Semiclassical dynamics of wave - packets in generalized mechanics

1. 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), 87-92 P. Horvathy, L. M., P. Stichel, Mod. Phys. Lett. A 20 (2005), 1177-1185 C. Duval, Z. Horvath, P. A. Horvathy, L. M., P.Stichel, Mod. Phys. Lett. B. 20 No. 7 (2006) 373-378 L. Martina, Fundam. Appl. Math. 12 (2006), 109-118

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

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

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

5. 2D - Model C. Duval,P. Horvathy Phys. Lett B479 (2000)284 J. Lukierski, P.C. Stichel, W.J. Zakrzewski, Ann. Phys. (1997)

6. Hamiltonian Structure B = const Canonical Variables

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

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

9. ‘ = ‘ = ‘ =

10. 6D-Orbits: Local Coord.: 4D-Orbits: Extremum of Local Coord.:

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

12. The gyromagnetic problem Unitary representations of the planar Lorentz algebra A free relativistic particle in the plane Coadjoint Orbits L. Feher, PhD Thesis (1987) J. Negro, A.M. del Olmo, J. Tosiek math-ph/0512007 2D 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

13. P. Horvathy, L. M., P. Stichel, Mod. Phys. Lett. A 20 (2005), 1177-1185 3D-Minkowski sp. (in w.f.l.)

14. General Model for Bloch electrons 2D D. Culcer et al. Phys. Rev. B 68 (2003) 045327, zincoblende Restricted orbits

15. 3D

16. 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

17. Future outlook B. Basu, S. Ghosh, hep-th/0503263 C. Zang et al , cond-mat/0507125