Fractal Nature of Defect States in the Hofstadter butterfly

[1] P G Harper. The general motion of conduction electrons in a uniform magnetic field, with application to the diamagnetism of metals. Proceedings of the Physical Society. Section A, 68:879–892, oct 1955.

[2] P G Harper. Single band motion of conduction electrons in a uniform magnetic field. Proceedings of the Physical Society. Section A, 68:874–878, 1955.

[3] Douglas R Hofstadter. Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields. Physical review B, 14(6):2239, 1976.

[4] Till Schl¨osser, Klaus Ensslin, Jo¨rg P Kotthaus, and Martin Holland. Landau sub- bands generated by a lateral electrostatic superlattice - chasing the hofstadter butterfly. Semiconductor Science and Technology, 11, nov 1996.

[5] 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. Phys. Rev. Lett., 86:147–150, Jan 2001.

[6] MC Geisler, JH Smet, V Umansky, K Von Klitzing, B Naundorf, R Ketzmerick, and H Schweizer. Detection of a landau band-coupling-induced rearrangement of the hofs- tadter butterfly. Phys. Rev. Lett., 92(25):256801, 2004.

[7] 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 moir´e super- lattices. Nature, 497:598–602, Jun 2013.

[8] B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watan- abe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori. Massive dirac fermions and hofstadter butterfly in a van der waals heterostructure. Science, 340, 2013.

[9] 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, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal’ko, and A. K. Geim. Cloning of dirac fermions in graphene superlattices. Nature, 497:594–597, nov 2013.

[10] D Jaksch and P Zoller. Creation of effective magnetic fields in optical lattices: the hofstadter butterfly for cold neutral atoms. New Journal of Physics, 5:56–56, Jun 2003.

[11] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch. Realiza- tion of the hofstadter hamiltonian with ultracold atoms in optical lattices. Phys. Rev. Lett., 111:185301, Oct 2013.

[12] Hirokazu Miyake, Georgios A. Siviloglou, Colin J. Kennedy, William Cody Burton, and Wolfgang Ketterle. Realizing the harper hamiltonian with laser-assisted tunneling in optical lattices. Phys. Rev. Lett., 111:185302, Oct 2013.

[13] 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. Jef- frey, 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 sig- natures of localization with interacting photons in superconducting qubits. Science, 358(6367):1175–1179, 2017.

[14] U. Kuhl and H.-J. St¨ockmann. Microwave realization of the hofstadter butterfly. Phys. Rev. Lett., 80:3232–3235, Apr 1998.

[15] O Richoux and V Pagneux. Acoustic characterization of the hofstadter butterfly with resonant scatterers. Europhysics Letters (EPL), 59:34–40, jul 2002.

[16] Xiang Ni, Kai Chen, Matthew Weiner, David J. Apigo, Camelia Prodan, Andrea Alu`, Emil Prodan, and Alexander B. Khanikaev. Observation of hofstadter butterfly and topological edge states in reconfigurable quasi-periodic acoustic crystals. Commun Phys, 2:256 – 261, 2019.

[17] Indubala I Satija. Butterfly in the Quantum World. Morgan & Claypool Publishers, 2016.

[18] G. H. Wannier, G. M. Obermair, and R. Ray. Manetoelectronic denisty of states for a model crystal. physica status solidi (b), 93:337–342, 1979.

[19] F. H. Claro and G. H. Wannier. Magnetic subband structure of electrons in hexagonal lattices. Phys. Rev. B, 19:6068–6074, Jun 1979.

[20] Martin C. Geisler. The hofstadter butterfly and quantum interferences in modulated 2- dimensional electron systems, phd thesis university of stuttgart doi:10.18419/opus-6610.

[21] A. L. C. Pereira and P. A. Schulz. Additional levels between landau bands due to vacancies in graphene: Towards defect engineering. Phys. Rev. B, 78:125402, Sep 2008.

[22] Selcen I˙slamo˘glu, M. O¨ . Oktel, and O g˘uz Gu¨lseren. Hofstadter butterfly of graphene with point defects. Phys. Rev. B, 85:235414, Jun 2012.

[23] Selcen I˙slamo˘glu, M. O¨ . Oktel, and Gu¨lseren. Hall conductance in graphene with point defects. Journal of Physics: Condensed Matter, 25(5):055302, 2013.

[24] Jesper Goor Pedersen and Thomas Garm Pedersen. Hofstadter butterflies and magneti- cally induced band-gap quenching in graphene antidot lattices. Phys. Rev. B, 87:235404, Jun 2013.

[25] Chenggang Zhou, Mona Berciu, and R. N. Bhatt. Effects of large disorder on the hofstadter butterfly. Phys. Rev. B, 71:125310, Mar 2005.

[26] E. N. Grishanov and I. Y. Popov. Spectral properties of graphene with periodic array of defects in a magnetic field. Russ. J. Math. Phys., 25:277–283, 2018.

[27] Yoshiyuki Matsuki and Kazuki Ikeda. Comments on the fractal energy spectrum of honeycomb lattice with defects. Journal of Physics Communications, 3(5):055003, may 2019.

[28] Seydou-Samba Diop, Lars Fritz, Matthias Vojta, and Stephan Rachel. Impurity bound states as detectors of topological band structures revisited. Phys. Rev. B, 101:245132, Jun 2020.

[29] Tsuneya Ando and Yasutada Uemura. Theory of quantum transport in a two- dimensional electron system under magnetic fields. i. characteristics of level broadening and transport under strong fields. Journal of the Physical Society of Japan, 36:959–967, 1974.

[30] Tsuneya Ando. Theory of quantum transport in a two-dimensional electron system under magnetic fields. iii. many-site approximation. Journal of the Physical Society of Japan, 37:622–630, 1974.

[31] Tsuneya Ando, Yukio Matsumoto, and Yasutada Uemura. Theory of hall effect in a two-dimensional electron system. Journal of the Physical Society of Japan, 39:279–288, 1975.

[32] R. E. Prange. Quantized hall resistance and the measurement of the fine-structure constant. Phys. Rev. B, 23:4802–4805, May 1981.

[33] Robert Joynt and R. E. Prange. Conditions for the quantum hall effect. Phys. Rev. B, 29:3303–3317, Mar 1984.

[34] S.A. Gredeskul, M. Zusman, Y. Avishai, and M.Ya. Azbel’. Spectral properties and lo- calization of an electron in a two-dimensional system with point scatterers in a magnetic field. Physics Reports, 288(1):223 – 257, 1997.

[35] Yoshiyuki Matsuki, Kazuki Ikeda, and Mikito Koshino. Fractal defect states in the hofstadter butterfly. Phys. Rev. B, 104:035305, Jul 2021.

[36] K. v. Klitzing, G. Dorda, and M. Pepper. New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Phys. Rev. Lett., 45(0):494–497, Aug 1980.

[37] D. C. Tsui, H. L. Stormer, and A. C. Gossard. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett., 48(0):1559–1562, May 1982.

[38] Daijiro Yoshioka. The Quantum Hall Effect. Number XII, 210 in Springer Series in Solid-State Sciences. Springer, 1 edition, 2002.

[39] Marvin E Cage, Kv Klitzing, AM Chang, F Duncan, M Haldane, RB Laughlin, AMM Pruisken, and DJ Thouless. The quantum Hall effect. Springer Science & Business Media, 2012.

[40] David Tong. Lectures on the quantum hall effect. arXiv preprint arXiv:1606.06687, 2016.

[41] Horst L. Stormer. Nobel lecture: The fractional quantum hall effect. Rev. Mod. Phys., 71(0):875–889, Jul 1999.

[42] R. B. Laughlin. Quantized hall conductivity in two dimensions. Phys. Rev. B, 23:5632– 5633, May 1981.

[43] B. I. Halperin. Quantized hall conductance, current-carrying edge states, and the ex- istence of extended states in a two-dimensional disordered potential. Phys. Rev. B, 25:2185–2190, Feb 1982.

[44] P. R. Wallace. The band theory of graphite. Phys. Rev., 71:622–634, May 1947.

[45] Nguyen Hong Shon and Tsuneya Ando. Hall conductivity of a two-dimensional graphite system. Journal of the Physical Society of Japan, 67:2421–2429, 1998.

[46] Yisong Zheng and Tsuneya Ando. Hall conductivity of a two-dimensional graphite system. Phys. Rev. B, 65:245420, Jun 2002.

[47] Yasuhiro Hatsugai. Chern number and edge states in the integer quantum hall effect. Phys. Rev. Lett., 71:3697–3700, Nov 1993.

[48] Yasuhiro Hatsugai. Edge states in the integer quantum hall effect and the riemann surface of the bloch function. Phys. Rev. B, 48:11851–11862, Oct 1993.

[49] P Streda. Theory of quantised hall conductivity in two dimensions. Journal of Physics C: Solid State Physics, 15(22):L717, 1982.

[50] A. Widom. Thermodynamic derivation of the Hall effect current. Physics Letters A, 90:474–474, August 1982.

[51] Felix Bloch. U¨ ber die quantenmechanik der elektronen in kristallgittern. Zeitschrift fu¨r Physik, 52(7):555–600, 1929.

[52] R. Peierls. Zur theorie des diamagnetismus von leitungselektronen. Zeitschrift fu¨r Physik, 80(11):763–791, Nov 1933.

[53] J. Zak. Magnetic translation group. Phys. Rev., 134:A1602–A1606, Jun 1964.

[54] B. Andrei Bernevig. Topological Insulators and Topological Superconductors. 2013.

[55] Eduardo Fradkin. Field Theories of Condensed Matter Physics. Cambridge University Press, 2 edition, 2013.

[56] A Rauh, GH Wannier, and G Obermair. Bloch electrons in irrational magnetic fields. physica status solidi (b), 63(1):215–229, 1974.

[57] X.G. Wen and A. Zee. Winding number, family index theorem, and electron hopping in a magnetic field. Nuclear Physics B, 316(3):641–662, 1989.

[58] G. H. Wannier. A result not dependent on rationality for bloch electrons in a magnetic field. physica status solidi (b), 88:757–765, 1978.

[59] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs. Quantized hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett., 49:405–408, Aug 1982.

[60] Di Xiao, Ming-Che Chang, and Qian Niu. Berry phase effects on electronic properties. Rev. Mod. Phys., 82:1959–2007, Jul 2010.

[61] Pilkyung Moon and Mikito Koshino. Energy spectrum and quantum hall effect in twisted bilayer graphene. Phys. Rev. B, 85:195458, May 2012.

[62] Gerardo G. Naumis and J. L. Arag´on. Substitutional disorder in a fibonacci chain: Resonant eigenstates and instability of the spectrum. Phys. Rev. B, 54:15079–15085, Dec 1996.

[63] Gerardo G. Naumis. Use of the trace map for evaluating localization properties. Phys. Rev. B, 59:11315–11321, May 1999.

[64] Gerardo G. Naumis. Effects of an impurity in the harper model. Physics Letters A, 309, 2003.

[65] A. H. MacDonald. Landau-level subband structure of electrons on a square lattice. Phys. Rev. B, 28:6713–6717, Dec 1983.

[66] Jun-Won Rhim and Kwon Park. Self-similar occurrence of massless dirac particles in graphene under a magnetic field. Phys. Rev. B, 86:235411, Dec 2012.

[67] Nobuyuki Yoshioka, Hiroyasu Matsuura, and Masao Ogata. Quantum hall effect of mass- less dirac fermions and free fermions in hofstadter’s butterfly. Journal of the Physical Society of Japan, 85(6):064712, 2016.

[68] Archana Mishra, S. R. Hassan, and R. Shankar. Translational symmetry breaking and the disintegration of the hofstadter butterfly. Phys. Rev. B, 95:035140, Jan 2017.

[69] Vibhuti Bhushan Jha, Garima Rani, and R. Ganesh. Impurity-induced current in a chern insulator. Phys. Rev. B, 95:115434, Mar 2017.

[70] R M Cavalcanti and C A A de Carvalho. Two-dimensional electron gas in a uniform magnetic field in the presence of a delta-impurity. Journal of Physics A: Mathematical and General, 31:2391–2399, Mar 1998.

[71] YT Li and R Wong. Integral and series representations of the dirac delta function. arXiv preprint arXiv:1303.1943, 2013.

[72] A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Richard A. Silverman, and George H. Weiss. Methods of quantum field theory in statistical physics. Physics Today, 17(4):78– 80, 1964.

[73] Y. Niimi, H. Kambara, T. Matsui, D. Yoshioka, and Hiroshi Fukuyama. Real-space imaging of alternate localization and extension of quasi-two-dimensional electronic states at graphite surfaces in magnetic fields. Phys. Rev. Lett., 97:236804, Dec 2006.

[74] Mikito Koshino and Hiroki Oka. Topological invariants in two-dimensional quasicrystals. arXiv preprint arXiv:2109.12859, 2021.

[75] Hiroki Oka and Mikito Koshino. Fractal energy gaps and topological invariants in hbn/graphene/hbn double moir´e systems. Phys. Rev. B, 104:035306, Jul 2021.

[76] Stampfli. P. A dodecagonal quasiperiodic lattice in two dimensions. Helvetica Physica Acta, 59:1260–1263, 1986.

[77] Sung Joon Ahn, Pilkyung Moon, Tae-Hoon Kim, Hyun-Woo Kim, Ha-Chul Shin, Eun Hye Kim, Hyun Woo Cha, Se-Jong Kahng, Philip Kim, Mikito Koshino, et al. Dirac electrons in a dodecagonal graphene quasicrystal. Science, 361(6404):782–786, 2018.

[78] Yuichiro Takesaki, Kenji Kawahara, Hiroki Hibino, Susumu Okada, Masaharu Tsuji, and Hiroki Ago. Highly uniform bilayer graphene on epitaxial cu–ni (111) alloy. Chemistry of Materials, 28(13):4583–4592, 2016.

[79] Wei Yao, Eryin Wang, Changhua Bao, Yiou Zhang, Kenan Zhang, Kejie Bao, Chun Kai Chan, Chaoyu Chen, Jose Avila, Maria C Asensio, Junyi Zhu, and Shuyun Zhou. Qua- sicrystalline 30° twisted bilayer graphene as an incommensurate superlattice with strong interlayer coupling. Proceedings of the National Academy of Sciences of the United States of America, 115(27):6928–6933, 2018.

[80] Xu-Dong Chen, Wei Xin, Wen-Shuai Jiang, Zhi-Bo Liu, Yongsheng Chen, and Jian-Guo Tian. High-precision twist-controlled bilayer and trilayer graphene. Advanced Materials, 28(13):2563–2570, 2016.

[81] Pilkyung Moon, Mikito Koshino, and Young-Woo Son. Quasicrystalline electronic states in 30◦ rotated twisted bilayer graphene. Phys. Rev. B, 99:165430, Apr 2019.

[82] Mikito Koshino and Tsuneya Ando. Hall plateau diagram for the hofstadter butterfly energy spectrum. Phys. Rev. B, 73:155304, Apr 2006.

[83] Izrail Solomonovich Gradshteyn and Iosif Moiseevich Ryzhik. Table of integrals, series, and products. 2014.