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Vietnam Academy of Science and Technology

Advances in Natural Sciences: Nanoscience and Nanotechnology

Adv. Nat. Sci.: Nanosci. Nanotechnol. 7 (2016) 015013 (11pp)

doi:10.1088/2043-6262/7/1/015013

Theory of Green functions of free Dirac
fermions in graphene
Van Hieu Nguyen1,2, Bich Ha Nguyen1,2 and Ngoc Dung Dinh1
1

Advanced Center of Physics and Institute of Materials Science, Vietnam Academy of Science and
Technology, 18 Hoang Quoc Viet, Cau Giay District, Hanoi, Vietnam
2
University of Engineering and Technology, Vietnam National University, 144 Xuan Thuy, Cau Giay
District, Hanoi, Vietnam
E-mail:
Received 10 November 2015
Accepted for publication 8 December 2015
Published 12 February 2016
Abstract

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.
Keywords: Dirac fermions, Heisenberg quantum equation of motions, Green functions
Classification numbers: 2.01, 3.00, 5.15

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.

1. 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
Original content from this work may be used under the terms
of the Creative Commons Attribution 3.0 licence. Any
further distribution of this work must maintain attribution to the author(s) and
the title of the work, journal citation and DOI.
2043-6262/16/015013+11$33.00

1

© 2016 Vietnam Academy of Science & Technology


Adv. Nat. Sci.: Nanosci. Nanotechnol. 7 (2016) 015013

V H Nguyen et al

Figure 1. Lattice structure (a) and the first Brillouin zone (b) of graphene.

2. 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
a
a
l1 = (3, 3 ) , l2 = (3, - 3 )
(1 )
2
2
where a is the distance between the two nearest carbon atoms
a≈1.42. The reciprocal lattice has the following lattice vectors
k1 =

2p
(1,
3a

3 ) , k2 =

2p
(1, - 3 ) .
3a

(2 )

Vectors li and ki satisfy the condition
ki l j = 2pdij .

Figure 2. Hexagonal lattice of the corners of all BZs in the reciprocal

lattice.

(3 )

The first Brillouin zone (BZ) is presented in figure 1(b).
Two inequivalent corners K and K′ with the coordinate vectors
K=

3p ⎛
1 ⎞
3p ⎛
1 ⎞
⎜1,
⎟ , K¢ =
⎜1, ⎟
2a ⎝
2a ⎝
3⎠
3⎠

(4 )

are called the Dirac points. Each of them is the common
vertex of two consecutive cone-like energy bands of Dirac
fermions.
The corners of all BZs in the reciprocal lattice form a new
hexagonal lattice of the points equivalent to the Dirac points
K and K′ in the first BZ (figure 2). This new hexagonal lattice
also consists of two interpenetrating triangular sublattices
with the lattice vectors
l1¢ =

3p
3p ⎛ 1
3⎞
(1, 0) , l ¢2 =
⎟⋅
⎜ ,
a
a ⎝2 2 ⎠

(5 )

As an example let us consider the sublattice of all points
equivalent to the corner K. They form a triangular lattice with
the natural parallelogram elementary cell drawn in the left
part of figure 3. For avoiding the presence of four equivalent
corners in each natural parallelogram elementary cell, in the
sequel we shall use the symmetric Wigner–Seitz elementary
cell drawn in the right part of figure 3 instead of the parallelogram one. The wave vector k is called to be near the
corner K if it is contained inside the symmetric Wigner–Seitz

Figure 3. Natural parallelogram elementary cell (left part) and

symmetric Wigner–Seitz elementary cell (right part) in the triangular
lattice of the points equivalent to the Dirac point K in the reciprocal
lattice.

elementary cell around this corner. With respect to the sublattice of all points equivalent to the corner K´ we also have a
similar result. We chose the length unit such that the area of
elementary cell is equal to 1.

2


Adv. Nat. Sci.: Nanosci. Nanotechnol. 7 (2016) 015013

V H Nguyen et al

3. Quantum field of free Dirac fermions

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

In order to establish explicit expressions of the quantum field
of free Dirac fermions it is necessary to have formulae of the
K,K ¢
wave functions of these quasiparticles. Denote Fk,
E (r) 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
K
⎧
iKr j K (r) ,
⎪ F k, E (r ) = e
k, E
⎨
¢
¢
K

i
K
r
K¢
⎪F
⎩ k, E (r ) = e j k, E (r ) ,

N=

in nuclear physics [4] and elementary particle physics [5–8].
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

(6 )

K,K ¢
where j k,
are the solutions of the 2D Dirac equations
E (r)

v F ( - it) j kK, E (r) = Ej kK, E (r) ,

(7 )

v F ( - it⁎) j kK,¢E (r) = Ej kK,¢E (r) ,

(8 )

where two components τ1 and τ2 of vector matrix τ are two
matrices

H0 = v F

( )

E(k ) =  v F k ,
(9 )
i

and two eigenfunctions
j kK,,EK¢(k ) (r) = eikruK , K ¢ (k) ,
uK =
uK ¢

1
2

⎛ e-iq (k) 2 ⎞
⎜ iq (k) 2 ⎟ h ,
⎝e
⎠

1 ⎛ eiq (k) 2 ⎞
=
⎜
⎟ h¢,
2 ⎝ e-iq (k) 2 ⎠

(10)

⎛k ⎞
q (k) = arctg ⎜ 1 ⎟ ,
⎝ k2 ⎠

(11)
i

¶Y K ¢ (r , t )
= v F ( - it⁎) Y K ¢ (r , t ) ,
¶t

(18)

K,K ¢

(19)

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 EF the Fermi
level and | Gñ the state vector of the ground state of the Dirac
fermion gas in which all levels with energies larger than EF
are empty and all those with energies less than EF are fully
occupied. The ground state | Gñ is expressed in terms of the
Dirac fermion creation operators and the state vector | 0ñ of the
vacuum

(13)

Y (r , t ) = eiKrY K (r , t ) + eiK ¢ r Y K ¢ (r , t )

¶Y K , K ¢ (r , t )
= - [H0, Y K , K ¢ (r , t )] .
¶t

(12)

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

Y K , K ¢ (r , t ) =

(17)

can be rewritten in the form of the Heisenberg quantum
equation of motion

where

with the following expansion of Y

¶Y K (r , t )
= v F ( - it) Y K (r , t ) ,
¶t

i

k12 + k 22 ,

(16)

From the expansion formula (15) and the canonical
anticommutation relations between destruction and creation
operators akKn, K ¢ and (akKn, K ¢ )+ it follows that Dirac equations

Equations (7) and (8) both have two solutions corresponding to two eigenvalues
k = | k| =

ò dr {YK (r, t )+(-it) YK (r, t )

+ Y K ¢ (r , t )+( - it⁎) Y K ¢ (r , t )}.

t1 = 0 1 , t2 = 0 -i ⋅
i 0
1 0

( )

( np)

| Gñ =

(14)

å

å

En (k)  EF En (k ¢)  EF

(r, t ):

1
å å ei [kr- En (k ) t] unK ,K ¢ (k) akKn,K ¢, (15)
Nc k n =

(akKn )+ (akK¢ n¢ ¢)+ | 0ñ .

(20)

With respect to the ground state | Gñ the destruction/
creation operator akKn, K ¢ (akKn, K ¢ )+ of the Dirac fermion with
energy less than EF 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.

where akKn, K ¢ 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 Nc 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
3


Adv. Nat. Sci.: Nanosci. Nanotechnol. 7 (2016) 015013

V H Nguyen et al

Figure 4. Energy bands when (a) EF=0, (b) EF>0 and (c) EF<0.

These are three different cases depending on the position
of the Fermi level EF (figure 4).

Case 3: EF<0 (figure 4(c))

Case 1: EF=0 (figure 4(a))

All states with energies E+(k ) are empty and for them we

All levels with energies E+(k ) are empty and all those
with energies E-(k ) are occupied. We set

set
E+(k ) = EF + Eh(2) (k ) ,

E+(k ) = Ee (k ) , u+(k) = u (k) , ak + = ak ,
E-(k ) = - Eh (k ) , u-(k) = v ( - k) , ak - = b-+k

ak + = ak(2),

and obtain
Y (r , t ) =
+

All states with energies E-(k ) > EF are also empty and
for them we set

1
å {ei [kr- Ee (k ) t] u (k) ak
Nc k
e-i [kr- Eh (k ) t ]

v (k) b k+}⋅

u+(k) = u(2) (k) .

E-(k ) = EF + Ee(1) (k ) ,
ak - = ak(1),

(21)

u-(k) = u(1) (k) .

All states with energies E-(k ) < EF are occupied and for
them we set

Case 2: EF>0 (figure 4(b))
All states with energies E+(k ) > EF are empty and for
them we set

E-(k ) = EF - Eh (k ) ,
ak - = b-+k , u-(k) = v ( - k) .

E+(k ) = EF + Ee (k ) ,
ak + = ak , u+(k) = u (k) .

In this case we obtain

All states with energies E+(k ) < EF are occupied and for
them we set

(2 )
1
{ei [kr- Ee (k ) t ] u(2) (k) ak(2)
å
Nc k

eiEF t Y (r , t ) =

+ q [E-(k ) - EF ] ei [kr- Ee

(1 )

E+(k ) = EF - Eh(1) (k ) ,
ak + = b-(1k) + ,

+ q [EF -

u+(k) = v (1) ( - k) .

(k ) t ] u(1) (k) a (1)
k
E-(k )] e-i [kr- Eh (k ) t ] v (k) b k+}⋅

(23)

All states with energies E-(k ) are occupied and for them
we set
E-(k ) =
ak - = b-(2k) + ,

Instead of the quantum fields Y (r, t ) we use the new
ones

EF - Eh(2) (k ) ,
u-(k) = v (2) ( - k) .

ˆ (r , t ) = eiEF t Y (r , t )⋅
Y

From formulae (20)–(23) it follows that the new fields
(24) satisfy the new Heisenberg quantum equation of motion

In this case we obtain
eiEF t Y (r , t )
1
=
å {q [E+(k ) - EF ] ei [kr- Ee (k ) t] u (k) ak
Nc k
(k ) t ] v (1) (k) b (1) +

k
(2 )
e-i [kr- Eh (k ) t ] v (2) (k) b k(2) +}⋅

i

ˆ (r , t )
¶Y
ˆ (r , t )]
= - [H0¢, Y
¶t

(25)

å {Ee (k) ak+ak + Eh (k) b k+b k},

(26)

where

+ q [EF - E+(k )] e-i [kr- Eh

(1 )

+

(24)

H0¢ =

(22)

k

4


Adv. Nat. Sci.: Nanosci. Nanotechnol. 7 (2016) 015013

V H Nguyen et al

in the case 1 with EF=0,
H0¢ = å {q [E+(k ) -

Introduce the Fourier transformation of Green functions
(29) and (30)

EF ] Ee (k ) ak+ak

k

Dab (r , t ) =

+ q [EF - E+(k )] Eh(1) (k ) b k(1) +b k(1)
+ Eh(2) (k ) b k(2) +b k(2)}

k

E-(k )] Eh (k ) b k+b k}

(28)

in the case 3 with EF<0.

(34)

In the second case with EF>0 we have
˜ ab (k, w ) = q [E+(k ) - EF ] ua (k) ub (k)*
D
w - Ee (k ) + i0

4. Green functions of Dirac fermions in the free Dirac
fermion gas at T=0

+

ˆ aK (r , t ) Y
ˆ bK (r¢ , t ¢)+]| G ñ ,
- r¢ , t - t ¢) = - i á G | T [Y
(29)

w + Eh(1) (k ) - i0

va(2) ( - k) vb(2) ( - k)*
w + Eh(2) (k ) - i0

,

(35)

while in the third case with EF<0

and
K¢

va(1) ( - k) vb(1) ( - k)*

+ q [EF - E+(k )]

Green functions of Dirac fermions in the free Dirac fermion
gas at T=0 are defined by the following formulae

˜ ab (k, w ) =
D

K¢

K¢
ˆ a (r , t ) Y
ˆ b (r¢ , t ¢)+]| G ñ .
Dab
(r - r¢ , t - t ¢) = - i á G | T [Y
(30)

ua(2) (k) ub(2) (k)*
w - Ee(2) (k ) + i0

+ q [E-(k ) - EF ]

ua(1) ( - k) ub(1) ( - k)*

w - Ee(1) (k ) + i0
va ( - k) vb ( - k)*
+ q [EF - E-(k )]
⋅
w + Eh (k ) - i0

Using the Heisenberg quantum equation of motion (25)
as well as the equal-time canonical anticommutation relations
between the quantum field operators Y aK , K ¢ (r, t ) and
Y bK , K ¢ (r¢ , t ¢), we derive the following inhomogeneous differential equations for these Green functions

(36)

5. Matsubara temperature Green functions of Dirac
fermions in the free Dirac fermion gas

¶ K
K
Dab (r - r¢ , t - t ¢) - v F ( - it )ag D gb
(r - r¢ , t - t ¢)
¶t
= dab d (t - t ¢) d (r - r¢) ,
(31)

Let us study the free Dirac fermion gas in the equilibrium
state at a non-vanishing temperature Ttemp. Instead of formulae (29) and (30) now we have the following definition of
Green functions of Dirac fermions:

and

i

ua (k) ub (k)*
va ( - k) vb ( - k)*
+
⋅
w - Ee (k ) + i0
w + Eh (k ) - i0

˜ ab (k, w ) =
D

+ q [E-(k ) - EF ] Ee(1) (k ) ak(1) +ak(1)

i

(33)

It is straightforward to derive the expressions of
˜ ab (k, w ) in all three cases. In the first case with EF=0 we
D
obtain

H0¢ = å {Ee(2) (k ) ak(2) +ak(2)

K
Dab
(r

ò dwD˜ ab (k, w).

(27)

in the case 2 with EF>0, and

+ q [EF -

1
1
åeikr 2p
Nc k

¶ K¢
K
Dab (r - r¢ , t - t ¢) - v F ( - it ⁎)ag D gb
(r - r¢ , t - t ¢)
¶t
= dab d (t - t ¢) d (r - r¢) .

K
Dab
(r - r¢ , t - t ¢)

(32)

= -i

Explicit expressions of Green functions (29) and (30)
depend on the position of the Fermi level EF. For simplifying
formulae let us omit again the indices K and K′ until the end

of this section. Depending on the value of EF there exist three
different cases. In the first case with EF=0 the operator
Yˆ a (r, t ) is expressed in terms of the components ua (k) and
va (k) by means of formula (21), in the second case with
EF>0 it is expressed in terms of the components ua (k),
va(1) (k) and va(2) (k) by means of formula (22), while in the third
case with EF <0 it is expressed in terms of the components
ua(2) (k), ua(1) (k) and va (k) by means of formula (23).

¢
ˆ aK (r , t ) Y
ˆ bK (r¢ , t ¢)+]}
Tr {e-b T H0 T [Y

(37)

¢

Tr {e-b T H0 }

and
K¢
Dab
(r - r¢ , t - t ¢)

= -i

ˆ aK ¢ (r , t ) Y
ˆ bK ¢ (r¢ , t ¢)+]}
Tr {e-b T H ¢0 T [Y

Tr {e-b T H ¢0 }

,

(38)

where
bT =

5

1
,
k B Ttemp

(39)


Adv. Nat. Sci.: Nanosci. Nanotechnol. 7 (2016) 015013

V H Nguyen et al

and kB is the Boltzmann constant. Note that formula (38) can
be obtained from formula (37) by means of the replacement
K,K ¢
K,K ¢
K  K ¢ . Field operators Yˆ a (r, t ) and Yˆ a (r, t )+ are
obtained from the corresponding operators at t=0 by means
of the action of the time translation operator eitH0 , namely
ˆ K , K ¢ (r , t ) = eitH0¢ Y K , K ¢ (r , 0) e-itH0¢

Y

direction of the real variable τ (the ‘chronological product’
with respect to the real ‘time’ variable), for example
¯ bK (r¢ , t ¢)M ]
Tt [Y aK (r , t )M Y
K

´ (r¢ , t ¢)M Y aK (r , t )M .

(40)

From homogeneous differential equations (45) and (46)
for the field operators YaK (r, t )M and Y aK ¢ (r, t )M it follows
that corresponding inhomogeneous differential equations
K¢
K
for the Green functions Dab
(r - r¢ , t - t ¢)M and Dab
(r - r¢ , t - t ¢)M are:

and
ˆ K , K ¢ (r , t )+ = eitH0¢ Y K , K ¢ (r , 0)+e-itH0¢ .
Y

(41)

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
¢

¢

Y aK , K ¢ (r , t )M = etH0 Y aK , K ¢ (r , 0)+e-tH0

K
¶Dab
(r - r¢ , t - t ¢)M

+ [v F ( - it)ag - EF dag ]
¶t
K
´ D gb
(r - r¢ , t - t ¢)M

(42)

= dab d (r - r¢) d (t - t ¢) ,

and
¯ aK , K ¢ (r , t )M = etH0¢ Y aK , K ¢ (r , 0)+e-tH0¢ .
Y

K¢
¶Dab
(r

(44)

⎧ q (t ) e-tEe (k ) - q ( - t ) e-(b T + t ) Ee (k )
ua (k) ub (k)*
´⎨
⎩
1 + e-b T Ee (k )
⎫
q (t ) e(t - b T ) Eh (k ) - q ( - t ) etEh (k )
va ( - k) vb ( - k)*⎬⋅
+
E
k
(
)
b
⎭
1+e T h

(46)

It is straightforward to extend this result to other cases
with non-vanishing EF. In the case 2 with EF >0 we have

(48)

Dab (r - r¢ , t )M =

K

=

Tr {e

¢

Tr {e-b T H0 }

1
åeik (r- r¢)
Nc k

⎧
q (t ) e-tEe (k ) - q ( - t ) e-(b T + t ) Ee (k )
´ ⎨q [E+(k ) - EF ]
⎩
1 + e-b T Ee (k )
⁎
´ ua (k) ub (k)

K
¯ b (r¢ , t ¢)M ] ñ
Dab
(r - r¢ , t - t ¢)M = á Tt [Y aK (r , t )M Y

¯ bK (r¢ , t ¢)M ]}
t )M Y

(52)

(47)

The Matsubara temperature Green functions of Dirac
fermions are defined by the following formula

Tt [Y aK (r ,

1
åeik (r- r¢)
Nc k

Dab (r - r¢ , t )M =

¯ aK ¢ (r , t )M
¶Y
¯ gK ¢ (r , t )M .
= - [v F ( - it)ag - EF dag ] Y
¶t

-b T H0¢

(51)

Now let us derive the explicit expressions of the Green
K¢
K
functions Dab
(r - r¢ , t )M . Since the
(r - r¢ , t )M and Dab
reasonings and calculations do not depend on the presence of
the indices K and K´, we shall omit both these indices until

the end of this section. There are three different cases
depending on the position of the Fermi level EF. By means of
standard calculations we obtain following result in the case 1
with EF=0:

¶Y aK (r , t )M
= - [v F ( - it)ag - EF dag ] Y gK (r , t )M , (45)
¶t

¯ aK (r , t )M
¶Y
= -[v F ( - it⁎)ag - EF dag ]
¶t
¯ gK (r , t )M ,
´Y

- r¢ , t - t ¢)M

= dab d (r - r¢) d (t - t ¢) .

From this common form it is easy to derive concrete
forms of the differential equations for different fields
K¢
K
YaK (r, t )M , Y aK ¢ (r, t )M and Y¯ a (r, t )M , Y¯ a (r, t )M . We obtain

¶Y aK ¢ (r , t )M
= -[v F ( - it⁎)ag - EF dag ]
¶t
´ Y gK ¢ (r , t )M ,

(50)

+ [v F ( - it⁎)ag - EF dag ]
¶t
K¢
´ D gb
(r - r¢ , t - t ¢)M

(43)

They obey the Heisenberg quantum equation of motion
¶Y aK , K ¢ (r , t )M
= [H0¢, Y aK , K ¢ (r , t )M ] ,
¶t
¯ aK , K ¢ (r , t )M
¶Y
¯ aK , K ¢ (r , t )M ] .
= [H0¢, Y
¶t

K

¯ b (r¢ , t ¢)M - q (t ¢ - t ) Y
¯b
= q (t - t ¢) Y aK (r , t )M Y

q (t ) e(t - b T ) Eh

(1 )

+ q [EF - E+(k )]
,

´

(49)

+

6

- q ( - t ) etEh

(1 )

(k )

(1 )
e-b T Eh (k )

va(1) ( - k) vb(1) ( - k)⁎
q (t ) e(t - b T ) Eh

(2 )

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

1+

(k )

1+

(k )

- q ( - t ) etEh

(2 )
e-b T Eh (k )

(2 )

(k )

⎫
va(2) ( - k) vb(2) ( - k)⁎⎬⋅
⎭
(53)


Adv. Nat. Sci.: Nanosci. Nanotechnol. 7 (2016) 015013

V H Nguyen et al

Similarly, in the case 3 with EF<0 the result is

6. Keldysh non-equilibrium Green functions of Dirac

fermions in the free Dirac fermion gas

1
eik (r- r¢)
å
Nc k

Dab (r - r¢ , t )M =

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’ TC 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 [12–14]:

⎧ q (t ) e-tEe(2) (k ) - q ( - t ) e-(b T + t ) Ee(2) (k )
´⎨
(2 )
⎩
1 + e-b T Ee (k )
´ ua(2) (k) ub(2) (k)*
q (t ) e-tEe

(1 )

+ q [E-(k ) - EF ]

(k )

- q ( - t ) e-(b T + t ) Ee

(1 )

1 + e-b T Ee

(1 )

(k )

(k )

´ ua(1) (k) ub(1) (k)*
+ q [EF - E-(k )]

q (t ) e(t - b T ) Eh (k ) - q ( - t ) etEh (k )
1 + e-b T Eh (k )

´ va ( - k) vb ( - k)*}⋅
(54)

K
¯ bK (r¢ , z¢)C] ñ
Dab
(r , r¢ ; z - z¢)C = - i á TC [Y aK (r , z)C Y

It is easy to verify that Green functions (52)–(54) satisfy
the following condition of antiperiodicity
Dab (r - r¢ , t + b T )M = -Dab (r - r¢ , t )M ,

(55)

which must be valid in any equilibrium quantum system, as
was demonstrated by Abrikosov et al [10].
The Matsubara temperature Green functions (52)–(54)
have the Fourier expansions of the form
Dab (r - r¢ , t )M =

´

ò0

K¢

= -i

˜ ab (k, en)M den .
eien t D

˜ ab (k, en)M = ua (k) ub (k)* + va ( - k) vb ( - k)*
D
ien + Ee (k )
ien - Eh (k )

Tr {e-b T H ¢0 }

,
(61)

where complex time-dependent field operators Y aK , K ¢ (r, z )C
K,K ¢
and Y¯ a (r, z )C have the form
K ,K ¢
⎧
⎪Y
(r , z)C = eizH ¢0 Y aK , K ¢ (r , 0)+e-izH ¢0 ,
⎨ aK , K ¢
⎪
¯ a (r , z)C = eizH ¢0 Y aK , K ¢ (r , 0)+e-izH ¢0 .
⎩Y

(57)

(62)

They satisfy the Heisenberg quantum equation of motion

˜ ab (k, en)M = q [E+(k ) - EF ] ua (k) ub (k)*
D
ien + Ee (k )

¶Y aK , K ¢ (r , z)C
= - [H0¢, Y aK , K ¢ (r , z)C] ,
¶z
¯ K , K ¢ (r , z ) C

¶Y
¯ aK , K ¢ (r , z)C] .
i a
= - [H0¢, Y
¶z

i

va(1) ( - k) vb(1) ( - k)*
ien - Eh(1) (k )

va(2) ( - k) vb(2) ( - k)*

From this common form it is easy to derive concrete
forms of the differential equations for different fields
K¢
K
YaK (r, z )C , Y aK ¢ (r, z )C and Y¯ a (r, z )C , Y¯ a (r, z )C . We obtain

in the case 2 with EF>0, and
ua(2) (k) ub(2) (k)⁎
ien + Ee(2) (k )

+ q [E-(k ) -

(63)

(58)

ien - Eh(2) (k )

˜ ab (k, en)M =
D

¯ b (r¢ , z¢)C ]}
Tr {e-b T H ¢0 TC [Y aK ¢ (r , z)C Y

(56)

in the case 1 with EF=0,

+

(60)

K¢
¯ bK ¢ (r¢ , z¢)C ] ñ
Dab
(r , r¢ ; z - z¢)C = - i á TC [Y aK ¢ (r , z)C Y

Inverting the expansion formulae of the functions (52)–
(54), we obtain

+ q [EF - E+(k )]

Tr {e-b T H ¢0 }

and

1

1
åeik (r- r¢) b
Nc k
T
bT

¯ b (r¢ , z¢)C ]}
Tr {e-b T H ¢0 TC [Y aK (r , z)C Y
K

= -i

i

ua(1) (k) ub(1) (k)⁎
EF ]
ien + Ee(1) (k )

+ q [EF - E-(k )]

( - k)⁎

va ( - k) vb
ien - Eh (k )

i
(59)

7

(64)

¶Y aK ¢ (r , z)C
= [v F ( - it*)ag - EF dag ] Y gK ¢ (r , z)C , (65)
¶z

i

in the case 3 with EF<0.

¶Y aK (r , z)C
= [v F ( - it)ag - EF dag ] Y gK (r , z)C ,
¶z

¯ aK (r , z)C
¶Y
¯ gK (r , z)C , (66)
= [v F ( - it*)ag - EF dag ] Y
¶z


Adv. Nat. Sci.: Nanosci. Nanotechnol. 7 (2016) 015013

V H Nguyen et al
K
(r - r¢ , t - t ¢)22 = - i {q (t ¢ - t ) áY aK (r , t - i0)
Dab

¯ bK (r¢ , t ¢ - i0)ñ
´Y

K

¯ b (r¢ , t ¢ - i0) Y aK (r , t - i0)ñ },
- q (t - t ¢) áY
(69)
K
Dab
(r - r¢ , t - t ¢)21 = -i { áY aK (r , t - i0)
K

¯ b (r¢ , t ¢ + i0)ñ ,
´Y

Figure 5. Contour C consists of four parts.

i

¯ aK ¢ (r ,
¶Y
¶z

z)C

(70)

K
¯ bK (r¢ , t ¢ - i0) Y aK (r , t + i0)ñ .
(r - r¢ , t - t ¢)12 = i { áY
Dab
(71)

¯ gK ¢ (r , z)C . (67)
= [v F ( - it)ag - EF dag ] Y

They satisfy following differential equations:
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
C = C1

i

- [v F ( - it)ag - EF dag ]
¶t
K
´ D gb
(r - r¢ , t - t ¢)11
= dab d (t - t ¢) d (r - r¢) ,

È C2 È C3 È C 4,
i

C1 being the part of the straight line over and infinitely
close to the real axis from some point t0 + i0 to infinity
+¥ + i0, C2 being the part of the straight line under and
infinitely close to the real axis from infinity +¥ - i0 to the
point t0 - i0, C3 and C4 being the segments
[t0 - i0, t0 - ibT ] and [+¥ + i0, +¥ - i0] (figure 5). The
contributions of the segment [+¥ + i0, +¥ - i0] 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
C1, C2 and C3. When both variables z and z′ belong to the line
C1, the functions (60) and (61) are the quantum statistical
average of the usual chronological products of the quantum
K,K ¢
field operators Y aK , K ¢ (r, t ) and Y¯ a (r, t ) in the Heisenberg
picture over a statistical ensemble. When both variables z and
z′ belong to the line C3, 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 t0  -¥ . Because the interaction
must satisfy the ‘adiabatic hypothesis’ and vanish at this limit,
the segment C3 also gives no contribution. In this case the
contourC can be considered to consist of only two lines C1
and C2, 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

i

i

(72)

K
¶Dab

(r - r¢ , t - t ¢)22

- [v F ( - it)ag - EF dag ]
¶t
K
´ D gb
(r - r¢ , t - t ¢)22
= - dab d (t - t ¢) d (r - r¢) ,

(73)

K
¶Dab
(r - r¢ , t - t ¢)21

- [v F ( - it)ag - EF dag ]
¶t
K
´ D gb
(r - r¢ , t - t ¢)21 = 0,

(74)

K
¶Dab
(r - r¢ , t - t ¢)12

- [v F ( - it)ag - EF dag ]
¶t
K

´ D gb
(r - r¢ , t - t ¢)12 = 0.

(75)

K¢
For the set of four functions Dab
(r - r¢ , t - t ¢)ij , with i,
j=1, 2, we have the definition obtained from formulae (68)–
(71) and four differential equations obtained from
equations (72)–(75) after the replacement K  K ¢ and
t  t⁎. Since in the sequel all reasonings and calculations
do not depend on the indices K and K′, we shall omit them for
simplifying the expressions.
It is straightforward to derive the explicit expressions of
four Green functions Dab (r - r¢ , t - t ¢)ij and obtain following result:
In the case 1 with EF=0

Dab (r - r¢ , t - t ¢)11
i
= - åeik (r- r¢) {e-iEe (k )(t - t ¢) ua (k) ub (k)*
Nc k
´ [q (t ¢ - t ) - n e (k)]

K
Dab
(r - r¢ , t - t ¢)11 = - i {q (t - t ¢) áY aK (r , t + i0)
K

¯ b (r¢ , t ¢ + i0)ñ

´Y
¯ bK (r¢ , t ¢ + i0) Y aK (r , t + i0)ñ },
- q (t ¢ - t ) áY

K
¶Dab
(r - r¢ , t - t ¢)11

+ eiEh (k )(t - t ¢) va ( - k) vb ( - k)*
´ [nh ( - k) - q (t - t ¢)]},

(68)

8

(76)


Adv. Nat. Sci.: Nanosci. Nanotechnol. 7 (2016) 015013

V H Nguyen et al

Dab (r - r¢ , t - t ¢)22
i
= - åeik (r- r¢) {e-iEe (k )(t - t ¢) ua (k) ub (k)*
Nc k
´ [q (t ¢ - t ) - n e (k)]

Dab (r - r¢ , t - t ¢)12
i

= åeik (r- r¢) {q (E+ - EF) e-iEe (k )(t - t ¢)
Nc k
´ ua (k) ub (k)*n e (k)

+ eiEh (k )(t - t ¢) va ( - k) vb ( - k)*[nh ( - k) - q (t - t ¢)]},

(1 )

+ q (EF - E+) eiEh

(77)

´ [1 - nh(1) ( - k)]
(2 )

+ eiEh

Dab (r - r¢ , t - t ¢)12
i
= åeik (r- r¢) {e-iEe (k )(t - t ¢) ua (k) ub (k)*n e (k)
Nc k
+ eiEh (k )(t - t ¢) va ( - k) vb ( - k)*[1 - nh ( - k)]},

+e

va ( - k) vb ( - k)*nh ( - k)},

(k )(t - t ¢) (2)
va ( - k) vb(2) ( - k)*

´ [1 - nh(2) ( - k)]},

(82)

Dab (r - r¢ , t - t ¢)21
i
= - åeik (r- r¢) {q (E+ - EF) e-iEe (k )(t - t ¢)
Nc k
´ ua (k) ub (k)*[1 - n e (k)]

(78)

Dab (r - r¢ , t - t ¢)21
i
= - åeik (r- r¢) {e-iEe (k )(t - t ¢) ua (k) ub (k)*
Nc k
´ [1 - n e (k)]
iEh (k )(t - t ¢)

(k )(t - t ¢) (1)
va ( - k) vb(1) ( - k)*

(1 )

+ q (EF - E+) eiEh
(2 )

+ eiEh
(79)

(k )(t - t ¢) (1)
va ( - k) vb(1) ( - k)*nh(1) ( - k)

(k )(t - t ¢) (2)
va ( - k) vb(2)

´ ( - k)*nh(2) ( - k)}⋅
(83)

where
e-b T Ee (k )
1 + e-b T Ee (k )

n e (k) =

In the case 3 with EF<0 the result is
Dab (r - r¢ , t - t ¢)11
(2 )
i
= - åeik (r- r¢) {e-iEe (k )(t - t ¢) ua(2) (k) ub(2) (k)*
Nc k

and
nh ( - k) =

e-b T Eh (k )
⋅
1 + e-b T Eh (k )

´ [q (t - t ¢) - ne(2) (k)]

+ q (E- - EF) e-iEe

(1 )

In the case 2 with EF>0 we have

(1 )

+ q (EF - E-) eiEh (k )(t - t ¢) va ( - k) vb ( - k)*
´ [nh ( - k) - q (t ¢ - t )]},

´ [nh(1) ( - k) - q (t ¢ - t )]
+
´

[nh(2) ( - k)

- q (t ¢ - t )]},

´ [q (t ¢ - t ) - ne(2) (k)]
+ q (E- - EF) e-iEe

(1 )

(80)

(1 )

+e
´

[nh(2) ( - k)

(85)

Dab (r - r¢ , t - t ¢)12
(2 )
i
= åeik (r- r¢) {e-iEe (k )(t - t ¢) ua(2) (k) ub(2) (k)*ne(2) (k)
Nc k

(k )(t - t ¢) (1)
va ( - k) vb(1) ( - k)*

+ q (E- - EF) e-iEe

(1 )

va(2) ( - k) vb(2) ( - k)*

- q (t - t ¢)]},

ua(1) (k) ub(1) (k)*

+ q (EF - E-) eiEh (k )(t - t ¢) va ( - k) vb ( - k)*
´ [nh ( - k) - q (t - t ¢)]},

´ [nh(1) ( - k) - q (t - t ¢)]
iEh(2) (k )(t - t ¢)

(k )(t - t ¢)

´ [q (t ¢ - t ) - ne(1) (k)]

Dab (r - r¢ , t - t ¢)22
i
= - åeik (r- r¢) {q (E+ - EF) e-iEe (k )(t - t ¢)
Nc k
´ ua (k) ub (k)*[q (t ¢ - t ) - n e (k)]
+ q (EF - E+) eiEh

(84)

Dab (r - r¢ , t - t ¢)22
(2 )
i
= - åeik (r- r¢) {e-iEe (k )(t - t ¢) ua(2) (k) ub(2) (k)*
Nc k

(k )(t - t ¢) (1)
va ( - k) vb(1) ( - k)*

(2 )
eiEh (k )(t - t ¢) va(2) ( - k) vb(2) ( - k)*

ua(1) (k) ub(1) (k)*

´ [q (t - t ¢) - ne(1) (k)]

Dab (r - r¢ , t - t ¢)11

i
= - åeik (r- r¢) {q (E+ - EF) e-iEe (k )(t - t ¢)
Nc k
´ ua (k) ub (k)*[q (t - t ¢) - n e (k)]
+ q (EF - E+) eiEh

(k )(t - t ¢)

(k )(t - t ¢)

ua(1) (k) ub(1) (k)*ne(1) (k)

+ q (EF - E-) eiEh (k )(t - t ¢) va ( - k) vb ( - k)*[1 - nh ( - k)]},
(81)

(86)
9


Adv. Nat. Sci.: Nanosci. Nanotechnol. 7 (2016) 015013

V H Nguyen et al

Dab (r - r¢ , t - t ¢)21
(2 )
i
= - åeik (r- r¢) {e-iEe (k )(t - t ¢) ua(2) (k) ub(2) (k)*
Nc k

˜ ab (k, w )22

D
= q (E+ - EF) ua (k) ub (k)*
⎡
⎤
1
´ ⎢+ 2p id (w - Ee (k )) n e (k)⎥
⎣ w - Ee (k ) - i0
⎦

´ [1 - ne(2) (k)]
+ q (E- - EF) e-iEe

(1 )

(k )(t - t ¢)

ua(1) (k) ub(1) (k)*

+ q (EF - E+) va(1) ( - k) vb(1) ( - k)*
⎡
⎤
1
´ ⎢- 2p id (w + Eh(1) (k )) nh(1) ( - k)⎥
(1)
⎣ w + Eh (k ) + i0
⎦

´ [1 - ne(1) (k)]
+ q (EF - E-) e

iEh (k )(t - t ¢)

va ( - k) vb ( - k)*nh ( - k)}⋅ (87)

+ va(2) ( - k) vb(2) ( - k)*

The Keldysh Green functions (76)–(87) have the Fourier
expansion of the form
Dab (r , t )ij =

1
1
å
Nc k 2p

ò ei (kr-wt ) D˜ ab (k, w)ij dw.

⎡
⎤
1
(2)
(2)
⎥,
p
d
w
´ ⎢+
(
(
))

(
)
2
i
E
k
n
k
h
h
⎣ w + Eh(2) (k ) + i0
⎦
(94)
˜
Dab (k, w )12

(88)

It is straightforward to derive the expressions of
˜ ab (k, w )ij in all three cases. In the first case with EF=0 we
D
obtain
⎡
1
˜ ab (k, w )11 = ua (k) ub (k)⁎ ⎢
D
⎣ w - Ee (k ) + i0
+ 2p id (w - Ee (k )) n e (k)]
+ va ( - k) vb ( - k)⁎
⎡

⎤
1
´⎢
- 2p id (w + Eh (k )) nh ( - k)⎥ ,
⎣ w + Eh (k ) - i0
⎦
⎡
1
˜ ab (k, w )22 = ua (k) ub (k)*⎢ D
⎣ w - Ee (k ) - i0
+ 2p id (w - Ee (k )) n e (k)]
⎡
1
+ va ( - k) vb ( - k)*⎢ ⎣ w + Eh (k ) + i0
- 2p id (w + Eh (k )) nh ( - k)] ,

= 2p i {q (E+ - EF) ua (k) ub (k)*d (w - Ee (k )) n e (k)
+ q (EF - E+) va(1) ( - k) vb(1) ( - k)*
´ d (w + Eh(1) (k ))[1 - nh(1) ( - k)]
+ va(2) ( - k) vb(2) ( - k)*d (w + Eh(2) (k ))[1 - nh(2) ( - k)]},
˜ ab (k, w )21
D
= - 2p i {q (E+ - EF) ua (k) ub (k)*
´ d (w - Ee (k ))[1 - n e (k)]
+ q (EF - E+) va(1) ( - k) vb(1) ( - k)*

(89)

´ d (w + Eh(1) (k )) nh(1) ( - k)
+ va(2) ( - k) vb(2) ( - k)*d (w + Eh(2) (k )) nh(2) ( - k)},

(96)

while in the third case with EF<0
˜ ab (k, w )11
D
= ua(2) (k) ub(2) (k)*

(90)

⎡
⎤
1
´⎢
+ 2p id (w - Ee(2) (k )) ne(2) (k)⎥
(2)
⎣ w - Ee (k ) + i0
⎦

˜ ab (k, w )12 = 2p i {ua (k) ub (k)*d (w - Ee (k )) n e (k)
D
+ va ( - k) vb ( - k)*d (w + Eh (k ))[1 - nh ( - k)]},

+ q (E- - EF) ua(1) (k) ub(1) (k)*

(91)
˜ ab (k, w )21 = - 2p i {ua (k) ub (k)*d (w - Ee (k ))
D
´ [1 - n e (k)]
+ va ( - k) vb ( - k)*d (w + Eh (k )) nh ( - k)}.

(95)

⎤
⎡
1
(1)
(1)
⎥
(
(
))
(
)
p
d
w
E
k
n
´⎢
+
2
i
k
e
e
⎦
⎣ w - Ee(1) (k ) + i0
+ q (EF - E-) va ( - k) vb ( - k)*

⎡
⎤
1
´⎢
- 2p id (w + Eh (k )) nh ( - k)⎥ ,
⎣ w + Eh (k ) - i0
⎦

(92)

In the second case with EF>0 we have
˜ ab (k, w )11
D

˜ ab (k, w )22
D

⎡
1
= q (E+ - EF) ua (k) ub (k)*⎢
⎣ w - Ee (k ) + i0
+ 2p id (w - Ee (k )) n e (k)]

= ua(2) (k) ub(2) (k)*

(97)

⎡
⎤
1

´ ⎢+ 2p id (w - Ee(2) (k )) ne(2) (k)⎥
(2)
⎣ w - Ee (k ) - i0
⎦

+ q (EF - E+) va(1) ( - k) vb(1) ( - k)*

+ q (E- - EF) ua(1) (k) ub(1) (k)*

⎡
⎤
1
(1)
(1)
⎥
´⎢
p
d
w
+
(
(
))
(
)
E
k
n
k
2

i
h
h
⎣ w + Eh(1) (k ) - i0
⎦

⎡
⎤
1
(1)
(1)
(
(
))
(
)
p
d
w
⎥
´ ⎢+
E
k
n
2
i
k
e
e
⎣ w - Ee(1) (k ) - i0

⎦

+ va(2) ( - k) vb(2) ( - k)*

+ q (EF - E-) va ( - k) vb ( - k)*
⎡
⎤
1
´ ⎢- 2p id (w + Eh (k )) nh ( - k)⎥ ,
⎣ w + Eh (k ) + i0
⎦

⎡
⎤
1
´⎢
- 2p id (w + Eh(2) (k )) nh(2) ( - k)⎥ ,
(2)
⎣ w + Eh (k ) - i0
⎦
(93)
10

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Adv. Nat. Sci.: Nanosci. Nanotechnol. 7 (2016) 015013

V H Nguyen et al

interaction between the quantized electromagnetic field and
the Dirac fermions in graphene, the second step in the
establishment of the basics of GQED.

˜ ab (k, w )12
D
=

2p i {ua(2) (k) ub(2) (k)*d (w

-

Ee(2) (k )) ne(2) (k)

+ q (E- - EF) ua(1) (k) ub(1) (k)*d (w - Ee(1) (k )) ne(1) (k)
+ q (EF - E-) va ( - k) vb ( - k)*d (w + Eh (k ))
´ [1 - nh ( - k)]},

Acknowledgment

(99)

˜ ab (k, w )21
D

The authors would like to express their deep gratitude to the
Vietnam Academy of Science and Technology for the
support.

= - 2p i {ua(2) (k) ub(2) (k)*d (w - Ee(2) (k ))

´ [1 - ne(2) (k)]
+ q (E+ - EF) ua(1) (k) ub(1) (k)*d (w - Ee(1) (k ))
´ [1 - ne(1) (k)]
+ q (EF - E+) va ( - k) vb ( - k)*d (w + Eh (k ))
´ nh ( - k)}⋅

References
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7. 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

11