Integral Quadratic Forms, Kac-Moody Algebras, and Fractional Quantum Hall Effect. An ADE- 𝒪 Classification

Jürg Fröhlich; Emmanuel Thiran

Recherche Coopérative sur Programme n°25 (1993)

  • Volume: 45, Issue: 1-2, page 71-149

How to cite

top

Fröhlich, Jürg, and Thiran, Emmanuel. "Integral Quadratic Forms, Kac-Moody Algebras, and Fractional Quantum Hall Effect. An ADE-$\mathcal {O}$ Classification." Recherche Coopérative sur Programme n°25 45.1-2 (1993): 71-149. <http://eudml.org/doc/274910>.

@article{Fröhlich1993,
author = {Fröhlich, Jürg, Thiran, Emmanuel},
journal = {Recherche Coopérative sur Programme n°25},
keywords = {abelian Chern-Simons theory; integral lattices; low-energy excitations; incompressible quantum Hall fluid},
language = {eng},
number = {1-2},
pages = {71-149},
publisher = {Institut de Recherche Mathématique Avancée - Université Louis Pasteur},
title = {Integral Quadratic Forms, Kac-Moody Algebras, and Fractional Quantum Hall Effect. An ADE-$\mathcal \{O\}$ Classification},
url = {http://eudml.org/doc/274910},
volume = {45},
year = {1993},
}

TY - JOUR
AU - Fröhlich, Jürg
AU - Thiran, Emmanuel
TI - Integral Quadratic Forms, Kac-Moody Algebras, and Fractional Quantum Hall Effect. An ADE-$\mathcal {O}$ Classification
JO - Recherche Coopérative sur Programme n°25
PY - 1993
PB - Institut de Recherche Mathématique Avancée - Université Louis Pasteur
VL - 45
IS - 1-2
SP - 71
EP - 149
LA - eng
KW - abelian Chern-Simons theory; integral lattices; low-energy excitations; incompressible quantum Hall fluid
UR - http://eudml.org/doc/274910
ER -

References

top
  1. [1] K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett.45, 494 (1980). 
  2. [2] D. C. Tsui, H. L. Stormer and A. C. Gossard, Phys. Rev. B 48, 1559 (1982). 
  3. [3] R. Tao and Y.-S. Wu, Phys. Rev. B 31, 6859 (1985). MR789945
  4. [4] R. L. Willet, J. P. Eisenstein, H. L. Stormer, D. C. Tsui, A. C. Gossard and J. H. English, Phys. Rev. Lett.59, 1776 (1987) 
  5. J. P. Eisenstein, R. L. Willet, H. L. Stormer, D. C. Tsui, A. C. Gossard and J. H. English, Phys. Rev. Lett.61, 997 (1988); and references therein 
  6. J. P. Eisenstein, R. L. Willet, H. L. Stormer, L. N. Pfeiffer and K. W. West, Surf. Science229, 31 (1990). 
  7. [5] Y. W. Suen, L. W. Engel, M. B. Santos, M. Shayegan and D. C. Tsui, Phys. Rev. Lett.68, 1379 (1992); 
  8. J. P. Eisenstein, G. S. Boeblinger, L. N. Pfeiffer, K. W. West and Song He, Phys. Rev. Lett.68, 1383 (1992). 
  9. [6] R. G. Clark, J. R. Mallet, S. R. Haynes, J. J. Harris and C. T. Foxon, Phys. Rev. Lett.60, 1747 (1988); 
  10. A. M. Chang and J. E. Cunningham, Solid State Commun.72, 652 (1989); 
  11. R. G. Clarket al., cited in [8] for σ H = 2 / 3 study 
  12. [7] J. A. Simmons, H. P. Wei, L. W. Engel, D. C. Tsui and M. Shayegan, Phys. Rev. Lett.63, 1731 (1989); 
  13. S. W. Hwang, J. A. Simmons, D. C. Tsui and M. Shayegan, Surf. Science263, 72 (1992). 
  14. [8] Experimental results for tilted magnetic field transition at σ H = 8 5 and σ H = 5 3 : J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer and K. W. West, Phys. Rev. Lett.62, 1540 (1989); 
  15. Experimental results for tilted magnetic field transition at σ H = 8 5 and σ H = 5 3 : J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer and K. W. WestSurf. Science229, 21 (1990) 
  16. at σ H = 4 3 : R. G. Clark, S. R. Haynes, A. M. Suckling, J. R. Mallet, P. A. Wright, J. J. Harris and C. T. Foxon, Phys. Rev. Lett.62, 1536 (1989) 
  17. at σ H = 2 3 : J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer and K. W. West, Phys. Rev. B 41, 7910 (1990); 
  18. R. G. Clark, S. R. Haynes, J. V. Branch, A. M. Suckling, P. A. Wright, P. M. W. Oswald, J. J. Harris and C. T. Foxon, Surf. Science229, 25 (1990) 
  19. at σ H = 5 2 : see [4]; See also D. A. Syphers and J. E. Furneaux, Surf. Science196, 252 (1988); Solid State Commun.65, 1513 (1988); 
  20. J. Haug, K. von Klitzing, R. J. Nicholas, J. C. Maan and G. Weimann, Phys. Rev. B 36, 4528 (1987). 
  21. [9] For reviews of the quantum Hall effect and comprehensive compilations of references, see e.g. R. E. Prange and S. M. Girvin, eds., "The Quantum Hall Effect", Second Edition, Graduate Texts in Contemporary Physics (Springer, New York, 1990); 
  22. G. Morandi, "Quantum Hall Effect" (Bibliopolis, Napoli, 1988); Zbl0708.53058MR1089877
  23. T. Chakraborty and P. Pietiläinen, "The Fractional Quantum Hall Effect: Properties of an Incompressible Quantum Fluid", Springer Series in Solid State Science85 (Springer, Berlin, 1988); 
  24. M. Stone, ed., "Quantum Hall Effect" (World Scientific, Singapore, 1992). See also: Zbl0875.82006MR1238757
  25. F. Wilczeck, "Fractional Statistics and Anyon Superconductivity" (World Seientific, Singapore, 1990). 
  26. [10] For IQH: R. B. Laughlin, Phys. Rev. B 23, 5632 (1981); 
  27. for FQH: R. B. Laughlin, Phys. Rev. Lett.50, 1395 (1983); 
  28. R. B. LaughlinPhys. Rev. B 27, 3383 (1983); 
  29. R. B. LaughlinSurf. Science141, 11 (1984); see also [28]. 
  30. X. G. Wen, Phys. Rev. B 40, 7387 (1989); 
  31. X. G. WenPhys. Rev. Lett.64, 2206 (1990); 
  32. X. G. WenPhys. Rev. B 41, 12838 (1990); 
  33. X. G. WenPhys. Rev. B 43, 11025 (1991); 
  34. X. G. WenPhys. Rev. Lett.66, 802 (1991); Zbl0968.81553
  35. X. G. WenInt. J. Mod. Phys. B 6, 1711 (1992); 
  36. X. G. Wen and Q. Niu, Phys. Rev., B 41, 9377 (1990); 
  37. B. Block and X. G. Wen, Phys. Rev. B 42, 8133, 8145 (1990); 
  38. X. G. Wen and A. Zee, Phys. Rev. B 46, 2290 (1992). 
  39. [12] J. Fröhlich and T. Kerler, Nucl. Phys. B 354, 369 (1991); 
  40. J. Fröhlich and A. Zee, Nucl. Phys. B 364, 517 (1991); 
  41. J. Fröhlich and U. M. Studer, Commun. Math. Phys.148, 553 (1992); Zbl0758.53048
  42. J. Fröhlich and U. M. StuderInt. J. Mod. Phys. B6, 2201 (1992); "Incompressible Quantum Fluids, Gauge Invariance and Current Algebra", in J. Fröhlich et al., eds., New Symmetry Principles in Quantum Field Theory, Cargèse Lectures 1991 (Plenum PressNew York, 1992); Zbl1221.81220
  43. J. Fröhlich and U. M. Studer "Gauge Invariance and Current Algebra in Non-Relativistic Many-Body Theory", Rev. Mod. Phys.65, 733 (1993). Zbl0947.81570
  44. [13] M. Stone, Int. J. Mod. Phys. B5, 509 (1991); 
  45. M. StoneAnn. Phys. (New York) 207, 38 (1991). 
  46. A. V. Balatsky, Phys. Rev. B 43, 1257 (1991). 
  47. [14] See J. Fröhlich and T. Kerler, cited in [12]. 
  48. [15] J. H. Conway, N. J. A. Sloane, "Sphere Packings, Lattices and Groups" (Springer-Verlag, New York, 1988); Zbl0634.52002
  49. J. H. Conway, F. R. S. Sloane and N. J. A. Sloane, Proc. R. Soc. Lond. A418, 17 (1988); ibid, 419, 29 (1988); ibid 418, 43 (1988); ibid 419, 259 (1988). 
  50. [16] J. W. S. Cassels, "Rational Quadratic Forms", (Academic Press, 1978). Zbl0395.10029
  51. [17] E. Fradkin, "Field theories of Condensed Matter Systems", Frontiers in Physics, Vol. 82 (Addison-Wesley, Redwood City, 1991). Zbl0984.82504
  52. [18] J. Fröhlich, U. M. Studer, cited in [12]. 
  53. [19] R. E. Prange and S. M. Girvin, eds. "The Quantum Hall Effect", cited in [9]. Zbl1140.81024
  54. [20] M. Stone, ed., "Quantum Hall Effect", cited in [9]. Zbl0875.82006
  55. [21] B. I. Halperin, Phys. Rev. B 25, 2185 (1982). 
  56. [22] P. Goddard and D. Olive, Int. J. Mod. Phys. A1, 303 (1986). 
  57. [23] P. Ginsparg, "Applied Conformai Field Theory" in E. Brézin and J. Zinn-Justin, eds., "Fields, Strings and Critical Phenomena", Les Houches, Session XLIX, 1988 
  58. P. Ginsparg (North Holland, Elsevier Science Publishers B. V., 1990). 
  59. [24] E. Witten, Commun. Math. Phys.121, 351 (1989). Zbl0667.57005
  60. [25] J. Fröhlich and C. King, Commun. Math. Phys.126, 167 (1989); Zbl0681.57019
  61. J. Fröhlich and C. KingInt. J. Mod. Phys. A4, 5321 (1989). 
  62. [26] S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, Nucl. Phys. B 326, 108 (1989). 
  63. [27] J. Fröhlich and T. Kerler (1991), cited in [12]; J. Fröhlich and Z. Zee (1991), ibid. 
  64. [28] R. B. Laughlin, Chap. 7 in R. E. Prange and S. M. Girvin, eds., cited in [9]; F. D. M. Haldane, Chap. 8, ibid. 
  65. [29] K. Gawedzki, in G. 't Hooft et al., eds., "Non perturbative Quantum Field Theory" (Plenum, New York, 1988). 
  66. [30] X. G. Wen, Int. J. Mod. Phys., B 4, 239 (1990). 
  67. [31] G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991); 
  68. G. Moore and N. ReadProg. Theor. Phys. Suppl.107, 157 (1992). 
  69. [32] X. G. Wen and A. Zee, Phys. Rev. Lett.69, 953 (1992). 
  70. [33] N. Read, Phys. Rev. Lett.65, 1502 (1990). 
  71. [34] V. V. Nikulin, Math. USSR-Izv.14 (1979), n° 1, 103-167 (1980). Zbl0427.10014
  72. [35] J. Fröhlich and P. A. Marchetti, Lett. Math. Phys. 16, 347 (1988); Zbl0695.58031
  73. J. Fröhlich and P. A. MarchettiCommun. Math. Phys.121, 177 (1989). Zbl0819.58045
  74. [36] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Phys. B 241, 333 (1984). Zbl0661.17013
  75. [37] J. Fröhlich, U. M. Studer and E. Thiran, "Gauge Symmetry, Integral Lattices and the Classification of Quantum Hall fluids", preprint KUL-TF-93/33; "An ADE-O Classification of Minimal Incompressible Quantum Hall fluids", to appear in the proceedings of the conference "On Three Levels", Leuven (Belgium), 1993, A. Verbeure et al., eds, Plenum. 
  76. [38] V. Kac, M. Wakimoto, Adv. Math.70, 156 (1988); Zbl0661.17016
  77. V. Kac, N. Sanielevici, Phys. Rev. D37, 2231 (1988); 
  78. M. A. Walton, Nucl. Phys. B322, 775 (1989); 
  79. D. Altschuler, M. Bauer and C. Itzykson, Commun. Math. Phys.132, 349 (1990); Zbl0709.17019
  80. D. Versiegen, Commun. Math. Phys.137, 567 (1991); for the classification of Conformal Embeddings see [43]. 
  81. [39] A. H. Chamseddine and J. Fröhlich, Commun. Math. Phys.147, 549 (1992). Zbl0755.53049
  82. [40] J. Fröhlich, T. Kerler and E. Thiran, in preparation. 
  83. [41] B. Block and X. G. Wen, Phys. Rev. B42, 8145 (1990). 
  84. [42] R. Slansky, Phys. Rep.79, 1 (1981). 
  85. [43] F. A. Bais and P. Bouwknegt, Nucl. Phys. B279, 561 (1987); 
  86. A. N. Schellekens and N. P. Warner, Phys. Rev. D34, 3092 (1986). 
  87. [44] P. Engel, L. Michel and M. Senechal, Lattice Geometry (1993), in preparation. 
  88. [45] F. D. M. Haldane, Phys. Rev. Lett.51, 605 (1983); 
  89. B. I. Halperin, Phys. Rev. Lett.51, 1583 (1983); 
  90. for a review see F. D. M. Haldane, cited in [28]; see also [33]; 
  91. J. K. Jain and V. J. Goldman, Phys. Rev. B45, 1255 (1992); 
  92. see also Y. J. Chen, Phys. Rev. B46, 7941 (1992). 
  93. [46] R. C. Aschoori et al., Phys. Rev. B 45, 3894 (1992). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.