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Weyl-Dirac Equation in CondensedMatter Physics: Graphene
Juan M GuerraUniversidad Nacional de Colombia
Mayo, 2011.
Escuela de Física-Matemática 2011Universidad de los andes
Contents
1. Dirac equation
2. Dirac Lagrangian and Transformations
3. Massless Weyl Equation
4. Weyl’s Formulation
5. Klein Paradox (Chiral Tunneling)
6. Graphene and Nanotubes
7. Research
8. References
Dirac Equation (1928)
Covariant form, Invariance under a Lorentz transformation*
0)(xmi
g2,
)(x 4-component spinor.Decoupled when m=0.
)()(' xUx
Dirac Lagrangian and Transformation
)()()( xmixxL
if )()()'(' xSx
DE retains its Lorentz-invariant form when:
)2/()(
ieS ,
4
1
i
Spin term (Gen. Of the Lorentz group)
Finite mass particle
Mass: Flux of the grav. Field through a surface enclosing the particle
Massless Weyl Equation
• Setting m=0 in DE Weyl Eq.
• 4-comp. Parity 2-comp. Maximal parityviolation (x2 eq).
• Invariant under the global (chirality) transformation:
5ie
)(12
1)( 5 xx 0)(xt
Two uncoupled Weyl equations (reduced representation). In momentum space => Chirality.
Weyl’s formulation(1)
• Weyl gauge theory: Invariance and rescaling.
• Conformal vs. Weyl gauge symmetry.
• Weyl-Cartan geometry: The length of a vector has no absolutegeometric meaning.
• Conf. Transformation in Riemann space: The mapping isconformal if
)(' xSg
dxdxxgxsdxdxxg )()'('')'(
Weyl’s formulation(2)
• Group Wr in V4 (Riemann space) is defined by:
Weyl (conformal) rescalings of the metric
and transf. All other dynamical variables
• := Weyl dimension of the field.
gegxsxgxg xr )(2)()()(
exsr 2/)]([
Klein Paradox (Chiral Tunneling)
• “…If the potential is on the order of the electron mass, thebarrier is nearly transparent.”
• RQM can be consistently formulated only in terms of fieldsrather than individual particles.
Graphene and Nanotubes
Graphene and Nanotubes (2)
• Electronic structure of graphene: Dirac points.
fviH 0ˆ smv f /106
Graphene and Nanotubes (3)
• New bridge between condensed matterphysics and QED.
• Dirac-like electrons => honecomb structure.
• Quasiparticles in graphene
• Pseudospin degree of freedom (Sublattice)
• e- and e+ (holes) are interconnected.
• Coherent transport in graphene (impuritiesdon’t scatter).
fkvE
Research
• Electric transport through C-nanotubes and graphene nanoribbons double quantum dotscoupled by a superconductor.
Non-equilibrium Green’s functions.
References
• [1] P.A.M. Dirac, The quantum theory of the electron. 1928.
• [2] E. Marshak, Conceptual foundations of modern particlephysics. 1993.
• [3] M. Blagojevic, Gravitation and Gauge symmetries. 2002.
• [4] H. Weyl, Gravitation and the electron. 1929.
• [5] M. Katsnelson, K. Novoselov, Graphene: New bridge between cond. Matt. Phys. And QED. 2007.
• [6] Castro et al. The electronic properties of graphene. 2007.
Tks!