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The 2D Dirac Equation 55 conduction band (λ = +) 0 η = + η = − K (ξ = +) η= − η= + K’ (ξ = −) valence band (λ = −) Figure 3.2: Relation between band index λ, valley isospin ξ, and chirality η in graphene. 3.2.2 Symmetries and Lorentz transformations Chirality for zero-mass particles In high-energy physics, one defines the helicity of a particle as the projection of its spin onto the direction of propagation, hp = p·σ, (3.26) |p| which is a Hermitian and unitary operator with the eigenvalues η = ±, hp|η = ±⟩ = ±|η = ±⟩. (3.27) Notice that σ describes, in this case, the true physical spin of the particle. In the absence of a mass term, the helicity operator commutes with the Dirac Hamiltonian, and the helicity is, therefore, a good quantum number, e.g. in the description of neutrinos, which have approximately zero mass. One finds indeed, in nature, that all neutrinos are “left-handed” (η = −), i.e. their spin is antiparallel to their momentum, whereas all anti-neutrinos are “right-handed” (η = +). For the case of graphene, described in terms of the 2D Dirac equation (3.13), one may use the same definition (3.26) for a Hermitian and unitary operator hp. Here, the Pauli matrices no longer define the true spin, but the sublattice isospin, as described in Sec. 2.2. Because of this difference, one also calls hp the chirality operator. It clearly commutes with the massless momentum energyPDF Image | Physical Properties of Graphene
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