×

zbMATH — the first resource for mathematics

The noncommutative geometry of the quantum Hall effect. (English) Zbl 0824.46086
Alain Connes ([Zbl 0592.46056] and [Zbl 0745.46067]) has succeeded in extending most of the tools of differential geometry to noncommutative \(C^{\ast}\)-algebras, where the principal new products were cyclic cohomology and the proof of an index theorem for elliptic operators on a foliated manifold. The major objective in this paper is to review various contributions to theoretical or mathematical analysis of the quantum Hall effect in a synthetic and detailed manner by using noncommutative geometry.

In the 19th century, E. H. Hall [JFM 11.0767.01] undertook a famous experiment, which is what is now called the Hall effect. About a century later, the integer quantum Hall effect (IQHE) was discovered by von Klitzing and others [Phys. Rev. Lett. 45, 494 (1980)], for which von Klitzing got the Nobel-prize laureateship in 1985. Due to the works by Laughlin [Phys. Rev. B 23, 5632 (1981)] and D. Thouless and others [Phys. Rev. Lett. 49, 405 (1982)], it turned out that the quantization of the Hall conductance at low temperature has a geometric origin, which gave a good explanation for the universality of this effect. On the other hand, thanks to Prange ([Phys. Rev. B 23, 4802 (1981)] and [Phys. Rev. B 29, 3303 (1984)]), Thousless [J. Phys. C 14, 3475 (1981)] and Halperin [Phys. Rev. B 25, 2185 (1982)], it turned out that the plateaux of the Hall conductance observed in changing the magnetic field or the charge-carrier density is due to localization.

How to reconcile universality and localization in a single model had been a challenging problem. Avron and others [Phys. Rev. Lett. 51, 51 (1983)] have succeeded in quantization, only to fail to show that these quantum numbers are insensitive to disorder. Kunz ([Zbl 1108.81314]) has endeavored to establish this for disorder small enough to dodge filling the gaps between Landau levels. The first author of this paper ([Zbl 0612.46066] and [Zbl 0677.46055]) has proposed to use noncommutative geometry to extend the arguments in [Phys. Rev. Lett. 49, 405 (1982)] to the case of arbitrary magnetic field and disordered crystal, making it turn out that the condition under which plateaux occur is precisely the finiteness of the localization length near the Fermi level. The work was rephrased by Avron and others [Zbl 0822.47056] in terms of charge transport and relative index, filling the remaining gaps between experimental observations and theoretical considerations.

The authors of this paper present a point of view on localization produced by quenched disorder, which is crucial for understanding the IQHE. They give a precise condition under which the Hall conductance is quantized, while they propose a model for electronic transport giving rise to what is called the relaxation time approximation and then allowing to derive a Kubo formula for the conductivity. The noncommutative approach enables us to deal with the case of aperiodic crystals and magnetic fields when Bloch theory fails. However, no attempt is made to extend the noncommutative framework to the fractional quantum Hall effect (FQHE).

MSC:
46L87 Noncommutative differential geometry
46L85 Noncommutative topology
46N50 Applications of functional analysis in quantum physics
81S10 Geometry and quantization, symplectic methods
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.2307/2369245 · JFM 11.0767.01 · doi:10.2307/2369245
[2] DOI: 10.1103/PhysRevLett.45.494 · doi:10.1103/PhysRevLett.45.494
[3] Connes A., Publ. IHES 62 pp 257– (1986)
[4] DOI: 10.1103/PhysRevB.23.5632 · doi:10.1103/PhysRevB.23.5632
[5] DOI: 10.1103/PhysRevLett.49.405 · doi:10.1103/PhysRevLett.49.405
[6] DOI: 10.1103/PhysRevB.23.4802 · doi:10.1103/PhysRevB.23.4802
[7] DOI: 10.1103/PhysRevB.29.3303 · doi:10.1103/PhysRevB.29.3303
[8] DOI: 10.1088/0022-3719/14/23/022 · doi:10.1088/0022-3719/14/23/022
[9] DOI: 10.1103/PhysRevLett.51.51 · doi:10.1103/PhysRevLett.51.51
[10] DOI: 10.1007/BF01217683 · Zbl 1108.81314 · doi:10.1007/BF01217683
[11] DOI: 10.1007/BF02102644 · Zbl 0822.47056 · doi:10.1007/BF02102644
[12] DOI: 10.1103/RevModPhys.54.437 · doi:10.1103/RevModPhys.54.437
[13] DOI: 10.1103/PhysRevB.35.8005 · doi:10.1103/PhysRevB.35.8005
[14] DOI: 10.1088/0022-3719/18/26/003 · doi:10.1088/0022-3719/18/26/003
[15] DOI: 10.1103/PhysRevB.33.2965 · doi:10.1103/PhysRevB.33.2965
[16] Gerhardts R.R., J. de Phys. Colloques C 5 pp 227– (1987)
[17] DOI: 10.1007/BF01397213 · doi:10.1007/BF01397213
[18] DOI: 10.1103/PhysRev.134.A1602 · doi:10.1103/PhysRev.134.A1602
[19] DOI: 10.1103/PhysRev.109.1492 · doi:10.1103/PhysRev.109.1492
[20] DOI: 10.1103/PhysRevLett.42.673 · doi:10.1103/PhysRevLett.42.673
[21] DOI: 10.1007/BF01135526 · Zbl 0368.34015 · doi:10.1007/BF01135526
[22] DOI: 10.1007/BF01942371 · Zbl 0449.60048 · doi:10.1007/BF01942371
[23] DOI: 10.1007/BF01209475 · Zbl 0519.60066 · doi:10.1007/BF01209475
[24] DOI: 10.1007/BF01212355 · Zbl 0573.60096 · doi:10.1007/BF01212355
[25] DOI: 10.1007/BF01217724 · Zbl 0576.60053 · doi:10.1007/BF01217724
[26] DOI: 10.1002/cpa.3160390105 · Zbl 0609.47001 · doi:10.1002/cpa.3160390105
[27] DOI: 10.1103/PhysRevB.33.641 · doi:10.1103/PhysRevB.33.641
[28] DOI: 10.1007/BF02099760 · Zbl 0782.60044 · doi:10.1007/BF02099760
[29] DOI: 10.1016/0370-1573(74)90029-5 · doi:10.1016/0370-1573(74)90029-5
[30] DOI: 10.1016/0550-3213(83)90260-2 · doi:10.1016/0550-3213(83)90260-2
[31] DOI: 10.1088/0022-3719/14/6/003 · doi:10.1088/0022-3719/14/6/003
[32] DOI: 10.1103/PhysRevLett.47.1546 · doi:10.1103/PhysRevLett.47.1546
[33] DOI: 10.1007/BF01578242 · doi:10.1007/BF01578242
[34] DOI: 10.1103/PhysRevLett.64.1437 · doi:10.1103/PhysRevLett.64.1437
[35] DOI: 10.1088/0305-4470/15/7/025 · Zbl 0492.60055 · doi:10.1088/0305-4470/15/7/025
[36] DOI: 10.1088/0022-3719/17/23/012 · doi:10.1088/0022-3719/17/23/012
[37] DOI: 10.1103/PhysRevB.27.5142 · doi:10.1103/PhysRevB.27.5142
[38] Hatsugai Y., Phys. Rev. B 48 pp 2185– (1993)
[39] DOI: 10.1103/PhysRevB.25.2185 · doi:10.1103/PhysRevB.25.2185
[40] Harper P., Proc. Phys. Soc. London, Ser. A 265 pp 317– (1955)
[41] Bellissard J., J. Funct. Anal. 49 pp 191– (1982)
[42] DOI: 10.1007/BF02278001 · Zbl 0724.47002 · doi:10.1007/BF02278001
[43] DOI: 10.1007/BF01234419 · Zbl 0665.46051 · doi:10.1007/BF01234419
[44] DOI: 10.1007/BF01218391 · Zbl 0658.53068 · doi:10.1007/BF01218391
[45] DOI: 10.1070/RM1979v034n02ABEH002908 · Zbl 0448.47032 · doi:10.1070/RM1979v034n02ABEH002908
[46] DOI: 10.1007/BF02719543 · doi:10.1007/BF02719543
[47] DOI: 10.1016/0039-6028(86)90962-3 · doi:10.1016/0039-6028(86)90962-3
[48] DOI: 10.1103/PhysRevB.48.11167 · doi:10.1103/PhysRevB.48.11167
[49] DOI: 10.1016/0038-1098(83)90441-6 · doi:10.1016/0038-1098(83)90441-6
[50] DOI: 10.1007/BF01075982 · Zbl 0244.35073 · doi:10.1007/BF01075982
[51] DOI: 10.1007/BF01222516 · Zbl 0429.60099 · doi:10.1007/BF01222516
[52] DOI: 10.1090/conm/050/841099 · doi:10.1090/conm/050/841099
[53] DOI: 10.1007/BF02161415 · Zbl 0707.46050 · doi:10.1007/BF02161415
[54] DOI: 10.1103/PhysRevB.39.8525 · doi:10.1103/PhysRevB.39.8525
[55] DOI: 10.1103/PhysRevB.27.7539 · doi:10.1103/PhysRevB.27.7539
[56] Mil’nikov G., P. Z. Eksp. Teor. Fiz. 48 pp 494– (1988)
[57] DOI: 10.1088/0022-3719/5/8/007 · doi:10.1088/0022-3719/5/8/007
[58] DOI: 10.1103/PhysRevLett.60.619 · doi:10.1103/PhysRevLett.60.619
[59] DOI: 10.1016/0039-6028(88)90673-5 · doi:10.1016/0039-6028(88)90673-5
[60] DOI: 10.1103/PhysRevLett.68.1375 · doi:10.1103/PhysRevLett.68.1375
[61] DOI: 10.1103/PhysRevB.49.2677 · doi:10.1103/PhysRevB.49.2677
[62] DOI: 10.1088/0022-3719/21/14/008 · doi:10.1088/0022-3719/21/14/008
[63] DOI: 10.1016/0039-6028(92)90323-X · doi:10.1016/0039-6028(92)90323-X
[64] DOI: 10.1103/PhysRevLett.54.831 · doi:10.1103/PhysRevLett.54.831
[65] DOI: 10.1103/PhysRevB.14.2239 · doi:10.1103/PhysRevB.14.2239
[66] DOI: 10.1088/0022-3719/15/22/005 · doi:10.1088/0022-3719/15/22/005
[67] DOI: 10.1103/PhysRevLett.48.1559 · doi:10.1103/PhysRevLett.48.1559
[68] DOI: 10.1103/PhysRevLett.51.605 · doi:10.1103/PhysRevLett.51.605
[69] DOI: 10.1103/PhysRevLett.52.1583 · doi:10.1103/PhysRevLett.52.1583
[70] DOI: 10.1103/PhysRevLett.63.199 · doi:10.1103/PhysRevLett.63.199
[71] DOI: 10.1103/PhysRevB.43.11025 · doi:10.1103/PhysRevB.43.11025
[72] DOI: 10.1016/0550-3213(91)90360-A · doi:10.1016/0550-3213(91)90360-A
[73] DOI: 10.1016/0550-3213(91)90275-3 · doi:10.1016/0550-3213(91)90275-3
[74] DOI: 10.1007/BF02096549 · Zbl 0758.53048 · doi:10.1007/BF02096549
[75] DOI: 10.1142/S0217751X86000149 · Zbl 0631.17012 · doi:10.1142/S0217751X86000149
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.