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The Hofstadter problem is the lattice analog of the quantum Hall effect and is the paradigmatic example of topology induced by an applied magnetic field.
Fragile Phases As Affine Monoids: Classification and Material Examples
Zhida Song, L. Elcoro, Nicolas Regnault, and B. Andrei Bernevig · 1905
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Zur Theorie des Diamagnetismus von Leitungselektronen
R. Peierls · 1933
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The effect of a magnetic field on electrons in a periodic potential
J. M. Luttinger · 1951
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The general motion of conduction electrons in a uniform magnetic field, with application to the diamagnetism of metals
P G Harper · 1955
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Magnetic translation group
J. Zak · 1964
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Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields
Douglas R. Hofstadter · 1976
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Quantized hall conductance in a two-dimensional periodic potential
D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs · 1982
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Quantised hall conductance in a perfect crystal
I Dana, Y Avron, and J Zak · 1985
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Evidence of hofstadter’s fractal energy spectrum in the quantized hall conductance
C. Albrecht, J. H. Smet, K. von Klitzing, D. Weiss, V. Umansky, and H. Schweizer · 2001
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Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
B. Andrei Bernevig, Taylor L. Hughes, and Shou-Cheng Zhang · 2006
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Time reversal polarization and a Z 2
Liang Fu and C. L. Kane · 2006
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Bilbao crystallographic server: I. databases and crystallographic computing programs
MI Aroyo, JM Perez-Mato, Cesar Capillas, Eli Kroumova, Svetoslav Ivantchev, Gotzon Madariaga, Asen Kirov, and Hans Wondratschek · 2006
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Exponential localization of wannier functions in insulators
Christian Brouder, Gianluca Panati, Matteo Calandra, Christophe Mourougane, and Nicola Marzari · 2007
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Topological Insulators in Three Dimensions
Liang Fu, C. L. Kane, and E. J. Mele · 2007
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Periodic table for topological insulators and superconductors
Alexei Kitaev · 2009
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Topological phases and the quantum spin Hall effect in three dimensions
Rahul Roy · 2009
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Landau Quantization of Topological Surface States in Bi 2
Peng Cheng, Canli Song, Tong Zhang, Yanyi Zhang, Yilin Wang, Jin-Feng Jia, Jing Wang, Yayu Wang, Bang-Fen Zhu, Xi Chen, Xucun Ma, Ke He, Lili Wang, Xi Dai, Zhong Fang, Xincheng Xie, Xiao-Liang Qi, Chao-Xing Liu, Shou-Cheng Zhang, and Qi-Kun Xue · 2010
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Boron nitride substrates for high-quality graphene electronics
C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone · 2010
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Topological insulators and superconductors: tenfold way and dimensional hierarchy
Shinsei Ryu, Andreas P. Schnyder, Akira Furusaki, and Andreas W. W. Ludwig · 2010
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Moiré bands in twisted double-layer graphene
Rafi Bistritzer and Allan H. MacDonald · 2011
Cited alongside, same era.
Equivalent expression of Z 2
Rui Yu, Xiao Liang Qi, Andrei Bernevig, Zhong Fang, and Xi Dai · 2011
Cited alongside, same era.
Nearly flatbands with nontrivial topology
Kai Sun, Zhengcheng Gu, Hosho Katsura, and S. Das Sarma · 2011
Cited alongside, same era.
Inversion-symmetric topological insulators
Taylor L. Hughes, Emil Prodan, and B. Andrei Bernevig · 2011
Cited alongside, same era.
Emergent many-body translational symmetries of abelian and non-abelian fractionally filled topological insulators
B. Andrei Bernevig and N. Regnault · 2012
Cited alongside, same era.
Topological Insulators and Topological Superconductors
B. Andrei Bernevig and Taylor L. Hughes · 2013
Cited alongside, same era.
Faithful Tight-binding Models and Fragile Topology of Magic-angle Bilayer Graphene
Hoi Chun Po, Liujun Zou, T. Senthil, and Ashvin Vishwanath · 2018
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Zhijun Wang, Benjamin J. Wieder, Jian Li, Binghai Yan, and B. Andrei Bernevig · 2018
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The Axion Insulator as a Pump of Fragile Topology
Benjamin J. Wieder and B. Andrei Bernevig · 2018
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The Landau Level of Fragile Topology
Biao Lian, Fang Xie, and B. Andrei Bernevig · 2018
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Building blocks of topological quantum chemistry: Elementary band representations
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Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator
Cui-Zu Chang, Jinsong Zhang, Xiao Feng, Jie Shen, Zuocheng Zhang, Minghua Guo, Kang Li, Yunbo Ou, Pang Wei, Li-Li Wang, Zhong-Qing Ji, Yang Feng, Shuaihua Ji, Xi Chen, Jinfeng Jia, Xi Dai, Zhong Fang, Shou-Cheng Zhang, Ke He, Yayu Wang, Li Lu, Xu-Cun Ma, and Qi-Kun Xue · 2013
Cited alongside, same era.
Hofstadter’s butterfly and the fractal quantum hall effect in moirésuperlattices
C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim · 2013
Cited alongside, same era.
Massive Dirac Fermions and Hofstadter Butterfly in a van der Waals Heterostructure
B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, and P. Jarillo-Herrero · 2013
Cited alongside, same era.
Cloning of Dirac fermions in graphene superlattices
L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, and J. R. Wallbank · 2013
Cited alongside, same era.
Massive dirac fermions and hofstadter butterfly in a van der waals heterostructure
B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori · 2013
Cited alongside, same era.
Fractional quantum Hall physics in topological flat bands
Siddharth A. Parameswaran, Rahul Roy, and Shivaji L. Sondhi · 2013
Cited alongside, same era.
Jennifer Cano, Barry Bradlyn, Zhijun Wang, L. Elcoro, M. G. Vergniory, C. Felser, M. I. Aroyo, and B. Andrei Bernevig · 2018
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Symmetry Indicators and Anomalous Surface States of Topological Crystalline Insulators
Eslam Khalaf, Hoi Chun Po, Ashvin Vishwanath, and Haruki Watanabe · 2018
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Second-order topological insulators and superconductors with an order-two crystalline symmetry
Max Geier, Luka Trifunovic, Max Hoskam, and Piet W. Brouwer · 2018
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Higher-order topological insulators and superconductors protected by inversion symmetry
Eslam Khalaf · 2018
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Higher-dimensional quasicrystalline approach to the Hofstadter butterfly topological-phase band conductances: Symbolic sequences and self-similar rules at all magnetic fluxes
Gerardo Naumis · 2019
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Higher-order topological insulators in a magnetic field
Yuria Otaki and Takahiro Fukui · 2019
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Observation of hofstadter butterfly and topological edge states in reconfigurable quasi-periodic acoustic crystals
Xiang Ni, Kai Chen, Matthew Weiner, David J. Apigo, Camelia Prodan, Andrea Alù, Emil Prodan, and Alexander B. Khanikaev · 2019
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Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene
Yonglong Xie, Biao Lian, Berthold Jäck, Xiaomeng Liu, Cheng-Li Chiu, Kenji Watanabe, Takashi Taniguchi, B. Andrei Bernevig, and Ali Yazdani · 2019
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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
J. Ahn, S. Park, and B.-J. Yang · 2019
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Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry
Adrien Bouhon, Annica M. Black-Schaffer, and Robert-Jan Slager · 2019
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Higher-order bulk-boundary correspondence for topological crystalline phases
Luka Trifunovic and Piet W. Brouwer · 2019
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Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi 2
Yujun Deng, Yijun Yu, Meng Zhu Shi, Zhongxun Guo, Zihan Xu, Jing Wang, Xian Hui Chen, and Yuanbo Zhang · 2020
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Xiaobo Lu, Biao Lian, Gaurav Chaudhary, Benjamin A. Piot, Giulio Romagnoli, Kenji Watanabe, Takashi Taniguchi, Martino Poggio, Allan H. MacDonald, B. Andrei Bernevig, and Dmitri K. Efetov · 2020
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Evidence of Emergent Symmetry and Valley Chern Number in Twisted Double-Bilayer Graphene
G. William Burg, Biao Lian, Takashi Taniguchi, Kenji Watanabe, B. Andrei Bernevig, and Emanuel Tutuc · 2020
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Twisted bulk-boundary correspondence of fragile topology
Zhi-Da Song, Luis Elcoro, and B. Andrei Bernevig · 2020
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