Abstract
The purpose of this paper is to present a rather comprehensive classification of incompressible quantum Hall states in the limit of large distance scales and low frequencies. In this limit, the description of low-energy excitations above the groundstate of an incompressible quantum Hall fluid is intimately connected to the theory of integral quadratic forms on certain lattices which we call quantum Hall lattices. This connection is understood with the help of the representation theory of algebras of gapless, chiral edge currents or, alternatively, from the point of view of the bulk effective Chern-Simons theory. Our main results concern the classification of quantum Hall lattices in terms of certain invariants and their enumeration in low dimensions and for a limited range of values of those invariants. Among physical consequences of our analysis we find explicit, discrete sets of plateau values of the Hall conductivity, as well as the quantum numbers of quasiparticles in fluids corresponding to any one among those quantum Hall lattices. Furthermore, we are able to predict transitions between structurally different quantum Hall fluids corresponding to the same filling factor. Our general results are illustrated by explicitly considering the following plateau values: σ H =N/(2N±1),N=1, 2, 3,..., σ N =1/2.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K. von Klitzing, G. Dorda, and M. Pepper,Phys. Rev. Lett. 45:494 (1980).
D. C. Tsui, H. L. Stormer, and A. C. Gossard,Phys. Rev. B 48:1559 (1982).
R. Tao and Y.-S. Wu,Phys. Rev. B 31:6859 (1985).
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. Sci. 229:31 (1990).
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).
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).
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. Sci. 263:72 (1992).
J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer, and K. W. West,Phys. Rev. Lett. 62:1540 (1989);Surf. Sci. 229:21 (1990).
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).
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. Sci. 229:25 (1990).
D. A. Syphers and J. E. Furneaux,Surf. Sci. 196:252 (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).
R. E. Prange and S. M. Girvin, eds.,The Quantum Hall Effect, 2nd ed. (Springer, New York, 1990).
G. Morandi,Quantum Hall Effect (Bibliopolis, Naples, 1988).
T. Chakraborty and P. Pietiläinen,The Fractional Quantum Hall Effect: Properties of an Incompressible Quantum Fluid (Springer, Berlin, 1988).
M. Stone, ed.,Quantum Hall Effect (World Scientific, Singapore, 1992); see also F. Wilczeck,Fractional Statistics and Anyon Superconductivity (World Scientific, Singapore, 1990).
R. B. Laughlin,Phys. Rev. B 23:5632 (1981).
R. B. Laughlin,Phys. Rev. Lett. 50:1395 (1983);Phys. Rev. B 27:3383 (1983);Surf. Sci. 141:11 (1984).
X. G. Wen,Phys. Rev. B 40:7387 (1989);Phys. Rev. Lett. 64:2206 (1990);Phys. Rev. B 41:12838 (1990);Phys. Rev. B 43:11025 (1991);Phys. Rev. Lett. 66:802 (1991);Int. 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).
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);Int. J. Mod. Phys. B 6:2201 (1992); Incompressible quantum fluids, gauge invariance and current algebra, in J. Fröhlichet al., eds.,New Symmetry Principles in Quantum Field Theory, Cargèse Lectures 1991 (Plenum Press, New York, 1992); Gauge invariance and current algebra in non-relativistic many-body theory,Rev. Mod. Phys. 65:733 (1993).
M. Stone,Int. J. Mod. Phys. B 5:509 (1991);Ann. Phys. (NY)207:38 (1991); A. V. Balatsky,Phys. Rev. B 43:1257 (1991).
J. H. Conway and N. J. A. Sloane,Spere Packings, Lattices and Groups (Springer-Verlag, New York, 1988); J. H. Conway and N. J. A. Sloane,Proc. R. Soc. Lond. A 418:17, (1988);419:29, 259 (1988).
J. W. S. Cassels,Rational Quadratic Forms (Academic Press, 1978).
E. Fradkin,Field Theories of Condensed Matter Systems (Addison-Wesley, Redwood City, California, 1991).
B. I. Halperin,Phys. Rev. B 25:2185 (1982).
P. Goddard and D. Olive,Int. J. Mod. Phys. A 1:303 (1986).
P. Ginsparg, Applied conformal field theory, in E. Brézin and J. Zinn-Justin, eds.,Fields, Strings and Critical Phenomena (North-Holland/Elsevier, Amsterdam, 1990).
E. Witten,Commun. Math. Phys. 121:351 (1989).
J. Fröhlich and C. King,Commun. Math. Phys. 126:167 (1989);Int. J. Mod. Phys. A 4:5321 (1989).
S. Elitzur, G. Moore, A. Schwimmer, and N. Seiberg,Nucl. Phys. B 326:108 (1989).
R. B. Laughlin, inThe Quantum Hall Effect, 2nd ed. R. E. Prange and S. M. Girvin, eds. (Springer, New York, 1990); Chapter 7.
F. D. Haldane, inThe Quantum Hall Effect, 2nd ed., R. E. Prange and S. M. Girvin, eds. (Springer, New York, 1990), Chapter 8.
K. Gawędzki, inNon perturbative Quantum Field Theory G. 't Hooft et al., eds. (Plenum, New York, 1988).
X. G. Wen,Int. J. Mod. Phys. B 4:239 (1990).
G. Moore and N. Read,Nucl. Phys. B 360:362 (1991);Prog. Theor. Phys. Suppl. 107:157 (1992).
X. G. Wen and A. Zee,Phys. Rev. Lett. 69:953 (1992).
N. Read,Phys. Rev. Lett. 65:1502 (1990).
V. V. Nikulin,Math. USSR-Izv. 14(1979):103–167 (1980).
J. Fröhlich and P. A. Marchetti,Lett. Math. Phys. 16:347 (1988);Commun. Math. Phys. 121:177 (1989).
A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov,Nucl. Phys. B 241:333 (1984).
J. Fröhlich, U. M. Studer, and E. Thiran, “Classification of quantum Hall fluids,” in preparation; AnO classification of minimal incompressible quantum Hall fluids, inProceedings of the conference “On Three Levels,” Leuven (Belgium), 1993, M. Fannes et al., eds. (Plenum Press, New York, 1994).
V. Kac and M. Wakimoto,Adv. Math. 70:156 (1988); V. Kac and N. Sanielevici,Phys. Rev. D 37:2231 (1988); M. A. Walton,Nucl. Phys. B 322:775 (1989); D. Altschuler, M. Bauer, and C. Itzykson,Commun. Math. Phys. 132:349 (1990); D. Verstegen,Commun. Math. Phys. 137:567 (1991).
A. H. Chamseddine and J. Fröhlich,Commun. Math. Phys. 147:549 (1992).
J. Fröhlich, T. Kerler, and E. Thiran, in preparation.
B. Block and X. G. Wen,Phys. Rev. B 42:8145 (1990).
R. Slansky,Phys. Rep. 79:1 (1981).
F. A. Bais and P. Bouwknegt,Nucl. Phys. B 279:561 (1987); A. N. Schellekens and N. P. Warner,Phys. Rev. D 34:3092 (1986).
P. Engel, L. Michel, and M. Senechal, Lattice geometry (1993), in preparation.
F. D. M. Haldane,Phys. Rev. Lett. 51:605 (1983); B. I. Halperin,Phys. Rev. Lett. 51:1583 (1983); J. K. Jain and V. J. Goldman,Phys. Rev. B 45:1255 (1992); see alsoY. J. Chen,Phys. Rev. B 46:7941 (1992).
R. C. Aschoori et al.,Phys. Rev. B 45:3894 (1992).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fröhlich, J., Thiran, E. Integral quadratic forms, Kac-Moody algebras, and fractional quantum Hall effect. AnADE-O classification. J Stat Phys 76, 209–283 (1994). https://doi.org/10.1007/BF02188661
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02188661