Killing the Hofstadter Butterfly

  • Adhip AgarwalaEmail author
Part of the Springer Theses book series (Springer Theses)


In the last chapter we looked at fractals, where the spatial dimension is itself not an integer. We looked at the construction of a topological model on such a system and found that one finds a fractal spectrum where the eigenenergies are self similar. Here we construct a system which is otherwise translationally invariant, but has a fractal spectrum. We then investigate—what happens to this system if we remove bonds randomly?


  1. 1.
    Anderson PW (1958) Absence of diffusion in certain random lattices. Phys Rev 109:1492–1505CrossRefADSGoogle Scholar
  2. 2.
    Lee PA, Ramakrishnan TV (1985) Disordered electronic systems. Rev Mod Phys 57:287–337CrossRefGoogle Scholar
  3. 3.
    Kramer B, MacKinnon A (1993) Localization: theory and experiment. Rep Prog Phys 56(12):1469CrossRefADSGoogle Scholar
  4. 4.
    Janssen M (1998) Statistics and scaling in disordered mesoscopic electron systems. Phys Rep 295(12):1–91CrossRefADSGoogle Scholar
  5. 5.
    Evers F, Mirlin AD (2008) Anderson transitions. Rev Mod Phys 80:1355–1417CrossRefADSGoogle Scholar
  6. 6.
    Abrahams E, Kravchenko SV, Sarachik MP (2001) Metallic behavior and related phenomena in two dimensions. Rev Mod Phys 73:251–266CrossRefADSGoogle Scholar
  7. 7.
    Klitzing KV, Dorda G, Pepper M (1980) New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Phys Rev Lett 45:494–497CrossRefADSGoogle Scholar
  8. 8.
    Stormer HL, Tsui DC, Gossard AC (1999) The fractional quantum hall effect. Rev Mod Phys 71:S298–S305MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hofstadter DR (1976) Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields. Phys Rev B 14:2239–2249CrossRefADSGoogle Scholar
  10. 10.
    Aidelsburger M, Atala M, Lohse M, Barreiro JT, Paredes B, Bloch I (2013) Realization of the hofstadter hamiltonian with ultracold atoms in optical lattices. Phys Rev Lett 111:185301CrossRefADSGoogle Scholar
  11. 11.
    Miyake H, Siviloglou GA, Kennedy CJ, Burton WC, Ketterle W (2013) Realizing the harper hamiltonian with laser-assisted tunneling in optical lattices. Phys Rev Lett 111:185302CrossRefADSGoogle Scholar
  12. 12.
    Hunt B, Sanchez-Yamagishi JD, Young AF, Yankowitz M, LeRoy BJ, Watanabe K, Taniguchi T, Moon P, Koshino M, Jarillo-Herrero P et al (2013) Massive dirac fermions and Hofstadter butterfly in a van der Waals heterostructure. Science 340(6139):1427–1430CrossRefADSGoogle Scholar
  13. 13.
    Yu GL, Gorbachev RV, Tu JS, Kretinin AV, Cao Y, Jalil R, Withers F, Ponomarenko LA, Piot BA, Potemski M et al (2014) Hierarchy of Hofstadter states and replica quantum hall ferromagnetism in graphene superlattices. Nat Phys 10:525–529CrossRefGoogle Scholar
  14. 14.
    Thouless DJ, Kohmoto M, Nightingale MP, den Nijs M (1982) Quantized hall conductance in a two-dimensional periodic potential. Phys Rev Lett 49:405–408CrossRefADSGoogle Scholar
  15. 15.
    Osadchy D, Avron JE (2001) Hofstadter butterfly as quantum phase diagram. J Math Phys 42(12):5665–5671MathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Chalker JT, Coddington PD (1988) Percolation, quantum tunnelling and the integer hall effect. J Phys C: Solid State Phys 21(14):2665CrossRefADSGoogle Scholar
  17. 17.
    Cain P, Römer RA, Schreiber M, Raikh ME (2001) Integer quantum hall transition in the presence of a long-range-correlated quenched disorder. Phys Rev B 64:235326CrossRefADSGoogle Scholar
  18. 18.
    Galstyan AG, Raikh ME (1997) Localization and conductance fluctuations in the integer quantum hall effect: real-space renormalization-group approach. Phys Rev B 56:1422–1429CrossRefADSGoogle Scholar
  19. 19.
    Kramer B, Ohtsuki T, Kettemann S (2005) Random network models and quantum phase transitions in two dimensions. Phys Rep 417(56):211–342MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    Huckestein B (1995) Scaling theory of the integer quantum hall effect. Rev Mod Phys 67:357–396CrossRefADSGoogle Scholar
  21. 21.
    Ortuño M, Somoza AM, Mkhitaryan VV, Raikh ME (2011) Phase diagram of the weak-magnetic-field quantum hall transition quantified from classical percolation. Phys Rev B 84:165314CrossRefADSGoogle Scholar
  22. 22.
    Dolgopolov VT (2014) Integer quantum hall effect and related phenomena. Phys-Uspekhi 57(2):105CrossRefADSGoogle Scholar
  23. 23.
    Abrahams E, Anderson PW, Licciardello DC, Ramakrishnan TV (1979) Scaling theory of localization: absence of quantum diffusion in two dimensions. Phys Rev Lett 42:673–676CrossRefADSGoogle Scholar
  24. 24.
    Khmelnitskii D (1984) Quantum hall effect and additional oscillations of conductivity in weak magnetic fields. Phys Lett A 106(4):182–183CrossRefADSGoogle Scholar
  25. 25.
    Laughlin RB (1984) Levitation of extended-state bands in a strong magnetic field. Phys Rev Lett 52:2304–2304CrossRefADSGoogle Scholar
  26. 26.
    Yang K, Bhatt RN (1996) Floating of extended states and localization transition in a weak magnetic field. Phys Rev Lett 76:1316–1319CrossRefADSGoogle Scholar
  27. 27.
    Sheng DN, Weng ZY (1997) Disappearance of integer quantum hall effect. Phys Rev Lett 78:318–321CrossRefADSGoogle Scholar
  28. 28.
    Pruisken AMM (1985) Dilute instanton gas as the precursor to the integral quantum hall effect. Phys Rev B 32:2636–2639CrossRefADSGoogle Scholar
  29. 29.
    Sheng DN, Weng ZY (1998) New universality of the metal-insulator transition in an integer quantum hall effect system. Phys Rev Lett 80:580–583CrossRefADSGoogle Scholar
  30. 30.
    Sheng DN, Weng ZY (2000) Phase diagram of the integer quantum hall effect. Phys Rev B 62:15363–15366CrossRefADSGoogle Scholar
  31. 31.
    Kirkpatrick S (1973) Percolation and conduction. Rev Mod Phys 45:574–588CrossRefGoogle Scholar
  32. 32.
    Mookerjee A, Saha-Dasgupta T, Dasgupta I (2009) Quantum transmittance through random media. In: Quantum and semi-classical percolation and breakdown in disordered solids, vol 762. Springer, Berlin, p 83Google Scholar
  33. 33.
    Koslowski T, von Niessen W (1990) Mobility edges for the quantum percolation problem in two and three dimensions. Phys Rev B 42:10342–10347CrossRefADSGoogle Scholar
  34. 34.
    Islam MF, Nakanishi H (2008) Localization-delocalization transition in a two-dimensional quantum percolation model. Phys Rev E 77:061109CrossRefADSGoogle Scholar
  35. 35.
    Gong L, Tong P (2009) Localization-delocalization transitions in a two-dimensional quantum percolation model: von Neumann entropy studies. Phys Rev B 80:174205CrossRefADSGoogle Scholar
  36. 36.
    Dillon SB, Nakanishi H (2014) Localization phase diagram of two-dimensional quantum percolation. Eur Phys J B 87(12):1–9CrossRefGoogle Scholar
  37. 37.
    Stauffer D, Aharony A (1991) Introduction to percolation theory. Taylor and Francis, LondonzbMATHGoogle Scholar
  38. 38.
    Isichenko MB (1992) Percolation, statistical topography, and transport in random media. Rev Mod Phys 64:961–1043MathSciNetCrossRefADSGoogle Scholar
  39. 39.
    Soukoulis CM, Grest GS (1991) Localization in two-dimensional quantum percolation. Phys Rev B 44:4685–4688CrossRefADSGoogle Scholar
  40. 40.
    Odagaki T, Lax M, Puri A (1983) Hopping conduction in the \(d\)-dimensional lattice bond-percolation problem. Phys Rev B 28:2755–2765CrossRefGoogle Scholar
  41. 41.
    Raghavan R, Mattis DC (1981) Eigenfunction localization in dilute lattices of various dimensionalities. Phys Rev B 23:4791–4793CrossRefADSGoogle Scholar
  42. 42.
    Shapir Y, Aharony A, Harris AB (1982) Localization and quantum percolation. Phys Rev Lett 49:486–489MathSciNetCrossRefADSGoogle Scholar
  43. 43.
    Taylor JPG, MacKinnon A (1989) A study of the two-dimensional bond quantum percolation model. J Phys: Condens Matter 1(49):9963ADSGoogle Scholar
  44. 44.
    Schmidtke D, Khodja A, Gemmer J (2014) Transport in tight-binding bond percolation models. Phys Rev E 90:032127CrossRefADSGoogle Scholar
  45. 45.
    Sanyal S, Damle K, Motrunich OI (2016) Vacancy-induced low-energy states in undoped graphene. Phys Rev Lett 117:116806CrossRefADSGoogle Scholar
  46. 46.
    Häfner V, Schindler J, Weik N, Mayer T, Balakrishnan S, Narayanan R, Bera S, Evers F (2014) Density of states in graphene with vacancies: Midgap power law and frozen multifractality. Phys Rev Lett 113:186802CrossRefADSGoogle Scholar
  47. 47.
    Ostrovsky PM, Protopopov IV, König EJ, Gornyi IV, Mirlin AD, Skvortsov MA (2014) Density of states in a two-dimensional chiral metal with vacancies. Phys Rev Lett 113:186803CrossRefADSGoogle Scholar
  48. 48.
    Zhu L, Wang X (2016) Singularity of density of states induced by random bond disorder in graphene. Phys Lett A 380:2233–2236CrossRefADSGoogle Scholar
  49. 49.
    Liu W-S, Lei X (2003) Integer quantum hall transitions in the presence of off-diagonal disorder. J Phys: Condens Matter 15(17):2693ADSGoogle Scholar
  50. 50.
    Meir Y, Aharony A, Harris AB (1986) Quantum percolation in magnetic fields. Phys Rev Lett 56:976–979CrossRefADSGoogle Scholar
  51. 51.
    Yi-Fu Z, Yun-You Y, Yan J, Li S, Rui S, Dong-Ning S, Ding-Yu X (2013) Coupling-matrix approach to the Chern number calculation in disordered systems. Chin Phys B 22(11):117312CrossRefADSGoogle Scholar
  52. 52.
    Analytis JG, Blundell SJ, Ardavan A (2004) Landau levels, molecular orbitals, and the Hofstadter butterfly in finite systems. Am J Phys 72(5):613–618CrossRefADSGoogle Scholar
  53. 53.
    Weik N, Schindler J, Bera S, Solomon GC, Evers F (2016). Graphene with vacancies: supernumerary zero modes. ArXiv e-prints, arXiv:1603.00212
  54. 54.
    Markoš P (2006) Numerical analysis of the anderson localization. Acta Phys Slovaca 56:561–685Google Scholar
  55. 55.
    Fradkin E (1991) Field theories of condensed matter systems, vol 7. Addison-Wesley, Redwood CityzbMATHGoogle Scholar
  56. 56.
    Niu Q, Thouless DJ, Wu Y-S (1985) Quantized hall conductance as a topological invariant. Phys Rev B 31:3372–3377MathSciNetCrossRefADSGoogle Scholar
  57. 57.
    Dutta P, Maiti SK, Karmakar SN (2012) Integer quantum hall effect in a lattice model revisited: Kubo formalism. J Appl Phys 112(4):044306CrossRefADSGoogle Scholar
  58. 58.
    Analytis JG, Blundell SJ, Ardavan A (2005) Magnetic oscillations, disorder and the Hofstadter butterfly in finite systems. Synth Metals 154(13):265–268. Proceedings of the international conference on science and technology of synthetic metals Part IIICrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.International Centre for Theoretical SciencesTata Institute of Fundamental ResearchBangaloreIndia

Personalised recommendations