Magnetic Excitations of Graphene in 8-Spinor Realization of Chiral Model

The simplest scalar chiral model of graphene suggested earlier and based on the SU(2) order parameter is generalized by including 8-spinor field as an additional order parameter for the description of spin (magnetic) excitations in graphene. As an illustration we study the interaction of the graphene layer with the external magnetic field. In the case of the magnetic field parallel to the graphene plane the diamagnetic effect is predicted, that is the weakening of the magnetic intensity in the volume of the material. However, for the case of the magnetic field orthogonal to the graphene plane the strengthening of the magnetic intensity is revealed in the central domain (at small r). Thus, the magnetic properties of the graphene prove to be strongly anisotropic.

Since the very discovery of mono-atomic carbon layers called graphenes [1,2] this material attracted deep interest of researchers due to its extraordinary properties concerning magnetism, stiffness and high electric and thermal conductivity [3][4][5].The interesting connection of graphene was revealed with nano-tubes and fullerenes [6].A very simple explanation of these unusual properties of graphene was suggested in [7], where the idea of massless Dirac-like excitations of honeycomb carbon lattice was discussed, the latter one being considered as a superposition of two triangular sub-lattices.Some phenomenological development of this idea was realized in [8,9].
As is well known, the carbon atom possesses of four valence electrons in the socalled hybridized  2 -states, the one of them being "free" in graphene lattice and all others forming -bonds with the neighbors.It appears natural to introduce scalar  0 and 3-vector a fields corresponding to the -orbital and the -orbital states of the "free" electron respectively.These two fields can be combined into the unitary matrix  ∈  (2) considered as the order parameter of the model in question, the long-wave approximation being adopted, i. e.
where  0 is the unit 2 × 2-matrix and  are the three Pauli matrices, with the  (2)condition being imposed.It is convenient to construct via the differentiation of the chiral field (1) the so-called left chiral current the index  running 0, 1, 2, 3 and denoting the derivatives with respect to the time  0 =  and the space coordinates   ,  = 1, 2, 3. Then the simplest Lagrangian density reads and corresponds to the sigma-model approach in the field theory with the mass term.
Here the constant model parameters  and  are introduced.Comparing the Lagrangian density (4) with that of the Landau-Lifshits theory corresponding to the quasiclassical long-wave approximation to the Heisenberg magnetic model [10], one can interpret the parameter  in (4) as the exchange energy between the atoms (per spacing).
Inserting (1) into ( 3) and ( 4) and taking into account the condition (2), one easily finds the following Lagrangian density: For the case of small a-excitations the equations of motion generated by ( 5) read as a − ( 2 /)a = 0 and correspond to the dispersion law , which in the high-frequency approximation has the linear photon-like form.
First we begin with the static 1 configuration corresponding to the ideal graphene plane, the normal being oriented along the -axis.In this case the order parameter has the form  = exp(Θ 3 ), Θ = Θ(), with the Lagrangian density being The Lagrangian (6) yields the equation of motion The solution to (7) satisfying the natural boundary conditions has the well-known kink-like (or domain-wall) form: with the characteristic thickness (length parameter) and the energy per unit area

Spinor Chiral Model of Graphene
Now we intend to include in the model the interaction with the electromagnetic field for the description of conductivity and magnetic properties.To this end, we suggest 8-spinor generalization of the scalar chiral model and use the gauge invariance principle for introducing the electromagnetic interaction.The motivation for such a generalization is the following.
For the description of spin and quasi-spin excitations in graphene, the latter ones corresponding to independent excitation modes of the two triangular sub-lattices of graphene, we introduce the two Dirac spinors  1 ,  2 and consider the combined spinor field Ψ as a new order parameter: where  stands for the first column of the unitary matrix (1).The Lagrangian density of the model contains the projector  =     on the positive energy states, where   = Ψ  ,  = 0, 1, 2, 3, designates the Dirac current, Ψ = Ψ +  0 and   stands for the Dirac matrix.The model contains the two constant parameters of the previous scalar model: the exchange energy  per lattice spacing and some characteristic inverse length √  .The interaction with the electromagnetic field is realized through the extension of the derivative: with  0 > 0 being the coupling constant and Γ  = (1 −  3 )/2 being the charge operator chosen in accordance with the natural boundary condition at infinity:  0 (∞) = 1.However, the additional interaction term of the Pauli type should be added to take into account the proper magnetic moments of the electrons.Here and  0 > 0 denotes the Bohr magneton per lattice spacing cubed.
Let us consider as an illustration the interaction of the mono-atomic carbon layer  = 0 with the static uniform magnetic field B 0 oriented along the  axis.We introduce first the vector potential   = (), with the intensity of the magnetic field being and the natural boundary condition at infinity:  → − 0 .
The model in question admits the evident symmetry  1 ⇔  2 ,  0 -invariance Ψ ⇒  0 Ψ and also the discrete symmetry: Therefore, one can introduce the chiral angle Θ(): and the real 2-spinor () = col(, −), where  1 =  2 = col(, −).As a result the new Lagrangian density takes the form: where the new variable is introduced:  = || 2 = 2 2 .Taking into account that  2 = 16 2 , one can deduce from (12) and the boundary conditions at infinity: the following "energy" integral: , that implies the Hamilton-Jacobi equation for the "action" : Here the following definitions of the Jacobi momentums are used: Let us study the behavior of solution to the equations ( 13) and ( 14) in the asymptotic domain  → ∞, where  ≈ − 0 .In the first approximation one gets: Inserting (15) into (14), one derives the differential equation with the evident integral 4  = cos −4 Θ corresponding to the boundary condition  (∞) = 1/4.In view of ( 14) this fact permits one to obtain the equation for Θ(): with the solution of the form: where Θ 0 stands for the integration constant.Finally, combining (16) and the last relation in ( 14), one can find the magnetic field intensity in the asymptotic domain  → ∞: As can be seen from ( 17), the effect of weakening of the magnetic field is revealed for the positive value of the constant  0  − 2 0 , this effect being similar to that of London "screening" caused by the second term in the electromagnetic current: The current (18) contains beyond the standard conduction term, the diamagnetic current and the Pauli magnetization-polarization one.As follows from (17), for the negative value of the constant  0  − 2 0 the paramagnetic behavior of the material takes place.

Interaction with Magnetic Field Orthogonal to Graphene Plane
Let us now study the case with the orientation of the magnetic field B 0 along the axis.Using the cylindrical coordinates , , , we introduce the vector potential   = , with the intensity of the magnetic field being and the natural boundary condition at infinity being imposed: The model in question admits the evident symmetry  1 ⇔  2 and  0 -invariance Ψ ⇒  0 Ψ, that permits one to introduce 2-spinor  by putting To simplify the calculations, let us suppose the smallness of the radial magnetic field: In this approximation the new discrete symmetry holds: As a result the new Lagrangian density takes the form: where the new variable is introduced:  =  2 and  ⊥ signifies the differentiation with respect to  and .The equations of motion corresponding to (19) read: [ Let us now search for solutions to the equations ( 20), ( 21), ( 22) in the asymptotic domain  → ∞, where Thus, the equation ( 21) takes the form: and its solution can be found by separating variables: with the following constant parameters: Inserting ( 23) into ( 20) and ( 22), one gets the inhomogeneous equations for  and : with the solutions of the form: where the radial functions  () and () satisfy the following equations: Let us now estimate the magnetic intensity: Taking into account that due to (29)  ≈ ( 2 0  2 0 ) −1 as  → ∞, one gets from (27): However, at small  → 0 one finds from (29) that  ≈  3 /8, and therefore the intensity of the magnetic field reads: As can be seen from ( 30)-( 33) , according to the sign of the multiplier  0  − 4 0 our graphene material reveals diamagnetic or paramagnetic behavior.Therefore, it would be interesting to obtain numerical estimates for the parameters of the model.In view of definitions adopted one has where the exchange energy is usually adopted as  exch = 2.9 eV and the lattice spacing as  = 3.56 • 10 −8 cm, with  being the absolute value of the electron charge.Finally, one can find the following numerical values: It means that the parameter  0  − 4 0 is positive and the weakening of the magnetic field inside the graphene is predicted in accordance with (17), ( 30) and (31) for large  and its strengthening for small  in accordance with (32) and (33).
In view of the importance of the latter conclusion it would be desirable to investigate the magnetic field behavior in the central domain of the graphene material, i. e. at small  but arbitrary .To this end, we consider the extrapolation of the configuration (23) to the domain wall structure of the form: Later it will be shown that this approximation is valid in the small field limit  0 → 0. To start with, we insert (34) and  =  0 /2 + ,  ≈ 1/4 into (23), this amounting to the equation: where  1 and  1 stand for the modified Bessel functions of the imaginary argument.
Taking into account their asymptotic behavior as  → 0: