Basics of quantum field theory of electromagnetic interaction processes in single-layer graphene

The content of this work is the study of electromagnetic interaction in single-layer graphene by means of the perturbation theory. The interaction of electromagnetic field with Dirac fermions in single-layer graphene has a peculiarity: Dirac fermions in graphene interact not only with the electromagnetic wave propagating within the graphene sheet, but also with electromagnetic field propagating from a location outside the graphene sheet and illuminating this sheet. The interaction Hamiltonian of the system comprising electromagnetic field and Dirac fermions fields contains the limits at graphene plane of electromagnetic field vector and scalar potentials which can be shortly called boundary electromagnetic field. The study of S-matrix requires knowing the limits at graphene plane of 2-point Green functions of electromagnetic field which also can be shortly called boundary 2-point Green functions of electromagnetic field. As the first example of the application of perturbation theory, the second order terms in the perturbative expansions of boundary 2-point Green functions of electromagnetic field as well as of 2-point Green functions of Dirac fermion fields are explicitly derived. Further extension of the application of perturbation theory is also discussed.


Introduction
Soon after the discovery of graphene by Geim and Novoselov [1][2][3][4], the research on graphene rapidly developed and became a wide interdisciplinary area of science and technology. It was shown [5] that even in the case when the electron spin plays no role, its quantum states are still described by two-component wave functions satisfying differential wave equations similar to relativistic Dirac equation for a massless particles in (2 + 1)-dimensional Minkowski space-time. Therefore the charge carriers in graphene are called Dirac fermions.
Denote K and K′ two nearest corners of the first Brillouin zone in the reciprocal lattice of the hexagonal crystalline structure of a graphene monolayer. They are called Dirac points. In the framework of the quantum field theory the spinless fermions in graphene are described by twocomponent quantum fields ( ) y t r, K and ( ) y ¢ t r, , Each of them can be considered as a spinor field of a new SU(2) symmetry group similar to the isospinors in theory of elementary particles [6][7][8][9]. Thus the two-component fields ( ) y t r, τ i , i = 1, 2, 3. It was shown [5] where v F is the speed of Dirac fermions.
In the study of the interaction between Dirac fermions and electromagnetic field we must consider electromagnetic field in the physical three-dimensional space. Let us chose to use the Cartesian coordinate system as follows: the plane of graphene monolayer is the xOy coordinate plane and, therefore, the Oz axis is perpendicular to this plane. Then the coordinate of a point in the physical three-dimensional space is denoted {r, z} = {x, y, z}. The electromagnetic field is described by the vector potential A(r, z, t) and scalar potential field j(r, z, t). From formula (1) it follows that the interaction Hamiltonian of the system of Dirac fermion fields and electromagnetic field has the expression The function A(r, o, t) and j(r, o, t) of variable r and t are vector field A(r, t) and scalar field j(r, t) on the graphene plane: Since they are the limits of the vector potential A(r, z, t) and the scalar potential j(r, z, t) of the electromagnetic field when the point {r, z} tends to the limit {r, o} in the xOy coordinate plane, which is the boundary of the upper or lower half-space above or under the single-layer graphene plane, we shortly call them vector potential and scalar potential of the boundary electromagnetic field on the graphene plane. In terms of A(r, t) and j(r, t) the interaction Hamiltonian (4) has the expression The charge current density J(r, t) and charge density ρ(r, t) are expressed in terms of H int (t) by the definition Using Dirac equations derived from Hamiltonian (1) we can demonstrate that charge density ρ(r, t) and charge current density J(r, t) satisfy well-known continuity equation Recently there arose a significant attention to the study of electromagnetic interaction processes in graphene such as nonlinear optical processes [10] and plasmon resonance [11][12][13][14][15][16][17]. The purpose of present work is to elaborate the basics of quantum field theory of electromagnetic interaction processes in the singlelayer graphene starting from the interaction Hamiltonian (4).
It was well-known that the most popular approach for the theoretical study of dynamical processes in any quantum system with a given interaction Hamiltonian H int (t) is to work in the interaction picture in which the field operators satisfy the Heisenberg quantum equation of motion of the free fields and, therefore, have the same expressions in terms of the destruction and creation operators of their quanta as those of the corresponding free fields. This special feature of the interaction picture permits to establish exact and clearly formulated mathematical rules in the calculation of physical quantities of the quantum system. These quantities are expressed in terms of interaction Hamiltonian (4).
Since we have chosen to work in the interaction picture, for the study of electromagnetic interaction processes in singlelayer graphene we must use the expressions determining the boundary free electromagnetic field on graphene as well as the boundary limits on the graphene plane of Green functions of the free electromagnetic field, which also briefly called the boundary Green functions of free electromagnetic field on graphene plane. The boundary electromagnetic field and the 2-point boundary Green function of free electromagnetic field are investigated in the subsequent section 2. In section 3 the 2-point Green functions of boundary electromagnetic field on the graphene plane and Dirac fermion fields of the interacting system of electromagnetic field and Dirac fermions in graphene are studied by means of the perturbation theory. The conclusion and discussion are presented in section 4.

Boundary free electromagnetic field and boundary 2-point Green functions of free electromagnetic field
In order to study electromagnetic interaction processes in graphene it is necessary to use explicit expressions of vector and scalar potentials A(r, t) and j(r, t) of the boundary free electromagnetic field as well as the boundary Green functions of free electromagnetic field on the graphene plane, and also Green functions of free Dirac fermion fields. The laters were studied in [18] and we shall use the results of this work. In the present section we study vector and scalar potentials A(r, t) and j(r, t), and boundary 2-point Green function of free electromagnetic field. The study of boundary 2n-point Green functions of electromagnetic field with n > 1 by means of the perturbation theory will be carried out in section 3.
Vector field A(r, t) and complex scalar field ij(r, t) are the components of the limit at z → 0 of a 4-vector field A μ (x) in the (3 + 1)-dimensional Minkowski space-time: For simplifying equations and calculations in classical electrodynamics [19] one frequently imposes on A μ (x) the Lorentz condition However, in quantum electrodynamics (QED) this condition cannot hold for the quantum vector field A μ (x). Instead of condition (10) it was reasonably proposed to assume another similar but weaker condition imposed on the state vectors |F ñ 1 and |F ñ 2 of all physical states of the system: In the fundamental research works on QED [20,21] it was demonstrated that due to the weak Lorentz condition (11) the electromagnetic waves in the states with longitudinal and scalar polarizations play no role in all physical processes. Therefore in the Hilbert space of state vectors of all physical states of the system the vector potential field A(x) = A(r, z, t) has following effective Fourier expansion formula It looks like a linear combination of an un-numerable set of vector quantum fields A(r, t) l , each of them being labeled by a value of the continuous index l: , e The vector field A(r, t) l with an index l ≠ 0 looks like a massive free vector field with the mass | | l in the (2 + 1)dimensional Minkowski space-time.
Note that matrix elements of scalar field j(r, t) of boundary electromagnetic field between two state vectors |F ñ 1 and |F ñ 2 of any pairs of two physical states of the system always vanish. Therefore there is no necessity to write the explicit expression of j(r, t).
The 2-point Green function of free electromagnetic field at T = 0 is defined as follows: where the symbol á ñ  denote the average of inserted expression (containing field operators) in the ground state | ñ 0 of the Dirac fermion gas This ground state can be considered as the vacuum of electromagnetic field.
In QED it was shown that due to the gauge invariance of the theory one always can chose to work in such a gauge that the boundary limit of the 2-point Green function (16) of free electromagnetic field has following simple formula [20,21] ( ) Thus formulae (14) and (

Perturbation theory
In the present section we develop perturbation theory for studying electromagnetic interaction processes in single-layer graphene. The S-matrix is expressed in terms of interaction Hamiltonian (4) as follows where the integration with respect to the time variable t is performed over the whole real axis from −∞ to +∞. By expanding the exponential function in rhs of formula (22) into power series, we express S-matrix in the form of a series ( ) As the first example of the application of perturbation theory let us study boundary 2-point Green functions of the interacting system comprising electromagnetic field in the whole three-dimensional physical space and the Dirac fermions moving only in the graphene plane. They are expressed in terms of the boundary limits at z → 0 of the vector potential field A(r, z, t) and scalar potential field j(r, z, t) and S-matrix as follows: Using expansion formula (23) of S-matrix, we write each of 2-point Green functions (26)-(28) in the form of the series: n running all non-negative integers n = 0, 1, 2 K In the present work we consider the simple case of the Dirac fermion gas at T = 0 with the Fermi level E F = 0. The extension to other more general cases will be done in subsequent works. The first term in the series (29) is a special case of function determined by formulae (18) and (19): The first term in the series (30) can be directly obtained also from formulae (18) and (19): The expressions of 2-point Green functions of free Dirac fermion fields can be easily obtained from results demonstrated in [18]:   Using formula (4) of the interaction Hamiltonian H int (t), we rewrite relation (42) in the form explicitly containing all quantum field operators of electromagnetic field and Dirac fermion fields:   According to formulae (26), in order to find ( ) ( ) -¢ -¢ D t t r r , ij 2 it is necessary to calculate also the average of 2nd order terms S (2) of the S-matrix. We have Using formula (4) of the interaction Hamiltonian H int (t) we rewrite this matrix element in the form explicitly containing field operators   In order to demonstrate the calculation method let us consider in detail the first term in rhs of equation (43). According to the well-known Wick theorem for the average of any product of quantum free fields in the ground state of the system we have   It is obvious that the second term in rhs of equation (43) vanishes. The third term comprises following expression and a similar one with the replacement K → K′.
The first term in rhs of formula (45) contains following expression and a similar one with the replacement K → K′, τ n → * t , The second term vanishes. The third term contains following expressions  According to the definition (26) we have In order to calculate ( ) ( ) -¢ -¢ D t t r r , 00 2 we must consider matrix element   ,  ,  i  2  d  d  d  d   ,  , , According to the definition (27) we have Let us calculate the first matrix element in rhs of relation (55)   Consider first term in rhs of equation (56). It contains expression  Second term vanishes and third term contains expression Using formulae (45), (49) and (50) and similar ones for determining ( ) á ñ S 2 together with relations (56)-(58), we obtain  The first matrix element in rhs equation (61)