Towards the fractional quantum Hall effect: a noncommutative geometry perspective
In this paper we give a survey of some models of the integer and fractional quantum Hall effect based on noncommutative geometry. We begin by recalling some classical geometry of electrons in solids and the passage to noncommutative geometry produced by the presence of a magnetic field. We recall how one can obtain this way a single electron model of the integer quantum Hall effect. While in the case of the integer quantum Hall effect the underlying geometry is Euclidean, we then discuss a model of the fractional quantum Hall effect, which is based on hyperbolic geometry simulating the multi-electron interactions. We derive the fractional values of the Hall conductance as integer multiples of orbifold Euler characteristics. We compare the results with experimental data.
KeywordsNoncommutative Geometry Fuchsian Group Spectral Projection Hall Conductance Fractional Quantum
Unable to display preview. Download preview PDF.
- J. Bellissard, The noncommutative geometry of aperiodic solids, in “Geometric and topological methods for quantum field theory (Villa de Leyva, 2001)”, 86–156, World Scientific, 2003.Google Scholar
- A. Carey, K. Hannabuss, V. Mathai, Quantum Hall effect and noncommutative geometry, arXiv:math.OA/0008115.Google Scholar
- T. Chakraborti, P. Pietilänen, The Quantum Hall Effects, Second Edition, Springer 1995.Google Scholar
- T.-S. Choy, J. Naset, J. Chen, S. Hershfield, and C. Stanton. A database of fermi surface in virtual reality modeling language (vrml), Bulletin of The American Physical Society, 45(1):L36 42, 2000.Google Scholar
- R.G. Clark, R.J. Nicholas, A. Usher, C.T. Foxon, J.J. Harris, Surf.Sci. 170 (1986) 141.Google Scholar
- J. Dodziuk, V. Mathai, S. Yates, Arithmetic properties of eigenvalues of generalized Harper operators on graphs, arXiv math.SP/0311315Google Scholar
- D. Gieseker, H. Knörrer, E. Trubowitz, The geometry of algebraic Fermi curves, Perspectives in Mathematics, Vol.14. Academic Press, 1993. viii+236 pp.Google Scholar
- D. Gieseker, H. Knörrer, E. Trubowitz, An overview of the geometry of algebraic Fermi curves, in “Algebraic geometry: Sundance 1988”, 19–46, Contemp. Math. Vol.116, Amer. Math. Soc. 1991.Google Scholar
- E.H. Hall, On a new action of the magnet on electric currents, Amer. J. of Math. Vol.287, (1879) N.2.Google Scholar
- Y. Kordyukov, V. Mathai and M.A. Shubin, Equivalence of spectral projections in semiclassical limit and a vanishing theorem for higher traces in K-theory, J.Reine Angew.Math. (Crelle), Vol.581 (2005) 44 pages (to appear).Google Scholar
- R.G. M∪Φ, The fractional quantum Hall effect, Chern-Simons theory, and integral lattices, in “Proceedings of the International Congress of Mathematicians”, Vol. 1,2 (Zrich, 1994), 75–105, Birkhäuser, 1995.Google Scholar
- H.L. Störmer, Advances in solid state physics, ed. P. Grosse, vol.24, Vieweg 1984.Google Scholar