf, Eslam; Liu, Shang
- Physical Review X, Vol. 10, Issue 3
Renormalization Group Study of Hidden Symmetry in Twisted Bilayer Graphene with Coulomb Interactions
journal, December 2020
- Vafek, Oskar; Kang, Jian
- Physical Review Letters, Vol. 125, Issue 25
Topological Superconductivity in Twisted Multilayer Graphene
journal, August 2018
- Xu, Cenke; Balents, Leon
- Physical Review Letters, Vol. 121, Issue 8
Non-Abelian Dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic-angle twisted bilayer graphene
journal, July 2020
- Kang, Jian; Vafek, Oskar
- Physical Review B, Vol. 102, Issue 3
Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene
journal, October 2019
- Lu, Xiaobo; Stepanov, Petr; Yang, Wei
- Nature, Vol. 574, Issue 7780
Kekulé valence bond order in an extended Hubbard model on the honeycomb lattice with possible applications to twisted bilayer graphene
journal, September 2018
- Xu, Xiao Yan; Law, K. T.; Lee, Patrick A.
- Physical Review B, Vol. 98, Issue 12
Kohn-Luttinger Superconductivity in Twisted Bilayer Graphene
journal, January 2019
- González, J.; Stauber, T.
- Physical Review Letters, Vol. 122, Issue 2
Intrinsic quantized anomalous Hall effect in a moiré heterostructure
journal, December 2019
- Serlin, M.; Tschirhart, C. L.; Polshyn, H.
- Science, Vol. 367, Issue 6480
Particle-Hole Duality, Emergent Fermi Liquids, and Fractional Chern Insulators in Moiré Flatbands
journal, March 2020
- Abouelkomsan, Ahmed; Liu, Zhao; Bergholtz, Emil J.
- Physical Review Letters, Vol. 124, Issue 10
Flatbands and Perfect Metal in Trilayer Moiré Graphene
journal, July 2019
- Mora, Christophe; Regnault, Nicolas; Bernevig, B. Andrei
- Physical Review Letters, Vol. 123, Issue 2
Helical network model for twisted bilayer graphene
journal, July 2018
- Efimkin, Dmitry K.; MacDonald, Allan H.
- Physical Review B, Vol. 98, Issue 3
Pseudo Landau level representation of twisted bilayer graphene: Band topology and implications on the correlated insulating phase
journal, April 2019
- Liu, Jianpeng; Liu, Junwei; Dai, Xi
- Physical Review B, Vol. 99, Issue 15
DMRG study of strongly interacting $\mathbb{Z}_2$ flatbands: a toy model inspired by twisted bilayer graphene
journal, January 2020
- Eugenio, Paul; Dag, Ceren
- SciPost Physics Core, Vol. 3, Issue 2
Twisted Bilayer Graphene: Moiré with a Twist
journal, August 2016
- Dai, Shuyang; Xiang, Yang; Srolovitz, David J.
- Nano Letters, Vol. 16, Issue 9
Designing flat bands by strain
journal, July 2019
- Bi, Zhen; Yuan, Noah F. Q.; Fu, Liang
- Physical Review B, Vol. 100, Issue 3
Large linear-in-temperature resistivity in twisted bilayer graphene
journal, August 2019
- Polshyn, Hryhoriy; Yankowitz, Matthew; Chen, Shaowen
- Nature Physics, Vol. 15, Issue 10
Maximized electron interactions at the magic angle in twisted bilayer graphene
journal, July 2019
- Kerelsky, Alexander; McGilly, Leo J.; Kennes, Dante M.
- Nature, Vol. 572, Issue 7767
Electrostatic effects, band distortions, and superconductivity in twisted graphene bilayers
journal, December 2018
- Guinea, Francisco; Walet, Niels R.
- Proceedings of the National Academy of Sciences, Vol. 115, Issue 52
Fragile Topology and Wannier Obstructions
journal, September 2018
- Po, Hoi Chun; Watanabe, Haruki; Vishwanath, Ashvin
- Physical Review Letters, Vol. 121, Issue 12
Collective Excitations of Quantum Anomalous Hall Ferromagnets in Twisted Bilayer Graphene
journal, January 2020
- Wu, Fengcheng; Das Sarma, Sankar
- Physical Review Letters, Vol. 124, Issue 4
Valence Bond Orders at Charge Neutrality in a Possible Two-Orbital Extended Hubbard Model for Twisted Bilayer Graphene
journal, October 2019
- Da Liao, Yuan; Meng, Zi Yang; Xu, Xiao Yan
- Physical Review Letters, Vol. 123, Issue 15
Tunable correlated states and spin-polarized phases in twisted bilayer–bilayer graphene
journal, May 2020
- Cao, Yuan; Rodan-Legrain, Daniel; Rubies-Bigorda, Oriol
- Nature
Disorder in twisted bilayer graphene
journal, June 2020
- Wilson, Justin H.; Fu, Yixing; Das Sarma, S.
- Physical Review Research, Vol. 2, Issue 2
Correlated states in twisted double bilayer graphene
journal, March 2020
- Shen, Cheng; Chu, Yanbang; Wu, QuanSheng
- Nature Physics, Vol. 16, Issue 5
Mechanism for Anomalous Hall Ferromagnetism in Twisted Bilayer Graphene
journal, April 2020
- Bultinck, Nick; Chatterjee, Shubhayu; Zaletel, Michael P.
- Physical Review Letters, Vol. 124, Issue 16
Correlations and electronic order in a two-orbital honeycomb lattice model for twisted bilayer graphene
journal, December 2018
- Venderbos, Jörn W. F.; Fernandes, Rafael M.
- Physical Review B, Vol. 98, Issue 24
Superconductivity from valley fluctuations and approximate SO(4) symmetry in a weak coupling theory of twisted bilayer graphene
journal, April 2019
- You, Yi-Zhuang; Vishwanath, Ashvin
- npj Quantum Materials, Vol. 4, Issue 1
Relaxation of moiré patterns for slightly misaligned identical lattices: graphene on graphite
journal, July 2015
- Wijk, M. M. van; Schuring, A.; Katsnelson, M. I.
- 2D Materials, Vol. 2, Issue 3
Flavour Hund’s coupling, Chern gaps and charge diffusivity in moiré graphene
journal, March 2021
- Park, Jeong Min; Cao, Yuan; Watanabe, Kenji
- Nature, Vol. 592, Issue 7852
Superfluid weight and Berezinskii-Kosterlitz-Thouless transition temperature of twisted bilayer graphene
journal, February 2020
- Julku, A.; Peltonen, T. J.; Liang, L.
- Physical Review B, Vol. 101, Issue 6
Chiral Spin Density Wave and Superconductivity in the Magic-Angle-Twisted Bilayer Graphene
journal, November 2018
- Liu, Cheng-Cheng; Zhang, Li-Da; Chen, Wei-Qiang
- Physical Review Letters, Vol. 121, Issue 21
Tunable correlated Chern insulator and ferromagnetism in a moiré superlattice
journal, March 2020
- Chen, Guorui; Sharpe, Aaron L.; Fox, Eli J.
- Nature, Vol. 579, Issue 7797
Phases of a phenomenological model of twisted bilayer graphene
journal, August 2018
- Dodaro, J. F.; Kivelson, S. A.; Schattner, Y.
- Physical Review B, Vol. 98, Issue 7
Theories for the correlated insulating states and quantum anomalous Hall effect phenomena in twisted bilayer graphene
journal, January 2021
- Liu, Jianpeng; Dai, Xi
- Physical Review B, Vol. 103, Issue 3
Antiferromagnetically ordered Mott insulator and superconductivity in twisted bilayer graphene: a quantum Monte Carlo study
journal, March 2019
- Huang, Tongyun; Zhang, Lufeng; Ma, Tianxing
- Science Bulletin, Vol. 64, Issue 5
Exact continuum model for low-energy electronic states of twisted bilayer graphene
journal, August 2019
- Carr, Stephen; Fang, Shiang; Zhu, Ziyan
- Physical Review Research, Vol. 1, Issue 1
Magic-angle semimetals
journal, October 2020
- Fu, Yixing; König, Elio J.; Wilson, Justin H.
- npj Quantum Materials, Vol. 5, Issue 1
Twisted bilayer graphene. VI. An exact diagonalization study at nonzero integer filling
journal, May 2021
- Xie, Fang; Cowsik, Aditya; Song, Zhi-Da
- Physical Review B, Vol. 103, Issue 20
Tunable spin-polarized correlated states in twisted double bilayer graphene
journal, July 2020
- Liu, Xiaomeng; Hao, Zeyu; Khalaf, Eslam
- Nature, Vol. 583, Issue 7815
Twisted bilayer graphene. V. Exact analytic many-body excitations in Coulomb Hamiltonians: Charge gap, Goldstone modes, and absence of Cooper pairing
journal, May 2021
- Bernevig, B. Andrei; Lian, Biao; Cowsik, Aditya
- Physical Review B, Vol. 103, Issue 20
Twisted bilayer graphene. IV. Exact insulator ground states and phase diagram
journal, May 2021
- Lian, Biao; Song, Zhi-Da; Regnault, Nicolas
- Physical Review B, Vol. 103, Issue 20
Twisted bilayer graphene. III. Interacting Hamiltonian and exact symmetries
journal, May 2021
- Bernevig, B. Andrei; Song, Zhi-Da; Regnault, Nicolas
- Physical Review B, Vol. 103, Issue 20
Ab initio theory of moiré superlattice bands in layered two-dimensional materials
journal, May 2014
- Jung, Jeil; Raoux, Arnaud; Qiao, Zhenhua
- Physical Review B, Vol. 89, Issue 20
Twisted bilayer graphene. II. Stable symmetry anomaly
journal, May 2021
- Song, Zhi-Da; Lian, Biao; Regnault, Nicolas
- Physical Review B, Vol. 103, Issue 20
Spin--Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems
book, January 2003
- Winkler, Roland; Höhler, Gerhard; Kühn, Johann H.
- Springer Tracts in Modern Physics, Vol. 191
Superconductivity in metallic twisted bilayer graphene stabilized by WSe2
journal, July 2020
- Arora, Harpreet Singh; Polski, Robert; Zhang, Yiran
- Nature, Vol. 583, Issue 7816
Efficient simulation of moiré materials using the density matrix renormalization group
journal, November 2020
- Soejima, Tomohiro; Parker, Daniel E.; Bultinck, Nick
- Physical Review B, Vol. 102, Issue 20
Independent superconductors and correlated insulators in twisted bilayer graphene
journal, June 2020
- Saito, Yu; Ge, Jingyuan; Watanabe, Kenji
- Nature Physics, Vol. 16, Issue 9
Strongly correlated Chern insulators in magic-angle twisted bilayer graphene
journal, December 2020
- Nuckolls, Kevin P.; Oh, Myungchul; Wong, Dillon
- Nature, Vol. 588, Issue 7839
Multiple topological transitions in twisted bilayer graphene near the first magic angle
journal, January 2019
- Hejazi, Kasra; Liu, Chunxiao; Shapourian, Hassan
- Physical Review B, Vol. 99, Issue 3
Entropic evidence for a Pomeranchuk effect in magic-angle graphene
journal, April 2021
- Rozen, Asaf; Park, Jeong Min; Zondiner, Uri
- Nature, Vol. 592, Issue 7853
Strong correlations and superconductivity in twisted bilayer graphene
journal, December 2018
- Kennes, Dante M.; Lischner, Johannes; Karrasch, Christoph
- Physical Review B, Vol. 98, Issue 24
Chern insulators, van Hove singularities and topological flat bands in magic-angle twisted bilayer graphene
journal, February 2021
- Wu, Shuang; Zhang, Zhenyuan; Watanabe, K.
- Nature Materials, Vol. 20, Issue 4
Strong Coupling Phases of Partially Filled Twisted Bilayer Graphene Narrow Bands
journal, June 2019
- Kang, Jian; Vafek, Oskar
- Physical Review Letters, Vol. 122, Issue 24
Ferromagnetic Mott state in Twisted Graphene Bilayers at the Magic Angle
journal, June 2019
- Seo, Kangjun; Kotov, Valeri N.; Uchoa, Bruno
- Physical Review Letters, Vol. 122, Issue 24
Ferromagnetism in magic-angle graphene
journal, August 2019
- Pixley, Jed H.; Andrei, Eva Y.
- Science, Vol. 365, Issue 6453
Possible correlated insulating states in magic-angle twisted bilayer graphene under strongly competing interactions
journal, August 2018
- Ochi, Masayuki; Koshino, Mikito; Kuroki, Kazuhiko
- Physical Review B, Vol. 98, Issue 8
Unconventional superconductivity in magic-angle graphene superlattices
journal, March 2018
- Cao, Yuan; Fatemi, Valla; Fang, Shiang
- Nature, Vol. 556, Issue 7699
Landau level of fragile topology
journal, July 2020
- Lian, Biao; Xie, Fang; Bernevig, B. Andrei
- Physical Review B, Vol. 102, Issue 4
Pairing symmetry of interacting fermions on a twisted bilayer graphene superlattice
journal, June 2018
- Guo, Huaiming; Zhu, Xingchuan; Feng, Shiping
- Physical Review B, Vol. 97, Issue 23
Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene
journal, July 2019
- Sharpe, Aaron L.; Fox, Eli J.; Barnard, Arthur W.
- Science, Vol. 365, Issue 6453
Isospin Pomeranchuk effect in twisted bilayer graphene
journal, April 2021
- Saito, Yu; Yang, Fangyuan; Ge, Jingyuan
- Nature, Vol. 592, Issue 7853
Tuning electron correlation in magic-angle twisted bilayer graphene using Coulomb screening
journal, March 2021
- Liu, Xiaoxue; Wang, Zhi; Watanabe, K.
- Science, Vol. 371, Issue 6535
Fractional Chern insulator states in twisted bilayer graphene: An analytical approach
journal, May 2020
- Ledwith, Patrick J.; Tarnopolsky, Grigory; Khalaf, Eslam
- Physical Review Research, Vol. 2, Issue 2
Evidence of a gate-tunable Mott insulator in a trilayer graphene moiré superlattice
journal, January 2019
- Chen, Guorui; Jiang, Lili; Wu, Shuang
- Nature Physics, Vol. 15, Issue 3
Chern bands of twisted bilayer graphene: Fractional Chern insulators and spin phase transition
journal, May 2020
- Repellin, Cécile; Senthil, T.
- Physical Review Research, Vol. 2, Issue 2
Structure of twisted and buckled bilayer graphene
journal, November 2016
- Jain, Sandeep K.; Juričić, Vladimir; Barkema, Gerard T.
- 2D Materials, Vol. 4, Issue 1
Superconductivity, correlated insulators, and Wess–Zumino–Witten terms in twisted bilayer graphene
journal, November 2020
- Christos, Maine; Sachdev, Subir; Scheurer, Mathias S.
- Proceedings of the National Academy of Sciences, Vol. 117, Issue 47
Twisted Bilayer Graphene: A Phonon-Driven Superconductor
journal, June 2019
- Lian, Biao; Wang, Zhijun; Bernevig, B. Andrei
- Physical Review Letters, Vol. 122, Issue 25
Theory of Phonon-Mediated Superconductivity in Twisted Bilayer Graphene
journal, December 2018
- Wu, Fengcheng; MacDonald, A. H.; Martin, Ivar
- Physical Review Letters, Vol. 121, Issue 25
Untying the insulating and superconducting orders in magic-angle graphene
journal, July 2020
- Stepanov, Petr; Das, Ipsita; Lu, Xiaobo
- Nature, Vol. 583, Issue 7816
Failure of Nielsen-Ninomiya Theorem and Fragile Topology in Two-Dimensional Systems with Space-Time Inversion Symmetry: Application to Twisted Bilayer Graphene at Magic Angle
journal, April 2019
- Ahn, Junyeong; Park, Sungjoon; Yang, Bohm-Jung
- Physical Review X, Vol. 9, Issue 2
Quasi-flat-band physics in a two-leg ladder model and its relation to magic-angle twisted bilayer graphene
journal, October 2020
- Huang, Yixuan; Hosur, Pavan; Pal, Hridis K.
- Physical Review B, Vol. 102, Issue 15
Cascade of phase transitions and Dirac revivals in magic-angle graphene
journal, June 2020
- Zondiner, U.; Rozen, A.; Rodan-Legrain, D.
- Nature, Vol. 582, Issue 7811
Spin magnetometry as a probe of stripe superconductivity in twisted bilayer graphene
journal, September 2020
- König, E. J.; Coleman, Piers; Tsvelik, A. M.
- Physical Review B, Vol. 102, Issue 10
Correlated Insulating States in Twisted Double Bilayer Graphene
journal, November 2019
- Burg, G. William; Zhu, Jihang; Taniguchi, Takashi
- Physical Review Letters, Vol. 123, Issue 19
Effective continuum model for relaxed twisted bilayer graphene and moiré electron-phonon interaction
journal, May 2020
- Koshino, Mikito; Nam, Nguyen N. T.
- Physical Review B, Vol. 101, Issue 19
Triangular antiferromagnetism on the honeycomb lattice of twisted bilayer graphene
journal, August 2018
- Thomson, Alex; Chatterjee, Shubhayu; Sachdev, Subir
- Physical Review B, Vol. 98, Issue 7
Competing phases of interacting electrons on triangular lattices in moiré heterostructures
journal, May 2019
- Classen, Laura; Honerkamp, Carsten; Scherer, Michael M.
- Physical Review B, Vol. 99, Issue 19
Nematicity with a twist: Rotational symmetry breaking in a moiré superlattice
journal, August 2020
- Fernandes, Rafael M.; Venderbos, Jörn W. F.
- Science Advances, Vol. 6, Issue 32
Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene
journal, July 2019
- Xie, Yonglong; Lian, Biao; Jäck, Berthold
- Nature, Vol. 572, Issue 7767
Ferromagnetism in Narrow Bands of Moiré Superlattices
journal, May 2020
- Repellin, Cécile; Dong, Zhihuan; Zhang, Ya-Hui
- Physical Review Letters, Vol. 124, Issue 18
Correlated insulating phases of twisted bilayer graphene at commensurate filling fractions: A Hartree-Fock study
journal, July 2020
- Zhang, Yi; Jiang, Kun; Wang, Ziqiang
- Physical Review B, Vol. 102, Issue 3
Nematic topological semimetal and insulator in magic-angle bilayer graphene at charge neutrality
journal, January 2021
- Liu, Shang; Khalaf, Eslam; Lee, Jong Yeon
- Physical Review Research, Vol. 3, Issue 1
Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene
journal, January 2021
- Saito, Yu; Ge, Jingyuan; Rademaker, Louk
- Nature Physics, Vol. 17, Issue 4
Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry
journal, November 2019
- Bouhon, Adrien; Black-Schaffer, Annica M.; Slager, Robert-Jan
- Physical Review B, Vol. 100, Issue 19
Nearly flat Chern bands in moiré superlattices
journal, February 2019
- Zhang, Ya-Hui; Mao, Dan; Cao, Yuan
- Physical Review B, Vol. 99, Issue 7
Band structure and insulating states driven by Coulomb interaction in twisted bilayer graphene
journal, July 2020
- Cea, Tommaso; Guinea, Francisco
- Physical Review B, Vol. 102, Issue 4
Coupled-wire description of the correlated physics in twisted bilayer graphene
journal, April 2019
- Wu, Xiao-Chuan; Jian, Chao-Ming; Xu, Cenke
- Physical Review B, Vol. 99, Issue 16
Model for the metal-insulator transition in graphene superlattices and beyond
journal, July 2018
- Yuan, Noah F. Q.; Fu, Liang
- Physical Review B, Vol. 98, Issue 4
Cascade of electronic transitions in magic-angle twisted bilayer graphene
journal, June 2020
- Wong, Dillon; Nuckolls, Kevin P.; Oh, Myungchul
- Nature, Vol. 582, Issue 7811
Electronic correlations in twisted bilayer graphene near the magic angle
journal, August 2019
- Choi, Youngjoon; Kemmer, Jeannette; Peng, Yang
- Nature Physics, Vol. 15, Issue 11
Faithful tight-binding models and fragile topology of magic-angle bilayer graphene
journal, May 2019
- Po, Hoi Chun; Zou, Liujun; Senthil, T.
- Physical Review B, Vol. 99, Issue 19
Transport across twist angle domains in moiré graphene
journal, September 2020
- Padhi, Bikash; Tiwari, Apoorv; Neupert, Titus
- Physical Review Research, Vol. 2, Issue 3
Correlation-Induced Insulating Topological Phases at Charge Neutrality in Twisted Bilayer Graphene
journal, January 2021
- Da Liao, Yuan; Kang, Jian; Breiø, Clara N.
- Physical Review X, Vol. 11, Issue 1
Hybrid Wannier Chern bands in magic angle twisted bilayer graphene and the quantized anomalous Hall effect
journal, March 2021
- Hejazi, Kasra; Chen, Xiao; Balents, Leon
- Physical Review Research, Vol. 3, Issue 1
Moire bands in twisted double-layer graphene
journal, July 2011
- Bistritzer, R.; MacDonald, A. H.
- Proceedings of the National Academy of Sciences, Vol. 108, Issue 30
Correlated insulator behaviour at half-filling in magic-angle graphene superlattices
journal, March 2018
- Cao, Yuan; Fatemi, Valla; Demir, Ahmet
- Nature, Vol. 556, Issue 7699
Unconventional Superconductivity and Density Waves in Twisted Bilayer Graphene
journal, December 2018
- Isobe, Hiroki; Yuan, Noah F. Q.; Fu, Liang
- Physical Review X, Vol. 8, Issue 4
Band structure of twisted bilayer graphene: Emergent symmetries, commensurate approximants, and Wannier obstructions
journal, August 2018
- Zou, Liujun; Po, Hoi Chun; Vishwanath, Ashvin
- Physical Review B, Vol. 98, Issue 8
Topology of Disconnected Elementary Band Representations
journal, June 2018
- Cano, Jennifer; Bradlyn, Barry; Wang, Zhijun
- Physical Review Letters, Vol. 120, Issue 26
Tuning superconductivity in twisted bilayer graphene
journal, January 2019
- Yankowitz, Matthew; Chen, Shaowen; Polshyn, Hryhoriy
- Science, Vol. 363, Issue 6431
Signatures of tunable superconductivity in a trilayer graphene moiré superlattice
journal, July 2019
- Chen, Guorui; Sharpe, Aaron L.; Gallagher, Patrick
- Nature, Vol. 572, Issue 7768
Symmetry-broken Chern insulators and Rashba-like Landau-level crossings in magic-angle bilayer graphene
journal, March 2021
- Das, Ipsita; Lu, Xiaobo; Herzog-Arbeitman, Jonah
- Nature Physics, Vol. 17, Issue 6
Geometric and Conventional Contribution to the Superfluid Weight in Twisted Bilayer Graphene
journal, December 2019
- Hu, Xiang; Hyart, Timo; Pikulin, Dmitry I.
- Physical Review Letters, Vol. 123, Issue 23
Origin of Mott Insulating Behavior and Superconductivity in Twisted Bilayer Graphene
journal, September 2018
- Po, Hoi Chun; Zou, Liujun; Vishwanath, Ashvin
- Physical Review X, Vol. 8, Issue 3
Symmetry, Maximally Localized Wannier States, and a Low-Energy Model for Twisted Bilayer Graphene Narrow Bands
journal, September 2018
- Kang, Jian; Vafek, Oskar
- Physical Review X, Vol. 8, Issue 3
Maximally Localized Wannier Orbitals and the Extended Hubbard Model for Twisted Bilayer Graphene
journal, September 2018
- Koshino, Mikito; Yuan, Noah F. Q.; Koretsune, Takashi
- Physical Review X, Vol. 8, Issue 3
Quantum Valley Hall Effect, Orbital Magnetism, and Anomalous Hall Effect in Twisted Multilayer Graphene Systems
journal, August 2019
- Liu, Jianpeng; Ma, Zhen; Gao, Jinhua
- Physical Review X, Vol. 9, Issue 3
Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene
journal, July 2019
- Jiang, Yuhang; Lai, Xinyuan; Watanabe, Kenji
- Nature, Vol. 573, Issue 7772
All Magic Angles in Twisted Bilayer Graphene are Topological
journal, July 2019
- Song, Zhida; Wang, Zhijun; Shi, Wujun
- Physical Review Letters, Vol. 123, Issue 3
Nature of the Correlated Insulator States in Twisted Bilayer Graphene
journal, March 2020
- Xie, Ming; MacDonald, A. H.
- Physical Review Letters, Vol. 124, Issue 9
Strange Metal in Magic-Angle Graphene with near Planckian Dissipation
journal, February 2020
- Cao, Yuan; Chowdhury, Debanjan; Rodan-Legrain, Daniel
- Physical Review Letters, Vol. 124, Issue 7
Landau levels in twisted bilayer graphene and semiclassical orbits
journal, July 2019
- Hejazi, Kasra; Liu, Chunxiao; Balents, Leon
- Physical Review B, Vol. 100, Issue 3