Theory of Green functions of free Dirac fermions in graphene

This work is the beginning of our research on graphene quantum electrodynamics (GQED), based on the application of the methods of traditional quantum field theory to the study of the interacting system of quantized electromagnetic field and Dirac fermions in single-layer graphene. After a brief review of the known results concerning the lattice and electronic structures of single-layer graphene we perform the construction of the quantum fields of free Dirac fermions and the establishment of the corresponding Heisenberg quantum equations of these fields. We then elaborate the theory of Green functions of Dirac fermions in a free Dirac fermion gas at vanishing absolute temperature T = 0, the theory of Matsubara temperature Green functions and the Keldysh theory of non-equilibrium Green functions.


Introduction
In the comprehensive review [1] on the rise of graphene as the emergence of a new bright star 'on the horizon of materials science and condensed matter physics', Geim and Novoselov have remarked exactly that, as a strictly two-dimensional (2D) material, graphene 'has already revealed a cornucopia of new physics'. It is the physics of graphene and graphene-based nanosystems, including graphene quantum electrodynamics (GQED). In the language of another work by Novoselov et al [2], GQED ('resulting from the merger' of the traditional quantum field theory with the dynamics of Dirac fermions in graphene) would 'provide a clear understanding' and a powerful theoretical tool for the investigation of a huge class of physical processes and phenomena talking place in the rich world of graphene-based nanosystems and their electromagnetic interaction processes. This work is the first step in the establishment of the basics of graphene quantum electrodynamics: the construction of the theory of Green functions of free Dirac fermions in graphene.
Since throughout the present work we often use knowledge of the lattice structure of graphene as well as expressions of the wave functions of Dirac fermions with the wave vectors near the corners of the Brillouin zones of the graphene lattice, first we present a brief review of this knowledge in section 2. In the subsequent section 3, the explicit expressions of the quantum field of free Dirac fermions in graphene and the corresponding Heisenberg quantum equations of motion are established. Section 4 is devoted to the study of Green functions of Dirac fermions in a free Dirac fermion gas at vanishing absolute temperature T=0. The theory of Matsubara temperature Green functions of free Dirac fermions is presented in section 5, and the content of section 6 is the Keldysh theory of nonequilibrium Green functions. The conclusion and discussions are presented in section 7. The unit system with  = = c 1 will be used.

Definitions and notations
According to the review [3] on the electronic properties of graphene, each graphene single layer is a 2D lattice of carbon atoms with the hexagonal structure presented in figure 1(a). It consists of two interpenetrating triangular sublattices with the lattice vectors where a is the distance between the two nearest carbon atoms a≈1.42. The reciprocal lattice has the following lattice vectors p p = = a a k k 2 3 1, 3 , 2 3 1, 3 . ( ) The first Brillouin zone (BZ) is presented in figure 1(b). Two inequivalent corners K and K′ with the coordinate vectors

Quantum field of free Dirac fermions
In order to establish explicit expressions of the quantum field of free Dirac fermions it is necessary to have formulae of the wave functions of these quasiparticles. Denote F ¢ r E K K k, , ( ) the wave function of the state with the wave vector k near the Dirac points K or K´and the energy E. It was known that , ( ) are the solutions of the 2D Dirac equations where two components τ 1 and τ 2 of vector matrix τ are two matrices (7) and (8) both have two solutions corresponding to two eigenvalues η and η′ are two arbitrary phase factors h h = ¢ = 1. | | | | The quantum field of free Dirac fermions in the hexagonal graphene lattice has the expression is the destruction operator of the Dirac fermion with the wave function being the plane wave whose wave vector k satisfies the periodic boundary condition for a very large square graphene lattice containing N c elementary cells.
Note that the role of the electron spin was omitted and electrons are considered as the spinless fermions. Twocomponent wave functions (11) and (12) are not the usual spinors (Pauli spinors) in the three-dimensional (3D) physical space with the Cartesian coordinate system. Being the spinors with respect to the rotations in some fictive 3D Euclidean space, they are similar to the isospinor called nucleon N with proton p and neutron n as its two components = N p n ( ) in nuclear physics [4] and elementary particle physics [5][6][7][8].
In order to distinguish the spinors (11) and (12) from the usual Pauli spinors let us call them Dirac spinors, quasispinors or pseudo-spinors. It is worth investigating the symmetry with respect to the rotations in the abovementioned fictive 3D Euclidean space. The Hamiltonian of the quantum field of free Dirac fermions is From the expansion formula (15) and the canonical anticommutation relations between destruction and creation operators n can be rewritten in the form of the Heisenberg quantum equation of motion Consider now the free Dirac fermion gas at vanishing absolute temperature T=0. In this case it is convenient to work in the electron hole formalism. Denote E F the Fermi level and ñ G | the state vector of the ground state of the Dirac fermion gas in which all levels with energies larger than E F are empty and all those with energies less than E F are fully occupied. The ground state ñ G | is expressed in terms of the Dirac fermion creation operators and the state vector ñ 0 | of the vacuum With respect to the ground state ñ G | the destruction/ creation operator n ) of the Dirac fermion with energy less than E F becomes the creation/destruction operator of the Dirac hole in the corresponding state with the momentum and energy which will be specified in each separate case. Since the reasonings for the states with wave vectors k near K and K′ are the same, until the end of this section we shall omit the indices K and K′ in the notations of field operators, destruction and creation operators as well as of the wave functions for simplifying the formulae.
All states with energies > are empty and for them we set are occupied and for them we set All states with energies -E k ( ) are occupied and for them we set In this case we obtain å q q Y = - All states with energies + E k ( ) are empty and for them we set are also empty and for them we set are occupied and for them we set In this case we obtain in the case 1 with E F =0, å q q ¢ = -+ -

Green functions of Dirac fermions in the free Dirac fermion gas at T=0
Green functions of Dirac fermions in the free Dirac fermion gas at T=0 are defined by the following formulae Using the Heisenberg quantum equation of motion (25) as well as the equal-time canonical anticommutation relations between the quantum field operators Y a ¢ t r, ) we derive the following inhomogeneous differential equations for these Green functions Introduce the Fourier transformation of Green functions (29) and (30) It is straightforward to derive the expressions of w D ab k, ( ) in all three cases. In the first case with E F =0 we obtain

Matsubara temperature Green functions of Dirac fermions in the free Dirac fermion gas
Let us study the free Dirac fermion gas in the equilibrium state at a non-vanishing temperature T temp . Instead of formulae (29) and (30) now we have the following definition of Green functions of Dirac fermions: Following Matsuraba [9] and Abrikosor et al [10] we consider t as an imaginary variable and set t=−iτ, where τ is a real variable. Instead of t-dependent field operators (40) and (41) we introduce corresponding τ-dependent ones

They obey the Heisenberg quantum equation of motion
From this common form it is easy to derive concrete forms of the differential equations for different fields t Y a r, , The Matsubara temperature Green functions of Dirac fermions are defined by the following formula and a similar one obtained from this formula after the replacement  ¢ K K , where Tτ denotes the operation of ordering the product of operators along the decreasing direction of the real variable τ (the 'chronological product' with respect to the real 'time' variable), for example Similarly, in the case 3 with E F <0 the result is

Keldysh non-equilibrium Green functions of Dirac fermions in the free Dirac fermion gas
With the purpose of extending the Green function theory for application to the study of non-equilibrium physical processes and phenomena in quantum systems, Keldysh [11] has developed the theory of Green functions of quantum fields depending on the complex time z=t+iτ, where t and τ are the real and imaginary components of z. These new Green functions were briefly called non-equilibrium Green functions. In the definition of Green functions of complex timedependent field operators it was proposed to define the 'extended chronological ordering' T C of two complex variables z and z′ as the ordering along some contourC passing through these two points in the complex plane. Thus the Keldysh non-equilibrium Green functions of Dirac fermions in the free Dirac fermion gas are defined as follows [12][13][14]: From this common form it is easy to derive concrete forms of the differential equations for different fields Y a z r, , For the application of Keldysh non-equilibrium Green functions to the study of physical quantum processes and phenomena it is convenient to choose the contourC to consist of four parts ] to all physical observables are negligibly small, because of its vanishing length. Therefore this segment plays no role, and the contour C can be considered to consist of only three parts C 1 , C 2 and C 3 . When both variables z and z′ belong to the line C 1 , the functions (60) and (61) are the quantum statistical average of the usual chronological products of the quantum field operators Y a ¢ t r, ( ) in the Heisenberg picture over a statistical ensemble. When both variables z and z′ belong to the line C 3 , the functions (60) and (61) are reduced to the Matsubara temperature Green function.
In the study of stationary physical processes one often used the complex time-dependent Green functions of the form (60) and (61) in the limit  -¥ t .

0
Because the interaction must satisfy the 'adiabatic hypothesis' and vanish at this limit, the segment C 3 also gives no contribution. In this case the contourC can be considered to consist of only two lines C 1 and C 2 , and each of the complex time-dependent Green functions (60) and (61) effectively becomes a set of four functions of real variables t and t′. For example, Green function (60) is equivalent to the set four functions They satisfy following differential equations:

Conclusion and discussions
In this paper we have presented the theory of the three most commonly used types of Green functions of Dirac fermions in a free Dirac fermion gas of single-layer graphene: real-time Green functions at vanishing absolute temperature T=0, imaginary-time Matsubara temperature Green functions and complex-time Keldysh non-equilibrium Green functions. In all three cases the expressions of corresponding Green functions were explicitly established. In the theoretical study of all quantum dynamical processes taking place in single-layer graphene it is necessary to use the expressions of corresponding Green functions established in the present work. However, for the comprehensive study of quantum dynamical processes with the participation of Dirac fermions in graphene it remains to study the interaction of Dirac fermions with photons as well as with phonons. We shall continue to study these topics. In particular, in the subsequent work we shall elaborate the quantum field theory of the interaction between the quantized electromagnetic field and the Dirac fermions in graphene, the second step in the establishment of the basics of GQED.