Integral Quadratic Forms, Kac-Moody Algebras, and Fractional Quantum Hall Effect. An ADE- Classification
Jürg Fröhlich; Emmanuel Thiran
Recherche Coopérative sur Programme n°25 (1993)
- Volume: 45, Issue: 1-2, page 71-149
Access Full Article
topHow to cite
topFröhlich, 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] K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett.45, 494 (1980).
- [2] D. C. Tsui, H. L. Stormer and A. C. Gossard, Phys. Rev. B 48, 1559 (1982).
- [3] R. Tao and Y.-S. Wu, Phys. Rev. B 31, 6859 (1985). MR789945
- [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)
- 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
- J. P. Eisenstein, R. L. Willet, H. L. Stormer, L. N. Pfeiffer and K. W. West, Surf. Science229, 31 (1990).
- [5] Y. W. Suen, L. W. Engel, M. B. Santos, M. Shayegan and D. C. Tsui, Phys. Rev. Lett.68, 1379 (1992);
- J. P. Eisenstein, G. S. Boeblinger, L. N. Pfeiffer, K. W. West and Song He, Phys. Rev. Lett.68, 1383 (1992).
- [6] R. G. Clark, J. R. Mallet, S. R. Haynes, J. J. Harris and C. T. Foxon, Phys. Rev. Lett.60, 1747 (1988);
- A. M. Chang and J. E. Cunningham, Solid State Commun.72, 652 (1989);
- R. G. Clarket al., cited in [8] for study
- [7] J. A. Simmons, H. P. Wei, L. W. Engel, D. C. Tsui and M. Shayegan, Phys. Rev. Lett.63, 1731 (1989);
- S. W. Hwang, J. A. Simmons, D. C. Tsui and M. Shayegan, Surf. Science263, 72 (1992).
- [8] Experimental results for tilted magnetic field transition at and : J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer and K. W. West, Phys. Rev. Lett.62, 1540 (1989);
- Experimental results for tilted magnetic field transition at and : J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer and K. W. WestSurf. Science229, 21 (1990)
- at : 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)
- at : J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer and K. W. West, Phys. Rev. B 41, 7910 (1990);
- 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)
- at : see [4]; See also D. A. Syphers and J. E. Furneaux, Surf. Science196, 252 (1988); Solid State Commun.65, 1513 (1988);
- J. Haug, K. von Klitzing, R. J. Nicholas, J. C. Maan and G. Weimann, Phys. Rev. B 36, 4528 (1987).
- [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);
- G. Morandi, "Quantum Hall Effect" (Bibliopolis, Napoli, 1988); Zbl0708.53058MR1089877
- 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);
- M. Stone, ed., "Quantum Hall Effect" (World Scientific, Singapore, 1992). See also: Zbl0875.82006MR1238757
- F. Wilczeck, "Fractional Statistics and Anyon Superconductivity" (World Seientific, Singapore, 1990).
- [10] For IQH: R. B. Laughlin, Phys. Rev. B 23, 5632 (1981);
- for FQH: R. B. Laughlin, Phys. Rev. Lett.50, 1395 (1983);
- R. B. LaughlinPhys. Rev. B 27, 3383 (1983);
- R. B. LaughlinSurf. Science141, 11 (1984); see also [28].
- X. G. Wen, Phys. Rev. B 40, 7387 (1989);
- X. G. WenPhys. Rev. Lett.64, 2206 (1990);
- X. G. WenPhys. Rev. B 41, 12838 (1990);
- X. G. WenPhys. Rev. B 43, 11025 (1991);
- X. G. WenPhys. Rev. Lett.66, 802 (1991); Zbl0968.81553
- X. G. WenInt. J. Mod. Phys. B 6, 1711 (1992);
- X. G. Wen and Q. Niu, Phys. Rev., B 41, 9377 (1990);
- B. Block and X. G. Wen, Phys. Rev. B 42, 8133, 8145 (1990);
- X. G. Wen and A. Zee, Phys. Rev. B 46, 2290 (1992).
- [12] J. Fröhlich and T. Kerler, Nucl. Phys. B 354, 369 (1991);
- J. Fröhlich and A. Zee, Nucl. Phys. B 364, 517 (1991);
- J. Fröhlich and U. M. Studer, Commun. Math. Phys.148, 553 (1992); Zbl0758.53048
- 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
- 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
- [13] M. Stone, Int. J. Mod. Phys. B5, 509 (1991);
- M. StoneAnn. Phys. (New York) 207, 38 (1991).
- A. V. Balatsky, Phys. Rev. B 43, 1257 (1991).
- [14] See J. Fröhlich and T. Kerler, cited in [12].
- [15] J. H. Conway, N. J. A. Sloane, "Sphere Packings, Lattices and Groups" (Springer-Verlag, New York, 1988); Zbl0634.52002
- 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).
- [16] J. W. S. Cassels, "Rational Quadratic Forms", (Academic Press, 1978). Zbl0395.10029
- [17] E. Fradkin, "Field theories of Condensed Matter Systems", Frontiers in Physics, Vol. 82 (Addison-Wesley, Redwood City, 1991). Zbl0984.82504
- [18] J. Fröhlich, U. M. Studer, cited in [12].
- [19] R. E. Prange and S. M. Girvin, eds. "The Quantum Hall Effect", cited in [9]. Zbl1140.81024
- [20] M. Stone, ed., "Quantum Hall Effect", cited in [9]. Zbl0875.82006
- [21] B. I. Halperin, Phys. Rev. B 25, 2185 (1982).
- [22] P. Goddard and D. Olive, Int. J. Mod. Phys. A1, 303 (1986).
- [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
- P. Ginsparg (North Holland, Elsevier Science Publishers B. V., 1990).
- [24] E. Witten, Commun. Math. Phys.121, 351 (1989). Zbl0667.57005
- [25] J. Fröhlich and C. King, Commun. Math. Phys.126, 167 (1989); Zbl0681.57019
- J. Fröhlich and C. KingInt. J. Mod. Phys. A4, 5321 (1989).
- [26] S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, Nucl. Phys. B 326, 108 (1989).
- [27] J. Fröhlich and T. Kerler (1991), cited in [12]; J. Fröhlich and Z. Zee (1991), ibid.
- [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.
- [29] K. Gawedzki, in G. 't Hooft et al., eds., "Non perturbative Quantum Field Theory" (Plenum, New York, 1988).
- [30] X. G. Wen, Int. J. Mod. Phys., B 4, 239 (1990).
- [31] G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991);
- G. Moore and N. ReadProg. Theor. Phys. Suppl.107, 157 (1992).
- [32] X. G. Wen and A. Zee, Phys. Rev. Lett.69, 953 (1992).
- [33] N. Read, Phys. Rev. Lett.65, 1502 (1990).
- [34] V. V. Nikulin, Math. USSR-Izv.14 (1979), n° 1, 103-167 (1980). Zbl0427.10014
- [35] J. Fröhlich and P. A. Marchetti, Lett. Math. Phys. 16, 347 (1988); Zbl0695.58031
- J. Fröhlich and P. A. MarchettiCommun. Math. Phys.121, 177 (1989). Zbl0819.58045
- [36] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Phys. B 241, 333 (1984). Zbl0661.17013
- [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.
- [38] V. Kac, M. Wakimoto, Adv. Math.70, 156 (1988); Zbl0661.17016
- V. Kac, N. Sanielevici, Phys. Rev. D37, 2231 (1988);
- M. A. Walton, Nucl. Phys. B322, 775 (1989);
- D. Altschuler, M. Bauer and C. Itzykson, Commun. Math. Phys.132, 349 (1990); Zbl0709.17019
- D. Versiegen, Commun. Math. Phys.137, 567 (1991); for the classification of Conformal Embeddings see [43].
- [39] A. H. Chamseddine and J. Fröhlich, Commun. Math. Phys.147, 549 (1992). Zbl0755.53049
- [40] J. Fröhlich, T. Kerler and E. Thiran, in preparation.
- [41] B. Block and X. G. Wen, Phys. Rev. B42, 8145 (1990).
- [42] R. Slansky, Phys. Rep.79, 1 (1981).
- [43] F. A. Bais and P. Bouwknegt, Nucl. Phys. B279, 561 (1987);
- A. N. Schellekens and N. P. Warner, Phys. Rev. D34, 3092 (1986).
- [44] P. Engel, L. Michel and M. Senechal, Lattice Geometry (1993), in preparation.
- [45] F. D. M. Haldane, Phys. Rev. Lett.51, 605 (1983);
- B. I. Halperin, Phys. Rev. Lett.51, 1583 (1983);
- for a review see F. D. M. Haldane, cited in [28]; see also [33];
- J. K. Jain and V. J. Goldman, Phys. Rev. B45, 1255 (1992);
- see also Y. J. Chen, Phys. Rev. B46, 7941 (1992).
- [46] R. C. Aschoori et al., Phys. Rev. B 45, 3894 (1992).
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.