References
A. Agazzi, J.-P. Eckmann, and G. M. Graf. The colored Hofstadter butterfly for the honeycomb lattice. Journal of Statistical Physics, 156(3):417–426, Aug 2014. URL: https://doi.org/10.1007/s10955-014-0992-0, doi:10.1007/s10955-014-0992-0.
M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch. Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices. Physical Review Letters, 111:185301, Oct 2013. URL: https://link.aps.org/doi/10.1103/PhysRevLett.111.185301, doi:10.1103/PhysRevLett.111.185301.
M. Aidelsburger. Artificial gauge fields with ultracold atoms in optical lattices. PhD thesis, Ludwig Maximilian University of Munich, 2014. URL: https://edoc.ub.uni-muenchen.de/18148/.
C. Albrecht, J. H. Smet, K. von Klitzing, D. Weiss, V. Umansky, and H. Schweizer. Evidence of Hofstadter's fractal energy spectrum in the quantized Hall conductance. Physical Review Letters, 86:147–150, Jan 2001. URL: https://link.aps.org/doi/10.1103/PhysRevLett.86.147, doi:10.1103/PhysRevLett.86.147.
B. Andrews and A. Soluyanov. Fractional quantum Hall states for moiré superstructures in the Hofstadter regime. Physical Review B, 101:235312, Jun 2020. URL: https://link.aps.org/doi/10.1103/PhysRevB.101.235312, doi:10.1103/PhysRevB.101.235312.
B. Andrews, M. Mohan, and T. Neupert. Abelian topological order of $\ensuremath \nu =2/5$ and $3/7$ fractional quantum Hall states in lattice models. Physical Review B, 103:075132, Feb 2021. URL: https://link.aps.org/doi/10.1103/PhysRevB.103.075132, doi:10.1103/PhysRevB.103.075132.
B. Andrews, M. Raja, N. Mishra, M. Zaletel, and R. Roy. Stability of fractional Chern insulators with a non-Landau level continuum limit. 2023. arXiv:2310.05758, doi:10.48550/arXiv.2310.05758.
A. Avila and S. Jitomirskaya. The Ten Martini Problem. Annals of Mathematics, 170(1):303–342, 2009. URL: http://www.jstor.org/stable/40345465, doi:10.4007/annals.2009.170.303.
J. E. Avron, D. Osadchy, and R. Seiler. A Topological Look at the Quantum Hall Effect. Physics Today, 56(8):38–42, Aug 2003. URL: https://doi.org/10.1063/1.1611351, doi:10.1063/1.1611351.
J. E. Avron, O. Kenneth, and G. Yehoshua. A study of the ambiguity in the solutions to the Diophantine equation for Chern numbers. Journal of Physics A: Mathematical and Theoretical, 47(18):185202, Apr 2014. URL: https://dx.doi.org/10.1088/1751-8113/47/18/185202, doi:10.1088/1751-8113/47/18/185202.
M. Y. Azbel. Energy spectrum of a conduction electron in a magnetic field. Journal of Experimental and Theoretical Physics, 19(3):634–645, 1964. URL: http://jetp.ras.ru/cgi-bin/e/index/e/19/3/p634?a=list.
D. Bauer, T. S. Jackson, and R. Roy. Quantum geometry and stability of the fractional quantum Hall effect in the Hofstadter model. Physical Review B, 93:235133, Jun 2016. URL: https://link.aps.org/doi/10.1103/PhysRevB.93.235133, doi:10.1103/PhysRevB.93.235133.
D. Bauer, S. Talkington, F. Harper, B. Andrews, and R. Roy. Fractional Chern insulators with a non-Landau level continuum limit. Physical Review B, 105:045144, Jan 2022. URL: https://link.aps.org/doi/10.1103/PhysRevB.105.045144, doi:10.1103/PhysRevB.105.045144.
D. Bodesheim, R. Biele, and G. Cuniberti. Hierarchies of Hofstadter butterflies in 2D covalent organic frameworks. npj 2D Materials and Applications, 7(1):16, Mar 2023. URL: https://doi.org/10.1038/s41699-023-00378-0, doi:10.1038/s41699-023-00378-0.
J. P. Chen and R. Guo. Spectral decimation of the magnetic Laplacian on the Sierpinski gasket: solving the Hofstadter–Sierpinski butterfly. Communications in Mathematical Physics, 380(1):187–243, Nov 2020. URL: https://doi.org/10.1007/s00220-020-03850-w, doi:10.1007/s00220-020-03850-w.
M. Claassen, C. H. Lee, R. Thomale, X.-L. Qi, and T. P. Devereaux. Position-momentum duality and fractional quantum Hall effect in Chern insulators. Physical Review Letters, 114:236802, Jun 2015. URL: https://link.aps.org/doi/10.1103/PhysRevLett.114.236802, doi:10.1103/PhysRevLett.114.236802.
N. R. Cooper, J. Dalibard, and I. B. Spielman. Topological bands for ultracold atoms. Reviews of Modern Physics, 91:015005, Mar 2019. URL: https://link.aps.org/doi/10.1103/RevModPhys.91.015005, doi:10.1103/RevModPhys.91.015005.
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. Hofstadter's butterfly and the fractal quantum Hall effect in moiré superlattices. Nature, 497(7451):598–602, May 2013. URL: https://doi.org/10.1038/nature12186, doi:10.1038/nature12186.
F. Di Colandrea, A. D’Errico, M. Maffei, H. M. Price, M. Lewenstein, L. Marrucci, F. Cardano, A. Dauphin, and P. Massignan. Linking topological features of the Hofstadter model to optical diffraction figures. New Journal of Physics, 24(1):013028, Jan 2022. URL: https://dx.doi.org/10.1088/1367-2630/ac4126, doi:10.1088/1367-2630/ac4126.
N. Regnault. DiagHam. Code repository at http://www.nick-ux.org/diagham, 2001.
L. Du, Q. Chen, A. D. Barr, A. R. Barr, and G. A. Fiete. Floquet Hofstadter butterfly on the kagome and triangular lattices. Physical Review B, 98:245145, Dec 2018. URL: https://link.aps.org/doi/10.1103/PhysRevB.98.245145, doi:10.1103/PhysRevB.98.245145.
A. Eckardt. Colloquium: atomic quantum gases in periodically driven optical lattices. Reviews of Modern Physics, 89:011004, Mar 2017. URL: https://link.aps.org/doi/10.1103/RevModPhys.89.011004, doi:10.1103/RevModPhys.89.011004.
F. T. P. Harper. The Hofstadter Model and Other Fractional Chern Insulators. PhD thesis, University of Oxford, 2015. URL: https://ora.ox.ac.uk/objects/uuid:4c4df19a-9bab-43c4-a845-ae170868913f.
T. Fukui, Y. Hatsugai, and H. Suzuki. Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances. Journal of the Physical Society of Japan, 74(6):1674–1677, 2005. URL: https://doi.org/10.1143/JPSJ.74.1674, doi:10.1143/JPSJ.74.1674.
R. Ghadimi, T. Sugimoto, and T. Tohyama. Higher-dimensional Hofstadter butterfly on the Penrose lattice. Physical Review B, 106:L201113, Nov 2022. URL: https://link.aps.org/doi/10.1103/PhysRevB.106.L201113, doi:10.1103/PhysRevB.106.L201113.
N. Goldman, G. Juzeliūnas, P. Öhberg, and I. B. Spielman. Light-induced gauge fields for ultracold atoms. Reports on Progress in Physics, 77(12):126401, Nov 2014. URL: https://dx.doi.org/10.1088/0034-4885/77/12/126401, doi:10.1088/0034-4885/77/12/126401.
K. Goudarzi, H. G. Maragheh, and M. Lee. Calculation of the Berry curvature and Chern number of topological photonic crystals. Journal of the Korean Physical Society, 81(5):386–390, Sep 2022. URL: https://doi.org/10.1007/s40042-022-00530-x, doi:10.1007/s40042-022-00530-x.
D. Gresch and A. Soluyanov. Calculating Topological Invariants with Z2Pack, pages 63–92. Springer International Publishing, Cham, 2018. URL: https://doi.org/10.1007/978-3-319-76388-0_3, doi:10.1007/978-3-319-76388-0_3.
P. G. Harper. Single band motion of conduction electrons in a uniform magnetic field. Proceedings of the Physical Society. Section A, 68(10):874, Oct 1955. URL: https://dx.doi.org/10.1088/0370-1298/68/10/304, doi:10.1088/0370-1298/68/10/304.
M. M. Hirschmann and J. Mitscherling. Symmetry-enforced double Weyl points, multiband quantum geometry, and singular flat bands of doping-induced states at the Fermi level. Physical Review Materials, 8:014201, Jan 2024. URL: https://link.aps.org/doi/10.1103/PhysRevMaterials.8.014201, doi:10.1103/PhysRevMaterials.8.014201.
D. R. Hofstadter. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Physical Review B, 14:2239–2249, Sep 1976. URL: https://link.aps.org/doi/10.1103/PhysRevB.14.2239, doi:10.1103/PhysRevB.14.2239.
B. Andrews. HofstadterTools: A Python package for analyzing the Hofstadter model. Journal of Open Source Software, 9(95):6356, 2024. URL: https://doi.org/10.21105/joss.06356, doi:10.21105/joss.06356.
T. S. Jackson, G. Möller, and R. Roy. Geometric stability of topological lattice phases. Nature Communications, 6(1):8629, Nov 2015. URL: https://doi.org/10.1038/ncomms9629, doi:10.1038/ncomms9629.
H. Jing-Min. Light-induced Hofstadter's butterfly spectrum of ultracold atoms on the two-dimensional kagomé lattice. Chinese Physics Letters, 26(12):123701, Dec 2009. URL: https://dx.doi.org/10.1088/0256-307X/26/12/123701, doi:10.1088/0256-307X/26/12/123701.
P. J. Ledwith, A. Vishwanath, and D. E. Parker. Vortexability: A unifying criterion for ideal fractional Chern insulators. Physical Review B, 108:205144, Nov 2023. URL: https://link.aps.org/doi/10.1103/PhysRevB.108.205144, doi:10.1103/PhysRevB.108.205144.
C. H. Lee, M. Claassen, and R. Thomale. Band structure engineering of ideal fractional Chern insulators. Physical Review B, 96:165150, Oct 2017. URL: https://link.aps.org/doi/10.1103/PhysRevB.96.165150, doi:10.1103/PhysRevB.96.165150.
B. Mera and J. Mitscherling. Nontrivial quantum geometry of degenerate flat bands. Physical Review B, 106:165133, Oct 2022. URL: https://link.aps.org/doi/10.1103/PhysRevB.106.165133, doi:10.1103/PhysRevB.106.165133.
H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle. Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices. Physical Review Letters, 111:185302, Oct 2013. URL: https://link.aps.org/doi/10.1103/PhysRevLett.111.185302, doi:10.1103/PhysRevLett.111.185302.
T. Neupert, L. Santos, C. Chamon, and C. Mudry. Fractional quantum Hall states at zero magnetic field. Physical Review Letters, 106:236804, Jun 2011. URL: https://link.aps.org/doi/10.1103/PhysRevLett.106.236804, doi:10.1103/PhysRevLett.106.236804.
X. Ni, K. Chen, M. Weiner, D. J. Apigo, C. Prodan, A. Alù, E. Prodan, and A. B. Khanikaev. Observation of Hofstadter butterfly and topological edge states in reconfigurable quasi-periodic acoustic crystals. Communications Physics, 2(1):55, Jun 2019. URL: https://doi.org/10.1038/s42005-019-0151-7, doi:10.1038/s42005-019-0151-7.
G.-Y. Oh. Energy spectrum of a triangular lattice in a uniform magnetic field: effect of next-nearest-neighbor hopping. Journal of the Korean Physical Society, 37:534–539, 11 2000. doi:10.3938/jkps.37.534.
K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewenstein. Cold atoms in non-Abelian gauge potentials: from the Hofstadter "moth" to lattice gauge theory. Physical Review Letters, 95:010403, Jun 2005. URL: https://link.aps.org/doi/10.1103/PhysRevLett.95.010403, doi:10.1103/PhysRevLett.95.010403.
S. A. Parameswaran, R. Roy, and S. L. Sondhi. Fractional quantum Hall physics in topological flat bands. Comptes Rendus Physique, 14(9):816–839, 2013. Topological insulators / Isolants topologiques. URL: https://www.sciencedirect.com/science/article/pii/S163107051300073X, doi:https://doi.org/10.1016/j.crhy.2013.04.003.
R. Peierls. Zur Theorie des Diamagnetismus von Leitungselektronen. Zeitschrift für Physik, 80(11):763–791, Nov 1933. URL: https://doi.org/10.1007/BF01342591, doi:10.1007/BF01342591.
A. Pena. Control of spectral, topological and charge transport properties of graphene via circularly polarized light and magnetic field. Results in Physics, 46:106257, 2023. URL: https://www.sciencedirect.com/science/article/pii/S2211379723000505, doi:https://doi.org/10.1016/j.rinp.2023.106257.
J. Lado. Pyqula. Code repository at https://github.com/joselado/pyqula, 2021.
R. Rammal. Landau level spectrum of Bloch electrons in a honeycomb lattice. Journal De Physique, 46(8):1345–1354, 1985. URL: https://doi.org/10.1051/jphys:019850046080134500, doi:10.1051/jphys:019850046080134500.
P. Roushan, C. Neill, J. Tangpanitanon, V. M. Bastidas, A. Megrant, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Fowler, B. Foxen, M. Giustina, E. Jeffrey, J. Kelly, E. Lucero, J. Mutus, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. White, H. Neven, D. G. Angelakis, and J. Martinis. Spectroscopic signatures of localization with interacting photons in superconducting qubits. Science, 358(6367):1175–1179, 2017. URL: https://www.science.org/doi/abs/10.1126/science.aao1401, doi:10.1126/science.aao1401.
R. Roy. Band geometry of fractional topological insulators. Physical Review B, 90:165139, Oct 2014. URL: https://link.aps.org/doi/10.1103/PhysRevB.90.165139, doi:10.1103/PhysRevB.90.165139.
S. Roy, M. Kolodrubetz, J. E. Moore, and A. G. Grushin. Chern numbers and chiral anomalies in Weyl butterflies. Physical Review B, 94:161107, Oct 2016. URL: https://link.aps.org/doi/10.1103/PhysRevB.94.161107, doi:10.1103/PhysRevB.94.161107.
R. Sahay, S. Divic, D. E. Parker, T. Soejima, S. Anand, J. Hauschild, M. Aidelsburger, A. Vishwanath, S. Chatterjee, N. Y. Yao, and M. P. Zaletel. Superconductivity in a topological lattice model with strong repulsion. 2023. arXiv:2308.10935, doi:10.48550/arXiv.2308.10935.
I. I. Satija. Butterfly in the Quantum World. 2053-2571. Morgan & Claypool Publishers, 2016. ISBN 978-1-6817-4117-8. URL: https://dx.doi.org/10.1088/978-1-6817-4117-8, doi:10.1088/978-1-6817-4117-8.
D. Shaffer, J. Wang, and L. H. Santos. Theory of Hofstadter superconductors. Physical Review B, 104:184501, Nov 2021. URL: https://link.aps.org/doi/10.1103/PhysRevB.104.184501, doi:10.1103/PhysRevB.104.184501.
B. Simon. Schrödinger operators in the twenty-first century, chapter, pages 283–288. World Scientific, 2000. URL: https://www.worldscientific.com/doi/abs/10.1142/9781848160224_0014, doi:10.1142/9781848160224_0014.
A. Soluyanov. Topological aspects of band theory. PhD thesis, Rutgers University, 2012. URL: https://rucore.libraries.rutgers.edu/rutgers-lib/39042/.
A. Stegmaier, L. K. Upreti, R. Thomale, and I. Boettcher. Universality of Hofstadter butterflies on hyperbolic lattices. Physical Review Letters, 128:166402, Apr 2022. URL: https://link.aps.org/doi/10.1103/PhysRevLett.128.166402, doi:10.1103/PhysRevLett.128.166402.
P. Streda. Theory of quantised Hall conductivity in two dimensions. Journal of Physics C: Solid State Physics, 15(22):L717, Aug 1982. URL: https://dx.doi.org/10.1088/0022-3719/15/22/005, doi:10.1088/0022-3719/15/22/005.
J. Hauschild and F. Pollmann. Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy). SciPost Physics Lecture Notes, pages 5, 2018. Code available from https://github.com/tenpy/tenpy. URL: https://scipost.org/10.21468/SciPostPhysLectNotes.5, arXiv:1805.00055, doi:10.21468/SciPostPhysLectNotes.5.
D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs. Quantized Hall conductance in a two-dimensional periodic potential. Physical Review Letters, 49:405–408, Aug 1982. URL: https://link.aps.org/doi/10.1103/PhysRevLett.49.405, doi:10.1103/PhysRevLett.49.405.
H. Tian, X. Gao, Y. Zhang, S. Che, T. Xu, P. Cheung, K. Watanabe, T. Taniguchi, M. Randeria, F. Zhang, C. N. Lau, and M. W. Bockrath. Evidence for Dirac flat band superconductivity enabled by quantum geometry. Nature, 614(7948):440–444, Feb 2023. URL: https://doi.org/10.1038/s41586-022-05576-2, doi:10.1038/s41586-022-05576-2.
J. Vidal, R. Mosseri, and B. Douçot. Aharonov-Bohm cages in two-dimensional structures. Physical Review Letters, 81:5888–5891, Dec 1998. URL: https://link.aps.org/doi/10.1103/PhysRevLett.81.5888, doi:10.1103/PhysRevLett.81.5888.
J. Wang, J. Cano, A. J. Millis, Z. Liu, and B. Yang. Exact Landau level description of geometry and interaction in a flatband. Physical Review Letters, 127:246403, Dec 2021. URL: https://link.aps.org/doi/10.1103/PhysRevLett.127.246403, doi:10.1103/PhysRevLett.127.246403.
G. H. Wannier. A result not dependent on rationality for Bloch electrons in a magnetic field. physica status solidi (b), 88(2):757–765, 1978. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/pssb.2220880243, doi:10.1002/pssb.2220880243.
Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and A. A. Soluyanov. WannierTools: An open-source software package for novel topological materials. Computer Physics Communications, 224:405–416, 2018. URL: http://www.sciencedirect.com/science/article/pii/S0010465517303442, doi:10.1016/j.cpc.2017.09.033.
Y. Xiao, V. Pelletier, P. M. Chaikin, and D. A. Huse. Landau levels in the case of two degenerate coupled bands: Kagomé lattice tight-binding spectrum. Physical Review B, 67:104505, Mar 2003. URL: https://link.aps.org/doi/10.1103/PhysRevB.67.104505, doi:10.1103/PhysRevB.67.104505.
Y. Yang, B. Zhen, J. D. Joannopoulos, and M. Soljačić. Non-Abelian generalizations of the Hofstadter model: spin–orbit-coupled butterfly pairs. Light: Science & Applications, 9(1):177, Oct 2020. URL: https://doi.org/10.1038/s41377-020-00384-7, doi:10.1038/s41377-020-00384-7.
F. Yilmaz and M. Ö. Oktel. Hofstadter butterfly evolution in the space of two-dimensional Bravais lattices. Physical Review A, 95:063628, Jun 2017. URL: https://link.aps.org/doi/10.1103/PhysRevA.95.063628, doi:10.1103/PhysRevA.95.063628.
J. Zak. Magnetic translation group. Physical Review, 134:A1602–A1606, Jun 1964. URL: https://link.aps.org/doi/10.1103/PhysRev.134.A1602, doi:10.1103/PhysRev.134.A1602.
Y. Zhang, N. Manjunath, G. Nambiar, and M. Barkeshli. Fractional disclination charge and discrete shift in the Hofstadter butterfly. Physical Review Letters, 129:275301, Dec 2022. URL: https://link.aps.org/doi/10.1103/PhysRevLett.129.275301, doi:10.1103/PhysRevLett.129.275301.
Y. Zhang, N. Manjunath, G. Nambiar, and M. Barkeshli. Quantized charge polarization as a many-body invariant in $(2+1)\mathrm D$ crystalline topological states and Hofstadter butterflies. Physical Review X, 13:031005, Jul 2023. URL: https://link.aps.org/doi/10.1103/PhysRevX.13.031005, doi:10.1103/PhysRevX.13.031005.
Y. Zhang, N. Manjunath, R. Kobayashi, and M. Barkeshli. Complete crystalline topological invariants from partial rotations in $(2+1)\mathrm D$ invertible fermionic states and Hofstadter's butterfly. Physical Review Letters, 131:176501, Oct 2023. URL: https://link.aps.org/doi/10.1103/PhysRevLett.131.176501, doi:10.1103/PhysRevLett.131.176501.
O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman. Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature, 553(7686):59–62, Jan 2018. URL: https://doi.org/10.1038/nature25011, doi:10.1038/nature25011.