Hamiltonian theory of anyons in crystals

L. Martina

Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 7, pp. 129-139.

Résumé

Semiclassical wave packets for electrons in crystals, subject to external electromagnetic field, satisfy Hamiltonian equations. In $(2+1)$-dimensions and in the limit of uniform fields, the symmetry group results a two-folded Galilei algebra, incorporating an “exotic” central charge. It has the physical meaning of the Berry-phase curvature. In the Hamiltonian scheme, we discuss possible deformations of that algebra and the physical meaning.

Bibliographie

[1] Ashcroft N. W., Mermin N. D., Solid State Physics, Saunders, Philadelphia, 1976

[2] Bacry H., “Wigner elementary particle in an external homogeneous field”, Lett. Math. Phys., 1, 1976, 295–299 | MR

[3] Bérard A., Mohrbach H., “Monopole and Berry phase in momentum space in noncommutative quantum mechanics”, Phys. Rev. D, 69:127701 (2004) | MR

[4] Bérard A., Mohrbach H., “Spin Hall effect and Berry phase of spinning particles”, Phys. Lett. A, 352 (2006), 190–195 | DOI | MR | Zbl

[5] Bohm A., Mostafazadeh A., Koizumi H., Niu Q., Zwanziger J., The Geometric Phase in Quantum Systems, Springer, 2003 | MR

[6] Brihaye Y., Gonera C., Giller S., Kosiński P., Galilean invariance in $2+1$ dimensions, , 1995 arXiv: hep-th/9503046 | MR

[7] Chang M. C., Niu Q., “Berry phase, hyperorbits, and the Hofstadter spectrum”, Phys. Rev. Lett., 75 (1995), 1348–1351 | DOI

[8] Chang M. C., Niu Q., “Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical dynamics in magnetic Bloch bands”, Phys. Rev. B, 53 (1996), 7010–7023 | DOI

[9] Culcer D., MacDonald A. H., Niu Q., “Anomalous Hall effect in paramagnetic two-dimensional systems”, Phys. Rev. B, 68:045327 (2003)

[10] Duval C., Thèse de 3e cycle, Marseille, 1972

[11] Duval C., Horváthy P. A., “The exotic Galilei group and the “Peierls substitution””, Phys. Lett. B, 479 (2000), 284–290 | DOI | MR | Zbl

[12] Duval C., Horváthy P. A., “Exotic Galilean symmetry in the noncommutative plane and the Hall effect”, J. Phys. A, 34 (2001), 10097–10107 | DOI | MR | Zbl

[13] Duval C., Horváthy P. A., “Anyons with anomalous gyromagnetic ratio and the Hall effect”, Phys. Lett. B, 594 (2004), 402–409 E-print hep-th/0402191. | DOI | MR

[14] Grignani G., Plyushchay M., Sodano P., “A pseudoclassical model for $P,T$-invariant planar fermions”, Nuclear Phys. B, 464 (1996), 189–212 ; E-print hep-th/9511072 | DOI | MR | Zbl

[15] Grigore D. R., “The projective unitary irreducible representations of the Galilei group in $1+2$ dimensions”, J. Math. Phys., 37:1 (1996), 460–473 | DOI | MR | Zbl

[16] Grigore D. R., “Transitive symplectic manifolds in $1+2$ dimensions”, J. Math. Phys., 37:1 (1996), 240–253 | DOI | MR | Zbl

[17] Horváthy P., Martina L., Stichel P., “Comments on spin-orbit interaction of anyons”, Modern Phys. Lett. A, 20 (2005), 1177–1185 | DOI | Zbl

[18] Horváthy P. A., Martina L., Stichel P. C., “Enlarged Galilean symmetry of anyons and the Hall effect”, Phys. Lett. B, 615 (2005), 87 | DOI | MR

[19] Horváthy P. A., Plyushchay M. S., Nonrelativistic anyons in external electromagnetic field, , 2005 arXiv:hep-th/0502040 | MR

[20] Horváthy P. A., Plyushchay M. S., Nonrelativistic anyons, exotic Galilean symmetry and noncommutative plane, , 2002 arXiv:hep-th/0201228 | MR

[21] Jungwirth T., Niu Q., MacDonald A. H., “Anomalous Hall effect in ferromagnetic semiconductors”, Phys. Rev. Lett., 88:207208 (2002)

[22] Karplus R., Luttinger J. M., “Hall effect in ferromagnetics”, Phys. Rev., 95 (1954), 1154–1160 | DOI | Zbl

[23] Laughlin R. B., “Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations”, Phys. Rev. Lett., 50 (1983), 1395–1398 | DOI

[24] Lévy-Leblond J.-M., Galilei group and Galilean invariance, Group Theory and Applications, Academic Press, New York, 1972, 222 pp. | MR

[25] 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., 260:2 (1997), 224–249 | DOI | MR | Zbl

[26] Negro J., Del Olmo M. A., Tosiek J., “Anyons, group theory and planar physics”, J. Math. Phys., 47:3 (2006), 033508, 19 pp. | DOI | MR

[27] Novoselov K. S., McCann E., Morozov S. V., Fal'ko V. I., Katsnelson M. I., Zeitler U., Jiang D., Schedin F., Geim A. K., “Unconventional quantum Hall effect and Berry's phase of 2 in bilayer graphene”, Nature Phys., 2 (2006), 177–180 | DOI

[28] Skagerstam B. S., Stern A., “Lagrangian descriptions of classical charged particles with spin”, Phys. Scripta, 24:3 (1981), 493–497 | DOI | MR

[29] Sundaram G., Niu Q., “Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects”, Phys. Rev. B, 59 (1999), 14915–14925 | DOI