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

Lattice constant

Wave – length of the modulations

x

xc

“Periodic” Hamiltonian Op.(xc)

Bloch Theory

txuex qnxqi

qn ,

xulxu qnqn

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2..,ˆˆ2

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c

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qnccnqncc tqxEtxH ,,,ˆ tqxE ccn ,,

Sundaram,Niu, Phys.Rev. B (1999)

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U(1) - Berry connection

Berry – curvature

Modulation by EM f.

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Anomalous Hall Effect (Karplus-Luttinger, Phys. Rev. (1954)

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0

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GaAs, ferro-, Antiferro- crystals

Chern numbers

2D - Model

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00 Bd tDH E

C. Duval,P. Horvathy Phys. Lett B479 (2000)284 DHKer

22

220 mm

L LSZ

txextxeBq

txemqxm

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E

J. Lukierski, P.C. Stichel, W.J. Zakrzewski, Ann. Phys. (1997)

22

0

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Hamiltonian Structure

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Canonical Variables

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m

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

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Constants of motion

Enlarged (2+1) – Galilei Group

anyons

Central Charges: , ,

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6D-Orbits:

Local Coord.:

4D-Orbits: Extremum of

Local Coord.:

0*m

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1

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Enlarged symmetry

Anomalous couplings

V.P. Nair, A.P. Policronakos, Phys. Lett B 505 (2001)

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2 0

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E

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( , generic)

Coadjoint Orbits

The gyromagnetic problem

A free relativistic particle in the plane Unitary representations of the planar Lorentz algebra 1,2o

1,2SO

L. Feher, PhD Thesis (1987)J. Negro, A.M. del Olmo, J. Tosiek math-ph/0512007 2D analog Pauli-Lubanski 4-vec

mc

kpjmcs

R. Jackiw, V.P. Nair, Phys. Lett B 480 (2000)237 2200 , m

c

sB

C. Chou, V.P. Nair, A. Polychronakos, Phys. Lett B304 (1993)105

2g

D.K. Maude et al., Phys. Rev. Lett. 77 (1996) 4604D.R. Leadley et al., Phys. Rev. Lett. 79 (1997) Experimental evidence 0g

CNP

P. Horvathy, L. M., P. Stichel, Mod. Phys. Lett. A 20 (2005), 1177-1185

(in w.f.l.)

3D-Minkowski sp.

d

ijji

ijji

ijji

eBm

mkk

Bkemm

mkr

km

mrr

*,

1*,*

,

,*

,

General Model for Bloch electrons

2D

2k

D. Culcer et al. Phys. Rev. B 68 (2003) 045327, zincoblende

Restricted orbits

3D

0,0 Brk

dkm

mu

* dkm

mx

uv

*1

0 vuut fvvvu t

m

m*

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uX

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E B

vE

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 packetsis provided. •Symmetry analysis and restricted Hall motions are characterized. •Boltzmann equation for 1 “exotic” particle distribution f. and is written.•Fluid equations

Future outlook

3r

rbB

B. Basu, S. Ghosh, hep-th/0503263

3k

kg

C. Zang et al , cond-mat/0507125