Hofstadter Topology: Noncrystalline Topological Materials at High Flux

Herzog-Arbeitman, Jonah; Song, Zhi-Da; Regnault, Nicolas; Bernevig, B. Andrei

doi:10.1103/physrevlett.125.236804

Title: Hofstadter Topology: Noncrystalline Topological Materials at High Flux

Journal Article · Wed Dec 02 00:00:00 EST 2020 · Physical Review Letters

DOI:https://doi.org/10.1103/physrevlett.125.236804· OSTI ID:1852152

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

Princeton Univ., NJ (United States)
Princeton Univ., NJ (United States); Sorbonne Univ., Paris (France)

he Hofstadter problem is the lattice analog of the quantum Hall effect and is the paradigmatic example of topology induced by an applied magnetic field. Conventionally, the Hofstadter problem involves adding ~10⁴ T magnetic fields to a trivial band structure. In this work, we show that when a magnetic field is added to an initially topological band structure, a wealth of possible phases emerges. Remarkably, we find topological phases that cannot be realized in any crystalline insulators. We prove that threading magnetic flux through a Hamiltonian with a nonzero Chern number or mirror Chern number enforces a phase transition at fixed filling and that a 2D Hamiltonian with a nontrivial Kane–Mele invariant can be classified as a 3D topological insulator (TI) or 3D weak TI phase in periodic flux. We then study fragile topology protected by the product of twofold rotation and time reversal and show that there exists a higher order TI phase where corner modes are pumped by flux. We show that a model of twisted bilayer graphene realizes this phase. Our results rely primarily on the magnetic translation group that exists at rational values of the flux. The advent of Moiré lattices renders our work relevant experimentally. Due to the enlarged Moiré unit cell, it is possible for laboratory-strength fields to reach one flux per plaquette and allow access to our proposed Hofstadter topological phase.

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Research Organization:: Princeton Univ., NJ (United States)

Sponsoring Organization:: USDOE Office of Science (SC); Schmidt Fund for Innovation Research; Packard Foundation; National Science Foundation (NSF); BSF Israel US Foundation

Grant/Contract Number:: SC0016239; 404513; DMR-1643312; DMR-142041; 2018226; N00014-20-1-2303

OSTI ID:: 1852152

Journal Information:: Physical Review Letters, Vol. 125, Issue 23; ISSN 0031-9007

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (63)

Equivalent expression of $Z_{2}$ topological invariant for band insulators using the non-Abelian Berry connection Yu, Rui; Qi, Xiao Liang; Bernevig, Andrei Physical Review B, Vol. 84, Issue 7 https://doi.org/10.1103/PhysRevB.84.075119	journal	August 2011
Time reversal polarization and a $Z_{2}$ adiabatic spin pump Fu, Liang; Kane, C. L. Physical Review B, Vol. 74, Issue 19 https://doi.org/10.1103/PhysRevB.74.195312	journal	November 2006
Symmetry Indicators and Anomalous Surface States of Topological Crystalline Insulators Khalaf, Eslam; Po, Hoi Chun; Vishwanath, Ashvin Physical Review X, Vol. 8, Issue 3 https://doi.org/10.1103/PhysRevX.8.031070	journal	September 2018
Landau level of fragile topology Lian, Biao; Xie, Fang; Bernevig, B. Andrei Physical Review B, Vol. 102, Issue 4 https://doi.org/10.1103/PhysRevB.102.041402	journal	July 2020
Higher-order topological insulators Schindler, Frank; Cook, Ashley M.; Vergniory, Maia G. Science Advances, Vol. 4, Issue 6 https://doi.org/10.1126/sciadv.aat0346	journal	June 2018
Higher-order topological insulators in a magnetic field Otaki, Yuria; Fukui, Takahiro Physical Review B, Vol. 100, Issue 24 https://doi.org/10.1103/PhysRevB.100.245108	journal	December 2019
Wilson-loop characterization of inversion-symmetric topological insulators Alexandradinata, A.; Dai, Xi; Bernevig, B. Andrei Physical Review B, Vol. 89, Issue 15 https://doi.org/10.1103/PhysRevB.89.155114	journal	April 2014
Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells Bernevig, B. A.; Hughes, T. L.; Zhang, S.-C. Science, Vol. 314, Issue 5806, p. 1757-1761 https://doi.org/10.1126/science.1133734	journal	December 2006
Higher-dimensional quasicrystalline approach to the Hofstadter butterfly topological-phase band conductances: Symbolic sequences and self-similar rules at all magnetic fluxes Naumis, Gerardo Physical Review B, Vol. 100, Issue 16 https://doi.org/10.1103/PhysRevB.100.165101	journal	October 2019
Second-order topological insulators and superconductors with an order-two crystalline symmetry Geier, Max; Trifunovic, Luka; Hoskam, Max Physical Review B, Vol. 97, Issue 20 https://doi.org/10.1103/PhysRevB.97.205135	journal	May 2018
Higher-Order Topology, Monopole Nodal Lines, and the Origin of Large Fermi Arcs in Transition Metal Dichalcogenides X Te 2 ( X = Mo , W ) Wang, Zhijun; Wieder, Benjamin J.; Li, Jian Physical Review Letters, Vol. 123, Issue 18 https://doi.org/10.1103/PhysRevLett.123.186401	journal	October 2019
Electronic properties of graphene-based heterostructures Skachkova, V. A.; Baranava, M. S.; Hvazdouski, D. C. Journal of Physics: Conference Series, Vol. 917 https://doi.org/10.1088/1742-6596/917/9/092012	journal	November 2017
Observation of Hofstadter butterfly and topological edge states in reconfigurable quasi-periodic acoustic crystals Ni, Xiang; Chen, Kai; Weiner, Matthew Communications Physics, Vol. 2, Issue 1 https://doi.org/10.1038/s42005-019-0151-7	journal	June 2019
The General Motion of Conduction Electrons in a Uniform Magnetic Field, with Application to the Diamagnetism of Metals Harper, P. G. Proceedings of the Physical Society. Section A, Vol. 68, Issue 10 https://doi.org/10.1088/0370-1298/68/10/305	journal	October 1955
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 Ahn, Junyeong; Park, Sungjoon; Yang, Bohm-Jung Physical Review X, Vol. 9, Issue 2 https://doi.org/10.1103/PhysRevX.9.021013	journal	April 2019
Fragile Phases as Affine Monoids: Classification and Material Examples Song, Zhi-Da; Elcoro, Luis; Xu, Yuan-Feng Physical Review X, Vol. 10, Issue 3 https://doi.org/10.1103/PhysRevX.10.031001	journal	July 2020
Higher-order topological insulators and superconductors protected by inversion symmetry Khalaf, Eslam Physical Review B, Vol. 97, Issue 20 https://doi.org/10.1103/PhysRevB.97.205136	journal	May 2018
Topological Insulators and Topological Superconductors Bernevig, B. Andrei Princeton University Press https://doi.org/10.1515/9781400846733	book	January 2013
Quantised Hall conductance in a perfect crystal Dana, I.; Avron, Y.; Zak, J. Journal of Physics C: Solid State Physics, Vol. 18, Issue 22 https://doi.org/10.1088/0022-3719/18/22/004	journal	August 1985
Periodic table for topological insulators and superconductors Kitaev, Alexei; Lebedev, Vladimir; Feigel’man, Mikhail ADVANCES IN THEORETICAL PHYSICS: Landau Memorial Conference, AIP Conference Proceedings https://doi.org/10.1063/1.3149495	conference	January 2009
Topological phases and the quantum spin Hall effect in three dimensions Roy, Rahul Physical Review B, Vol. 79, Issue 19 https://doi.org/10.1103/PhysRevB.79.195322	journal	May 2009
Building blocks of topological quantum chemistry: Elementary band representations Cano, Jennifer; Bradlyn, Barry; Wang, Zhijun Physical Review B, Vol. 97, Issue 3 https://doi.org/10.1103/PhysRevB.97.035139	journal	January 2018
Boron nitride substrates for high-quality graphene electronics Dean, C. R.; Young, A. F.; Meric, I. Nature Nanotechnology, Vol. 5, Issue 10, p. 722-726 https://doi.org/10.1038/nnano.2010.172	journal	August 2010
Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene Xie, Yonglong; Lian, Biao; Jäck, Berthold Nature, Vol. 572, Issue 7767 https://doi.org/10.1038/s41586-019-1422-x	journal	July 2019
Higher-Order Bulk-Boundary Correspondence for Topological Crystalline Phases Trifunovic, Luka; Brouwer, Piet W. Physical Review X, Vol. 9, Issue 1 https://doi.org/10.1103/PhysRevX.9.011012	journal	January 2019
Topological insulators and superconductors: tenfold way and dimensional hierarchy Ryu, Shinsei; Schnyder, Andreas P.; Furusaki, Akira New Journal of Physics, Vol. 12, Issue 6 https://doi.org/10.1088/1367-2630/12/6/065010	journal	June 2010
Nearly Flatbands with Nontrivial Topology Sun, Kai; Gu, Zhengcheng; Katsura, Hosho Physical Review Letters, Vol. 106, Issue 23 https://doi.org/10.1103/PhysRevLett.106.236803	journal	June 2011
Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry Bouhon, Adrien; Black-Schaffer, Annica M.; Slager, Robert-Jan Physical Review B, Vol. 100, Issue 19 https://doi.org/10.1103/PhysRevB.100.195135	journal	November 2019
Landau Quantization of Topological Surface States in ${Bi}_{2} {Se}_{3}$ Cheng, Peng; Song, Canli; Zhang, Tong Physical Review Letters, Vol. 105, Issue 7 https://doi.org/10.1103/PhysRevLett.105.076801	journal	August 2010
Quantized Hall Conductance in a Two-Dimensional Periodic Potential Thouless, D. J.; Kohmoto, M.; Nightingale, M. P. Physical Review Letters, Vol. 49, Issue 6 https://doi.org/10.1103/PhysRevLett.49.405	journal	August 1982
Graph theory data for topological quantum chemistry Vergniory, M. G.; Elcoro, L.; Wang, Zhijun Physical Review E, Vol. 96, Issue 2 https://doi.org/10.1103/PhysRevE.96.023310	journal	August 2017
Classification of Topological Phase Transitions and van Hove Singularity Steering Mechanism in Graphene Superlattices Wang, Jian; Santos, Luiz H. Physical Review Letters, Vol. 125, Issue 23 https://doi.org/10.1103/PhysRevLett.125.236805	journal	December 2020
Exponential Localization of Wannier Functions in Insulators Brouder, Christian; Panati, Gianluca; Calandra, Matteo Physical Review Letters, Vol. 98, Issue 4 https://doi.org/10.1103/PhysRevLett.98.046402	journal	January 2007
Evidence of Hofstadter's Fractal Energy Spectrum in the Quantized Hall Conductance Albrecht, C.; Smet, J. H.; von Klitzing, K. Physical Review Letters, Vol. 86, Issue 1 https://doi.org/10.1103/PhysRevLett.86.147	journal	January 2001
Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator Chang, C. -Z.; Zhang, J.; Feng, X. Science, Vol. 340, Issue 6129 https://doi.org/10.1126/science.1234414	journal	March 2013
Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices Dean, C. R.; Wang, L.; Maher, P. Nature, Vol. 497, Issue 7451 https://doi.org/10.1038/nature12186	journal	May 2013
Cloning of Dirac fermions in graphene superlattices Ponomarenko, L. A.; Gorbachev, R. V.; Yu, G. L. Nature, Vol. 497, Issue 7451 https://doi.org/10.1038/nature12187	journal	May 2013
$Z_{2}$ Topological Order and the Quantum Spin Hall Effect Kane, C. L.; Mele, E. J. Physical Review Letters, Vol. 95, Issue 14, Article No. 146802 https://doi.org/10.1103/PhysRevLett.95.146802	journal	September 2005
Inversion-symmetric topological insulators Hughes, Taylor L.; Prodan, Emil; Bernevig, B. Andrei Physical Review B, Vol. 83, Issue 24 https://doi.org/10.1103/PhysRevB.83.245132	journal	June 2011
Faithful tight-binding models and fragile topology of magic-angle bilayer graphene Po, Hoi Chun; Zou, Liujun; Senthil, T. Physical Review B, Vol. 99, Issue 19 https://doi.org/10.1103/PhysRevB.99.195455	journal	May 2019
Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators Benalcazar, Wladimir A.; Bernevig, B. Andrei; Hughes, Taylor L. Physical Review B, Vol. 96, Issue 24 https://doi.org/10.1103/PhysRevB.96.245115	journal	December 2017
Bilbao Crystallographic Server: I. Databases and crystallographic computing programs Aroyo, Mois Ilia; Perez-Mato, Juan Manuel; Capillas, Cesar Zeitschrift für Kristallographie - Crystalline Materials, Vol. 221, Issue 1 https://doi.org/10.1524/zkri.2006.221.1.15	journal	January 2006
Twisted bulk-boundary correspondence of fragile topology Song, Zhi-Da; Elcoro, Luis; Bernevig, B. Andrei Science, Vol. 367, Issue 6479 https://doi.org/10.1126/science.aaz7650	journal	February 2020
Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groups Aroyo, Mois I.; Kirov, Asen; Capillas, Cesar Acta Crystallographica Section A Foundations of Crystallography, Vol. 62, Issue 2 https://doi.org/10.1107/S0108767305040286	journal	March 2006
Zur Theorie des Diamagnetismus von Leitungselektronen Peierls, R. Zeitschrift f�r Physik, Vol. 80, Issue 11-12 https://doi.org/10.1007/BF01342591	journal	November 1933
Topological quantum chemistry Bradlyn, Barry; Elcoro, L.; Cano, Jennifer Nature, Vol. 547, Issue 7663 https://doi.org/10.1038/nature23268	journal	July 2017
Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi ₂ Te ₄ Deng, Yujun; Yu, Yijun; Shi, Meng Zhu Science, Vol. 367, Issue 6480 https://doi.org/10.1126/science.aax8156	journal	January 2020
Topological Insulators in Three Dimensions Fu, Liang; Kane, C. L.; Mele, E. J. Physical Review Letters, Vol. 98, Issue 10, Article No. 106803 https://doi.org/10.1103/PhysRevLett.98.106803	journal	March 2007
Moire bands in twisted double-layer graphene Bistritzer, R.; MacDonald, A. H. Proceedings of the National Academy of Sciences, Vol. 108, Issue 30 https://doi.org/10.1073/pnas.1108174108	journal	July 2011
Correlated insulator behaviour at half-filling in magic-angle graphene superlattices Cao, Yuan; Fatemi, Valla; Demir, Ahmet Nature, Vol. 556, Issue 7699 https://doi.org/10.1038/nature26154	journal	March 2018
$(d - 2)$ -Dimensional Edge States of Rotation Symmetry Protected Topological States Song, Zhida; Fang, Zhong; Fang, Chen Physical Review Letters, Vol. 119, Issue 24 https://doi.org/10.1103/PhysRevLett.119.246402	journal	December 2017
Massive Dirac Fermions and Hofstadter Butterfly in a van der Waals Heterostructure Hunt, B.; Sanchez-Yamagishi, J. D.; Young, A. F. Science, Vol. 340, Issue 6139 https://doi.org/10.1126/science.1237240	journal	May 2013
Quantized electric multipole insulators Benalcazar, Wladimir A.; Bernevig, B. Andrei; Hughes, Taylor L. Science, Vol. 357, Issue 6346 https://doi.org/10.1126/science.aah6442	journal	July 2017
Graphene field-effect transistors based on boron nitride gate dielectrics Meric, Inanc; Dean, Cory; Young, Andrea 2010 IEEE International Electron Devices Meeting (IEDM), 2010 International Electron Devices Meeting https://doi.org/10.1109/IEDM.2010.5703419	conference	December 2010
Fractional quantum Hall physics in topological flat bands Parameswaran, Siddharth A.; Roy, Rahul; Sondhi, Shivaji L. Comptes Rendus Physique, Vol. 14, Issue 9-10 https://doi.org/10.1016/j.crhy.2013.04.003	journal	November 2013
Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields Hofstadter, Douglas R. Physical Review B, Vol. 14, Issue 6 https://doi.org/10.1103/PhysRevB.14.2239	journal	September 1976
Magnetic Translation Group Zak, J. Physical Review, Vol. 134, Issue 6A https://doi.org/10.1103/PhysRev.134.A1602	journal	June 1964
Emergent many-body translational symmetries of Abelian and non-Abelian fractionally filled topological insulators Bernevig, B. Andrei; Regnault, N. Physical Review B, Vol. 85, Issue 7 https://doi.org/10.1103/PhysRevB.85.075128	journal	February 2012
All Magic Angles in Twisted Bilayer Graphene are Topological Song, Zhida; Wang, Zhijun; Shi, Wujun Physical Review Letters, Vol. 123, Issue 3 https://doi.org/10.1103/PhysRevLett.123.036401	journal	July 2019
Topological Winding Number Change and Broken Inversion Symmetry in a Hofstadter’s Butterfly Wang, Peng; Cheng, Bin; Martynov, Oleg Nano Letters, Vol. 15, Issue 10 https://doi.org/10.1021/acs.nanolett.5b01568	journal	September 2015
Zur Theorie des Diamagnetismus von Leitungselektronen. II Starke Magnetfelder Peierls, R. Zeitschrift f�r Physik, Vol. 81, Issue 3-4 https://doi.org/10.1007/bf01338364	journal	March 1933
Evidence of Hofstadter's fractal energy spectrum in the quantized Hall conductance Albrecht, C.; Smet, J. H.; von Klitzing, K. Physica E: Low-dimensional Systems and Nanostructures, Vol. 20, Issue 1-2 https://doi.org/10.1016/j.physe.2003.09.031	journal	December 2003
Spin-Orbit-Free Topological Insulators Alexandradinata, A.; Bernevig, B. Andrei arXiv https://doi.org/10.48550/arxiv.1411.3338	preprint	January 2014

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Title: Hofstadter Topology: Noncrystalline Topological Materials at High Flux

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