Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 146, 50 pages      arXiv:2006.06785      https://doi.org/10.3842/SIGMA.2020.146
Contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi

The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

Giuseppe De Nittis a and Maximiliano Sandoval b
a) Facultad de Matemáticas & Instituto de Física, Pontificia Universidad Católica de Chile, Santiago, Chile
b) Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile

Received June 12, 2020, in final form December 22, 2020; Published online December 28, 2020

Abstract
This work provides a first step towards the construction of a noncommutative geometry for the quantum Hall effect in the continuum. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the $C^*$-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.

Key words: Landau Hamiltonian; spectral triple; Dixmier trace; first Connes' formula.

pdf (766 kb)   tex (64 kb)  

References

  1. Alberti P.M., Matthes R., Connes' trace formula and Dirac realization of Maxwell and Yang-Mills action, in Noncommutative Geometry and the Standard Model of Elementary Particle Physics (Hesselberg, 1999), Lecture Notes in Phys., Vol. 596, Springer, Berlin, 2002, 40-74, arXiv:math-ph/9910011.
  2. Avron J., Herbst I., Simon B., Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J. 45 (1978), 847-883.
  3. Avron J.E., Seiler R., Simon B., Charge deficiency, charge transport and comparison of dimensions, Comm. Math. Phys. 159 (1994), 399-422, arXiv:physics/9803014.
  4. Azamov N., McDonald E., Sukochev F., Zanin D., A Dixmier trace formula for the density of states, Comm. Math. Phys. 377 (2020), 2597-2628, arXiv:1910.12380.
  5. Bellissard J., The noncommutative geometry of aperiodic solids, in Geometric and Topological Methods for Quantum Field Theory (Villa de Leyva, 2001), World Sci. Publ., River Edge, NJ, 2003, 86-156.
  6. Bellissard J., van Elst A., Schulz-Baldes H., The noncommutative geometry of the quantum Hall effect, J. Math. Phys. 35 (1994), 5373-5451.
  7. Blackadar B., $K$-theory for operator algebras, 2nd ed., Mathematical Sciences Research Institute Publications, Vol. 5, Cambridge University Press, Cambridge, 1998.
  8. Bourne C., Carey A.L., Rennie A., A non-commutative framework for topological insulators, Rev. Math. Phys. 28 (2016), 1650004, 51 pages, arXiv:1509.07210.
  9. Bourne C., Rennie A., Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases, Math. Phys. Anal. Geom. 21 (2018), 16, 62 pages, arXiv:1611.06016.
  10. Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics. 1. $C^*$- and $W^*$-algebras symmetry groups decomposition of states, Texts and Monographs in Physics, Springer-Verlag, Berlin - Heidelberg, 1987.
  11. Brezis H., Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011.
  12. Carey A.L., Gayral V., Rennie A., Sukochev F.A., Index theory for locally compact noncommutative geometries, Mem. Amer. Math. Soc. 231 (2014), vi+130 pages, arXiv:1107.0805.
  13. Carey A.L., Hannabuss K.C., Mathai V., Quantum Hall effect and noncommutative geometry, J. Geom. Symmetry Phys. 6 (2006), 16-37, arXiv:math.OA/0008115.
  14. Comtet A., On the Landau levels on the hyperbolic plane, Ann. Physics 173 (1987), 185-209.
  15. Connes A., $C^{\ast} $ algèbres et géométrie différentielle, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), A599-A604.
  16. Connes A., Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985), 257-360.
  17. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  18. Connes A., Moscovici H., The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), 174-243.
  19. Damak M., Georgescu V., Self-adjoint operators affiliated to $C^*$-algebras, Rev. Math. Phys. 16 (2004), 257-280.
  20. De Nittis G., Gomi K., Moscolari M., The geometry of (non-Abelian) Landau levels, J. Geom. Phys. 152 (2020), 103649, 34 pages, arXiv:1911.04786.
  21. De Nittis G., Lein M., Linear response theory: an analytic-algebraic approach, SpringerBriefs in Mathematical Physics, Vol. 21, Springer, Cham, 2017.
  22. de Pagter B., Sukochev F., Commutator estimates and $\mathbb R$-flows in non-commutative operator spaces, Proc. Edinb. Math. Soc. 50 (2007), 293-324.
  23. Dixmier J., von Neumann algebras, North-Holland Mathematical Library, Vol. 27, North-Holland Publishing Co., Amsterdam - New York, 1981.
  24. Edwards C.M., Lewis J.T., Twisted group algebras. I, Comm. Math. Phys. 13 (1969), 119-130.
  25. Folland G.B., Harmonic analysis in phase space, Annals of Mathematics Studies, Vol. 122, Princeton University Press, Princeton, NJ, 1989.
  26. Folland G.B., Real analysis: modern techniques and their applications, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999.
  27. Gayral V., Gracia-Bondía J.M., Iochum B., Schücker T., Várilly J.C., Moyal planes are spectral triples, Comm. Math. Phys. 246 (2004), 569-623, arXiv:hep-th/0307241.
  28. Gayral V., Wulkenhaar R., Spectral geometry of the Moyal plane with harmonic propagation, J. Noncommut. Geom. 7 (2013), 939-979, arXiv:1108.2184.
  29. Germinet F., Klein A., Schenker J.H., Dynamical delocalization in random Landau Hamiltonians, Ann. of Math. 166 (2007), 215-244, arXiv:math-ph/0412070.
  30. Germinet F., Klein A., Schenker J.H., Quantization of the Hall conductance and delocalization in ergodic Landau Hamiltonians, Rev. Math. Phys. 21 (2009), 1045-1080, arXiv:0812.0392.
  31. Goffeng M., Index formulas and charge deficiencies on the Landau levels, J. Math. Phys. 51 (2010), 023509, 18 pages, arXiv:1003.2865.
  32. Gracia-Bondía J.M., Várilly J.C., Algebras of distributions suitable for phase-space quantum mechanics. I, J. Math. Phys. 29 (1988), 869-879.
  33. Gracia-Bondía J.M., Várilly J.C., Figueroa H., Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston, Inc., Boston, MA, 2001.
  34. Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007.
  35. Greenleaf F.P., Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, Vol. 16, Van Nostrand Reinhold Co., New York - Toronto, Ont. - London, 1969.
  36. Greiner P.C., On the Laguerre calculus of left-invariant convolution (pseudodifferential) operators on the Heisenberg group, in Goulaouic-Meyer-Schwartz Seminar, 1980-1981, École Polytech., Palaiseau, 1981, Exp. No. XI, 40 pages.
  37. Grosse H., Wulkenhaar R., 8D-spectral triple on 4D-Moyal space and the vacuum of noncommutative gauge theory, J. Geom. Phys. 62 (2012), 1583-1599, arXiv:0709.0095.
  38. Halmos P.R., Sunder V.S., Bounded integral operators on $L^{2}$ spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 96, Springer-Verlag, Berlin - New York, 1978.
  39. Johnson M.H., Lippmann B.A., Motion in a constant magnetic field, Phys. Rev. 76 (1949), 828-832.
  40. Kammerer J.-B., Analysis of the Moyal product in a flat space, J. Math. Phys. 27 (1986), 529-535.
  41. Kato T., Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
  42. Kunz H., The quantum Hall effect for electrons in a random potential, Comm. Math. Phys. 112 (1987), 121-145.
  43. Langmann E., Noncommutative integration calculus, J. Math. Phys. 36 (1995), 3822-3835.
  44. Laughlin R.B., Quantized Hall conductivity in two dimensions, Phys. Rev. B 23 (1981), 5632-5633.
  45. Lenz D.H., Random operators and crossed products, Math. Phys. Anal. Geom. 2 (1999), 197-220, arXiv:math-ph/9902030.
  46. Lieb E.H., Loss M., Analysis, 2nd ed., Graduate Studies in Mathematics, Vol. 14, Amer. Math. Soc., Providence, RI, 2001.
  47. Lord S., Sukochev F., Zanin D., Singular traces: theory and applications, De Gruyter Studies in Mathematics, Vol. 46, De Gruyter, Berlin, 2013.
  48. Loring T.A., Schulz-Baldes H., Finite volume calculation of $K$-theory invariants, New York J. Math. 23 (2017), 1111-1140, arXiv:1701.07455.
  49. Loring T.A., Schulz-Baldes H., Spectral flow argument localizing an odd index pairing, Canad. Math. Bull. 62 (2019), 373-381, arXiv:1802.04518.
  50. Loring T.A., Schulz-Baldes H., The spectral localizer for even index pairings, J. Noncommut. Geom. 14 (2020), 1-23, arXiv:1802.04517.
  51. Maillard J.-M., On the twisted convolution product and the Weyl transformation of tempered distributions, J. Geom. Phys. 3 (1986), 230-261.
  52. McCann P.J., Carey A.L., A discrete model of the integer quantum Hall effect, Publ. Res. Inst. Math. Sci. 32 (1996), 117-156.
  53. McDonald E., Sukochev F., Xiong X., Quantum differentiability on quantum tori, Comm. Math. Phys. 371 (2019), 1231-1260, arXiv:1909.03659.
  54. McDonald E., Sukochev F., Xiong X., Quantum differentiability on noncommutative Euclidean spaces, Comm. Math. Phys. 379 (2020), 491-542, arXiv:1909.03581.
  55. Nelson E., Notes on non-commutative integration, J. Funct. Anal. 15 (1974), 103-116.
  56. Prodan E., A computational non-commutative geometry program for disordered topological insulators, SpringerBriefs in Mathematical Physics, Vol. 23, Springer, Cham, 2017.
  57. Prodan E., Schulz-Baldes H., Bulk and boundary invariants for complex topological insulators: from $K$-theory to physics, Mathematical Physics Studies, Springer, Cham, 2016.
  58. Raikov G.D., Warzel S., Quasi-classical versus non-classical spectral asymptotics for magnetic Schrödinger operators with decreasing electric potentials, Rev. Math. Phys. 14 (2002), 1051-1072, arXiv:math-ph/0201006.
  59. Segal I.E., A non-commutative extension of abstract integration, Ann. of Math. 57 (1953), 401-457.
  60. Sukochev F., Zanin D., The Connes character formula for locally compact spectral triples, arXiv:1803.01551.
  61. Terp S., $L^p$-spaces associated with von Neumann algebras, Copenhagen University, 1981.
  62. Thangavelu S., Lectures on Hermite and Laguerre expansions, Mathematical Notes, Vol. 42, Princeton University Press, Princeton, NJ, 1993.
  63. Thouless D.J., Kohmoto M., Nightingale M.P., den Nijs M., Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49 (1982), 405-408.
  64. van den Dungen K., Paschke M., Rennie A., Pseudo-Riemannian spectral triples and the harmonic oscillator, J. Geom. Phys. 73 (2013), 37-55, arXiv:1207.2112.
  65. Várilly J.C., Gracia-Bondía J.M., Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra, J. Math. Phys. 29 (1988), 880-887.
  66. Veselić I., Existence and regularity properties of the integrated density of states of random Schrödinger operators, Lecture Notes in Mathematics, Vol. 1917, Springer-Verlag, Berlin, 2008.
  67. Wulkenhaar R., Non-compact spectral triples with finite volume, in Quanta of Maths, Clay Math. Proc., Vol. 11, Amer. Math. Soc., Providence, RI, 2010, 617-648, arXiv:0907.1351.
  68. Xia J., Geometric invariants of the quantum Hall effect, Comm. Math. Phys. 119 (1988), 29-50.
  69. Zak J., Magnetic translation group, Phys. Rev. 134 (1964), A1602-A1606.
  70. Zak J., Magnetic translation group. II. Irreducible representations, Phys. Rev. 134 (1964), A1607-A1611.
  71. Zeller-Meier G., Produits croisés d'une $C^{\ast} $-algèbre par un groupe d'automorphismes, J. Math. Pures Appl. 47 (1968), 101-239.

Previous article  Contents of Volume 16 (2020)