Advertisement

The European Physical Journal B

, Volume 58, Issue 2, pp 137–148 | Cite as

Semiclassical diagonalization of quantum Hamiltonian and equations of motion with Berry phase corrections

  • P. Gosselin
  • A. Bérard
  • H. MohrbachEmail author
Mathematical Structures in Statistical and Condensed Matter Physics

Abstract.

It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at order ħ for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.

PACS.

03.65.-w Quantum mechanics 03.65.Sq Semiclassical theories and applications 03.65.Vf Phases: geometric; dynamic or topological 04.20.Cv Fundamental problems and general formalism 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.V. Berry, Proc. R. Soc. A 392, 45 (1984) zbMATHCrossRefADSGoogle Scholar
  2. S. Murakami, N. Nagaosa, S.C. Zhang, Science 301, 1348 (2003) CrossRefADSGoogle Scholar
  3. A. Bérard, H. Mohrbach, Phys. Rev. D 69, 127701 (2004); A. Bérard, H. Mohrbach, Phys. lett. A 352, 190 (2006) CrossRefADSGoogle Scholar
  4. K.Y. Bliokh, Europhys. Lett. 72, 7 (2005) CrossRefADSGoogle Scholar
  5. K.Y. Bliokh, Y.P. Bliokh, Phys. Lett. A 333, 181 (2004); K.Y. Bliokh, Y.P. Bliokh, Phys. Rev. E 70, 026605 (2004); K.Y. Bliokh, Y.P. Bliokh, Phys. Rev. Lett. 96, 073903 (2006) CrossRefADSGoogle Scholar
  6. M. Onoda, S. Murakami, N. Nagasoa, Phys. Rev. Lett. 93, 083901 (2004) CrossRefADSGoogle Scholar
  7. C. Duval, Z. Horváth, P.A. Horváthy, Phys. Rev. D 74, 021701 (2006), and J. Geom. Phys. 57, 925 (2007) CrossRefADSGoogle Scholar
  8. N.W. Ashcroft, N.D. Mermin, Solid State Physics (Saunders, Philadelphia, 1976) Google Scholar
  9. M.C. Chang, Q. Niu, Phys. Rev. Lett. 75, 1348 (1995); M.C. Chang, Q. Niu, Phys. Rev. B 53, 7010 (1996); G. Sundaram, Q. Niu, Phys. Rev. B 59, 14915 (1999) CrossRefADSGoogle Scholar
  10. R. Shindou, K.I. Imura, Nucl. Phys. B 720, 399 (2005); D. Culcer, Y. Yao, Q. Niu, Phys. Rev. B 72, 085110 (2005) zbMATHCrossRefADSGoogle Scholar
  11. D. Xiao, J. Shi, Q. Niu, Phys. Rev. Lett. 95, 137204 (2005) CrossRefADSGoogle Scholar
  12. C. Duval, Z. Horvath, P. Horvathy, L. Martina, P. Stichel, Phys. Rev. Lett. 96, 099701 (2006); D. Xiao, J. Shi, Q. Niu, Phys. Rev. Lett. 96, 099702 (2006) CrossRefADSGoogle Scholar
  13. K.Y. Bliokh, Phys. Lett. A 351, 123 (2006) CrossRefADSGoogle Scholar
  14. S. Ghosh, Phys. Lett. B 638, 350 (2006); B. Basu, S. Ghosh, S. Dhar, Europhys. Lett. 76, 395 (2006) CrossRefADSGoogle Scholar
  15. P. Gosselin, F. Ménas, A. Bérard, H. Mohrbach, Europhys. Lett. 76, 651 (2006) CrossRefADSGoogle Scholar
  16. P. Gosselin, A. Bérard, H. Mohrbach, Phys. Rev. D 75, 084035 (2007); P. Gosselin, A. Bérard, H. Mohrbach, Phys. Lett. A, in press. CrossRefADSGoogle Scholar
  17. Y.N. Obukhov, Phys. Rev. Lett. 86, 192 (2001) CrossRefADSGoogle Scholar
  18. A.J. Silenko, O.V. Teryaev, Phys. Rev. D 71, 064016 (2005) CrossRefADSGoogle Scholar
  19. L.L. Foldy, S.A. Wouthuysen, Phys. Rev. 78, 29 (1949) CrossRefADSGoogle Scholar
  20. K.Y. Bliokh, Y.P. Bliokh, Ann. Phys. (NY) 319, 13 (2005) zbMATHCrossRefADSGoogle Scholar
  21. C. Kittel, Quantum Therory of Solids (Wiley, New York, 1963) Google Scholar
  22. E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics (Pergamon Press, 1981), Vol 9 Google Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institut Fourier, UMR 5582 CNRS-UJF, UFR de Mathématiques, Université Grenoble ISaint Martin d'Hères CedexFrance
  2. 2.Laboratoire de Physique Moléculaire et des Collisions, ICPMB-FR CNRS 2843, Université Paul Verlaine-MetzMetz Cedex 3France

Personalised recommendations