**2. Fundamentals**

Quasi-relativistic graphene model has been derived in [19] as a consequent account of the effect of relativistic exchange interactions, while being grounded on truly secondary quantized relativistic consideration of the problem within the known Dirac-Hartree-Fock self-consistent field approximation. In subsequent publications [23, 27, 28], it has been established that the model admits a form as a Majorana-like system of equations as well as a two-dimensional Dirac-like equation with an additional "Majorana-force correction" term [23]. It reads

$$\left| \boldsymbol{\nu}\_{F} \left[ \overrightarrow{\boldsymbol{\sigma}}\_{\text{2D}}^{\text{AB}} \cdot \overrightarrow{\boldsymbol{p}}\_{\text{BA}} - \boldsymbol{c}^{-1} \boldsymbol{M}\_{\text{BA}} \right] \left| \boldsymbol{\nu}\_{\text{BA}}^{\*} \right. \right. \\ \left. \left. \left. \left. \left. \boldsymbol{\nu} \right|\_{\text{BA}} \right. \right. \right. \right. \tag{1}$$

and the same equation with labels ð Þ *AB*, *BA* exchanged for another sublattice. Here, ð Þ *AB*, *BA* are related to sublattices and refer to the quantities that are obtained by similar transformations with a relativistic exchange matrix Σ*<sup>x</sup> rel*; for example, for the momentum operator *p* ! one gets *p* ! *BA* ¼ Σ*BA p* ! Σ�<sup>1</sup> *BA*; *vF* is the Fermi velocity. The vector of two-dimensional (2D) Pauli matrixes comprises two matrixes *σ*2*<sup>D</sup>* ¼ *σx*, *σ<sup>y</sup>* � �. The term *MBA* ¼ � <sup>1</sup> *cvF* Σ*BA*Σ*AB* is a Majorana-like mass term, where *c* is the speed of light. It turns out to be zero in the Dirac point *K K*<sup>0</sup> ð Þ and gives a very small momentumdependent correction outside of *K K*<sup>0</sup> ð Þ. The relativistic exchange operator for tightbinding approximation and accounting of nearest lattice neighbors is given by its action on secondary quantized wave functions on sublattices *A B*ð Þ of the system [19, 27, 28]

Σ*x rel* ^*χ*�*σ<sup>A</sup>* † *r* !� � ^*χ*† *<sup>σ</sup><sup>B</sup> r* !� � 0 BB@ 1 CCA ∣0, � *σ*i∣0, *σ*i ¼ 0 Σ*AB* Σ*BA* 0 0 @ 1 A ^*χ*† �*σ<sup>A</sup> r* !� � ^*χ*† *<sup>σ</sup><sup>B</sup> r* !� � 0 BB@ 1 CCA ∣0, � *σ*i∣0, *σ*i, (2) Σ*AB*^*χ*† *<sup>σ</sup><sup>B</sup> r* !� �∣0, *<sup>σ</sup>*i ¼ *N* X*v N i*¼1 ð *d r*! *i*^*χ*† *σi <sup>B</sup> r* !� �∣0, *<sup>σ</sup>*iΔ*AB* 0, �*σ<sup>i</sup>* ^*χ*† �*σ<sup>A</sup> i r* ! *i* � �*V r*! *<sup>i</sup>* � *r* ! � �^*χ*�*σ<sup>B</sup> <sup>r</sup>* ! *i* � � � � � � � �0, �*σ<sup>i</sup>* 0 D E, (3) Σ*BA*^*χ*�*σ<sup>A</sup>* † *r* !� �∣0, � *<sup>σ</sup>*i ¼ *N* X*v N i* 0 ¼1 ð *d r*! *i* <sup>0</sup>^*χ*�*σ<sup>A</sup> i* 0 † *r* !� �∣0, � *<sup>σ</sup>*iΔ*BA* 0, *σ<sup>i</sup>* <sup>0</sup> ^*χ*† *σB i*0 *r* ! *i* 0 � �*V r*! *i* <sup>0</sup> � *r* ! � �^*χσ<sup>A</sup> <sup>r</sup>* ! *i* 0 � � � � � � � � � � 0, *σ<sup>i</sup>* � �*:* (4)

Here, interaction (2 � 2)-matrices Δ*AB* and Δ*BA* are gauge fields (or components of a gauge field). Vector-potentials for these gauge fields are introduced by the phases *α*<sup>0</sup> *Anomalous Charge Transport Properties and Band Flattening in Graphene: A Quasi… DOI: http://dx.doi.org/10.5772/intechopen.106144*

and *α*�,*k*, *k* ¼ 1, 2, 3 of *π pz* -electron wave functions *<sup>ψ</sup> pz <sup>r</sup>* ! and *<sup>ψ</sup>pz*,�*<sup>δ</sup>* ! *k r* ! attributed to a given lattice site and its three nearest neighbors (see details in [28]), *V r*! is the three-dimensional (3D) Coulomb potential, and the summation is performed on all lattice sites and number of electrons. The introduction of these three non-Abelian gauge fields was stipulated by a requirement of the reality of eigenvalues of the Hamiltonian operator as gauge conditions. In this case, the operator of relativistic exchange gains an additional implicit *k* ! -dependence upon momentum in the case of nonzero values of gauge fields.
