Semiclassical dynamics of wave - packets in generalized mechanics Outline Semiclassical Approximations in Condensed Matter Physics Berry Phase in Cond. Matt. Dynamical Systems for Wave Packets: Hamiltonian and Lagrangian formulations Symmetries Papers: P. Horvathy, L. M., P. Stichel, Phys. Lett B 615 (2005), P. Horvathy, L. M., P. Stichel, Mod. Phys. Lett. A 20 (2005), C. Duval, Z. Horvath, P. A. Horvathy, L. M., P.Stichel, Mod. Phys. Lett. B. 20 No. 7 (2006) L. Martina, Fundam. Appl. Math. 12 (2006),

Lattice constant Wave – length of the modulations x xcxc “Periodic” Hamiltonian Op.(x c ) Bloch Theory Sundaram,Niu, Phys.Rev. B (1999) Dispersion of the w-p

n fixed U(1) - Berry connection Berry – curvature

Modulation by EM f. Anomalous Hall Effect (Karplus-Luttinger, Phys. Rev. (1954) Thouless et al, Phys. Rev. Lett. 82 GaAs, ferro-, Antiferro- crystals Chern numbers

2D - Model C. Duval,P. Horvathy Phys. Lett B479 (2000)284 J. Lukierski, P.C. Stichel, W.J. Zakrzewski, Ann. Phys. (1997)

Hamiltonian Structure Canonical Variables B = const

J. Negro, MA del Olmo, J. Math. Phys. 31 (1990) 2811, P. Horvathy et al Phys. Lett B 615 (2005), 87-92

Constants of motion Enlarged (2+1) – Galilei Group anyons Central Charges:,,

‘ =

6D-Orbits: Local Coord.: 4D-Orbits:Extremum of Local Coord.:

Enlarged symmetry Anomalous couplings V.P. Nair, A.P. Policronakos, Phys. Lett B 505 (2001) (, const unif.)(, generic)

Coadjoint Orbits The gyromagnetic problem A free relativistic particle in the plane Unitary representations of the planar Lorentz algebra L. Feher, PhD Thesis (1987) J. Negro, A.M. del Olmo, J. Tosiek math-ph/ D analog Pauli-Lubanski 4-vec R. Jackiw, V.P. Nair, Phys. Lett B 480 (2000)237 C. Chou, V.P. Nair, A. Polychronakos, Phys. Lett B304 (1993)105 D.K. Maude et al., Phys. Rev. Lett. 77 (1996) 4604 D.R. Leadley et al., Phys. Rev. Lett. 79 (1997) Experimental evidence

P. Horvathy, L. M., P. Stichel, Mod. Phys. Lett. A 20 (2005), (in w.f.l.) 3D-Minkowski sp.

General Model for Bloch electrons 2D D. Culcer et al. Phys. Rev. B 68 (2003) , zincoblende Restricted orbits

3D

Conclusions Semiclassical dynamics of electron in crystals involves Berry phase effects They are Hamiltonian systems Enlarged formulations allows to embody the presence of external fields In 2D the Enlarged – two folded Galilei Group symmetry defines the “exotic” free model DH The exotic charge has a physical realization (constant Berry curvature) The group orbit method has been used to describe the phase space and its singular foliations Anomalous gyromagnetic effects can be considered by simple generalizations The exotic model (g=0) is not a relativistic limit of relativistic anyons. The anomalous gyromagnetic problem can be addressed for relativistic anyons, by a non-standard S·F contribution to the mass. Hamiltonian formulation, both in 2D and 3D, of semiclassical electron wave packets is provided. Symmetry analysis and restricted Hall motions are characterized. Boltzmann equation for 1 “exotic” particle distribution f. and is written. Fluid equations

Future outlook B. Basu, S. Ghosh, hep-th/ C. Zang et al, cond-mat/