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Continuum Limit 35 and the Pauli matrices σx=0 1, and 0 −i. 10 i0 Furthermore, we have introduced the valley isospin ξ = ±, where ξ = + denotes the K point at +K and ξ = − the K′ point at −K. The low-energy Hamiltonian (2.33) does not take into account nnn hopping corrections, which are proportional to |γk|2 and, thus, occur only in the second-order expansion of the energy dispersion [at order (|q|a)2]. The energy dispersion (2.22), therefore, reads ǫλq,ξ=± = λh ̄vF |q|, (2.35) independent of the the valley isospin ξ. We have already alluded to this twofold valley degeneracy in Sec. 2.1.3, in the framework of the discussion of the zero-energy states at the BZ corners. The twofold valley degeneracy, thus, survives when considering the low-energy excitations in the vicinity of the Dirac points. From Eq. (2.35) it is apparent that the continuum limit |q|a ≪ 1 coincides with the limit |ǫ| ≪ |t|, as described above, because |ǫq| = 3ta|q|/2 ≪ |t|. It is convenient to invert the spinor components at the K′ point (for ξ = −), Ψ = k,+ , Ψ = k,− , ψA k,ξ=+ ψB ψB k,ξ=− ψA k,− i.e. to invert the role of the two sublattices. In this case, the effective low- k,+ energy Hamiltonian may be represented as Heff,ξ =ξh ̄v (q σx +q σy)=h ̄v τz ⊗q·σ, qFxyF where we have introduced the four-spinor representation ψA k,+ ψB Ψ= k,+ k ψB k,− ψA k,− τz⊗σ=σ 0 , 0 −σ (2.36) in the last line via the 4 × 4 matricesPDF Image | Physical Properties of Graphene
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