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


SIGMA 2 (2006), 090, 9 pages      cond-mat/0609571      http://dx.doi.org/10.3842/SIGMA.2006.090
Contribution to the Proceedings of the O'Raifeartaigh Symposium

Non-Commutative Mechanics in Mathematical & in Condensed Matter Physics

Peter A. Horváthy
Laboratoire de Mathématiques et de Physique Théorique, Université de Tours, Parc de Grandmont, F-37200 Tours, France

Received September 25, 2006, in final form November 27, 2006; Published online December 14, 2006

Abstract
Non-commutative structures were introduced, independently and around the same time, in mathematical and in condensed matter physics (see Table 1). Souriau's construction applied to the two-parameter central extension of the planar Galilei group leads to the ''exotic'' particle, which has non-commuting position coordinates. A Berry-phase argument applied to the Bloch electron yields in turn a semiclassical model that has been used to explain the anomalous/spin/optical Hall effects. The non-commutative parameter is momentum-dependent in this case, and can take the form of a monopole in momentum space.

Key words: non-commutative mechanics; semiclassical models; Hall effect.

pdf (268 kb)   ps (269 kb)   tex (175 kb)

References

  1. Bargmann V., On unitary ray representations of continuous groups, Ann. Math., 1954, V.59, 1-46.
  2. Lévy-Leblond J.-M., Galilei group and Galilean invariance, in Group Theory and Its Applications, Vol. 2, Editor E.M. Loebl, New York, Academic Press, 1971, 221-299.
  3. Duval C., Exotic Galilei group, IQHE and Chern-Simons electrodynamics, 1995 (unpublished draft).
  4. Brihaye Y., Gonera C., Giller S., Kosi\'nski P., Galilean invariance in 2+1 dimensions, hep-th/9503046.
  5. Grigore D.R., Transitive symplectic manifolds in 1+2 dimensions, J. Math. Phys., 1996, V.37, 240-253.
  6. Grigore D.R., The projective unitary irreducible representations of the Galilei group in 1+2 dimensions, J. Math. Phys., 1996, V.37, 460-473, hep-th/9312048.
  7. Lukierski J., Stichel P.C., Zakrzewski W.J., Galilean-invariant (2+1)-dimensional models with a Chern-Simons-like term and d=2 noncommutative geometry, Ann. Phys., 1997, V.260, 224-249, hep-th/9612017.
  8. Ballesteros A., Gadella M., del Olmo M., Moyal quantization of 2+1 dimensional Galilean systems, J. Math. Phys., 1992, V.33, 3379-3386.
  9. Duval C., Horváthy P.A., The exotic Galilei group and the "Peierls substitution", Phys. Lett. B, 2000, V.479, 284-290, hep-th/0002233.
  10. Duval C., Horváthy P.A., Exotic Galilean symmetry in the non-commutative plane, and the Hall effect, J. Phys. A: Math. Gen., 2001, V.34, 10097-10108, hep-th/0106089.
  11. Horváthy P.A., The non-commutative Landau problem, Ann. Phys., 2002, V.299, 128-140, hep-th/0201007.
  12. Duval C., Horváthy P.A., Noncommuting coordinates, exotic particles, & anomalous anyons in the Hall effect, Theoret. and Math. Phys., 2005, V.144, 899-906, hep-th/0407010.
  13. Souriau J.-M., Structure des systèmes dynamiques, Paris, Dunod, 1970 (English transl.: Structure of dynamical systems: a symplectic view of physics, Dordrecht, Birkhäuser, 1997).
  14. Dixon W.G., On a classical theory of charged particles with spin and the classical limit of the Dirac equation, Il Nuovo Cimento, 1965, V.38, 1616-1643.
  15. Souriau J.-M., Modèle de particule à spin dans le champ électromagnétique et gravitationnel, Ann. Inst. H. Poincaré A, 1974, V.20, 315-364.
  16. Duval Ch., The general relativistic Dirac-Pauli particle: an underlying classical model, Ann. Inst. H. Poincaré A, 1976, V.25, 345-362.
  17. Dunne G., Jackiw R., Trugenberger C.A., "Topological" (Chern-Simons) quantum mechanics, Phys. Rev. D, 1990, V.41, 661-666.
  18. Dunne G., Jackiw R., "Peierls substitution" and Chern-Simons quantum mechanics. Nuclear Phys. B Proc. Suppl., 1993, V.33, 114-118.
  19. Kirchhoff G., Vorlesungen über mathematischen Physik. Mechanik, 3rd ed., Leipzig, G.B. Teubner, 1883, 251-272.
  20. Horváthy P., Noncommuting coordinates in the Hall effect and in vortex dynamics, Talk given at the COSLAB-VORTEX-BEC 2000 and Workshop Bilbao'03, hep-th/0307175.
  21. Laughlin R.B., Anomalous quantum Hall effect: an Incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett., 1983, V.50, 1395-1398.
  22. Stone M. (Editor), Quantum Hall effect, Singapore, World Scientific, 1992.
  23. Chang M.C., Niu Q., Berry phase, hyperorbits, and the hofstadter spectrum, Phys. Rev. Lett, 1995, V.75, 1348-1351.
  24. Bohm A., Mostafazadeh A., Koizumi H., Niu Q., Zwanziger J., The geometric phase in quantum systems, Chapter 12, Springer Verlag, 2003.
  25. Duval C., Horváth Z., Horváthy P.A., Martina L., Stichel P., Berry phase correction to electron density in solids and exotic dynamics Modern Phys. Lett. B, 2006, V.20, 373-378, cond-mat/0506051.
  26. Duval C., Horváth Z., Horváthy P.A., Martina L., Stichel P., Comment on "Berry phase correction to electron density in solids" by Xiao et al., Phys. Rev. Lett., 2006, V.96, 099701, 2 pages, cond-mat/0509806.
  27. Xiao D., Shi J., Niu Q., Berry phase correction to electron density of states in solids, Phys. Rev. Lett., 2005, V.95, 137204, 4 pages, cond-mat/0502340.
  28. Stone M., Urbana lectures, 2005, available from http://w3.physics.uiuc.edu/~m-stone5/mmb/mmb.html.
  29. Bliokh K.Yu., On the Hamiltonian nature of semiclassical equations of motion in the presence of an electromagnetic field and Berry curvature, Phys. Lett. A, 2006, V.351, 123-124, cond-mat/0507499.
  30. Ghosh S. A novel "magnetic" field and its dual non-commutative phase space, hep-th/0511302.
  31. Gosselin P., Menas F., Bérard A., Mohrbach H., Semiclassical dynamics of electrons in magnetic Bloch bands: a Hamiltonian approach, cond-mat/0601472.
  32. Olson J., Ao P., Nonequilibrium approach to Bloch-Peierls-Berry dynamics, physics/0605101.
  33. Jungwirth T., Niu Q., MacDonald A.H., Anomalous Hall effect in ferromagnetic semiconductors, Phys. Rev. Lett., 2002, V.90, 207208, 4 pages, cond-mat/0110484.
  34. Culcer D., MacDonald A.H., Niu Q., Anomalous Hall effect in paramagnetic two dimensional systems, Phys. Rev. B, 2003, V.68, 045327, 9 pages, cond-mat/0311147.
  35. Fang Z., Nagaosa N., Takahashi K.S., Asamitsu A., Mathieu R., Ogasawara T., Yamada H., Kawasaki M., Tokura Y., Terakura K., Anomalous Hall effect and magnetic monopoles in momentum-space, Science, 2003, V.302, 92-95, cond-mat/0310232.
  36. Murakami S., Nagaosa N., Zhang S.-C., Dissipationless quantum spin current at room temperature, Science, 2003, V.301, 1348-1351, cond-mat/0308167.
  37. Sinova J., Culcer D., Niu Q., Sinitsyn N.A., Jungwirth T., MacDonald A.H., Universal intrinsic spin-Hall effect, Phys. Rev. Lett., 2004, V.92, 126603, 4 pages.
  38. Murakami S., Intrinsic spin Hall effect, Adv. in Solid State Phys., 2005, V.45, 197-209, cond-mat/0504353.
  39. Karplus R., Luttinger J.M., Hall effect in ferromagnetics, Phys. Rev., 1954, V.95, 1154-1160.
  40. Kats Y., Genish I., Klein L., Reiner J.W., Beasley M.R., Testing the Berry phase model for extraordinary Hall effect in SrRuO3, Phys. Rev. B, 2004, V.70, 180407, 4 pages, cond-mat/0405645.
  41. Bérard A., Mohrbach H., Monopole and Berry phase in momentum space in noncommutative quantum mechanics, Phys. Rev. D, 2004, V.69, 127701, 4 pages, hep-th/0310167.
  42. Horváthy P.A., Anomalous Hall effect in non-commutative mechanics, Phys. Lett. A, 2006, V.359, 705-706, cond-mat/0606472.
  43. Onoda M., Murakami S., Nagaosa N., Hall effect of light, Phys. Rev. Lett., 2004, V.93, 083901, 4 pages, cond-mat/0405129.
  44. Liberman V.S., Zeldovich B.Ya., Spin-orbit interaction of a photon in an inhomogeneous medium, Phys. Rev. A, 1992, V.46, 5199-5207.
  45. Bliokh K.Yu., Bliokh Yu.P., Topological spin transport of photons: the optical Magnus effect and Berry phase, Phys. Lett. A, 2004, V.333, 181-186, physics/0402110.
  46. Bérard A., Mohrbach H., Spin Hall effect and Berry phase of spinning particles, Phys. Lett. A, 2006, V.352, 190-195, hep-th/0404165.
  47. Duval C., Horváth Z., Horváthy P.A., Geometrical spinoptics and the optical Hall effect, J. Geom. Phys., 2007, V.57, 925-941, math-ph/0509031.
  48. Duval C., Horváth Z., Horváthy P.A., Fermat principle for polarized light and the optical Hall effect, Phys. Rev. D, 2006, V.74, 021701, 5 pages, cond-mat/0509636.

Previous article   Next article   Contents of Volume 2 (2006)