Theory of Green functions of free Dirac fermions in graphene

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Theory of Green functions of free Dirac fermions in graphene

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2016 Adv Nat Sci: Nanosci Nanotechnol 7 015013

(http://iopscience.iop.org/2043-6262/7/1/015013)

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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:nvhieu@iop.vast.ac.vn

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

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

quantumfield 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 fer-mions 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 func-tions of free Dirac fermions is presented in section 5, and the content of section 6 is the Keldysh theory of non-equilibrium Green functions The conclusion and discussions are presented in section 7 The unit system with c==1 will be used

|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

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

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 infigure1(a) It

consists of two interpenetrating triangular sublattices with the

lattice vectors

1 ( ) 2 ( ) ( )

where a is the distance between the two nearest carbon atoms

a≈1.42 The reciprocal lattice has the following lattice vectors

3 1, 3 ,

2

1 ( ) 2 ( ) ( )

Vectorsliandkisatisfy the condition

pd

=

k li j 2 ij ( )3 Thefirst Brillouin zone (BZ) is presented in figure1(b)

Two inequivalent corners K and K′ with the coordinate vectors

-⎝

⎜ ⎞⎠⎟ ⎛⎝⎜ ⎞⎠⎟

2 1,

1

3 ,

3

2 1,

1

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 (figure2) This new hexagonal lattice

also consists of two interpenetrating triangular sublattices

with the lattice vectors

⎝

⎠

⎟

1, 0 , 3 1

2,

3

1 ( ) 2 ( )

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 offigure3 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 offigure 3 instead of the

paral-lelogram one The wave vector k is called to be near the

corner K if it is contained inside the symmetric Wigner–Seitz

elementary cell around this corner With respect to the sub-lattice 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

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

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

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

Adv Nat Sci.: Nanosci Nanotechnol 7 (2016) 015013 V H Nguyen et al

3 Quantumfield of free Dirac fermions

In order to establish explicit expressions of the quantumfield

of free Dirac fermions it is necessary to have formulae of the

wave functions of these quasiparticles Denote FK Kk,,E¢( ) ther

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

j j

⎪

⎪

⎧

⎨

⎩

E K

E K

E

E K

k

K k

, i , ,

i ,

( )

wherej K Kk,,E¢( ) are the solutions of the 2D Dirac equationsr

t

vF( i ) Kk,E( )r E kK,E( )r, ( )7

t 

vF( i ⁎ ) Kk,E( )r E Kk,E( )r, ( )8

where two componentsτ1and τ2of vector matrix τ are two

matrices

t = 0 1 t = - ⋅

1 0 ,

0 i

i 0

1 ( ) 2 ( )

Equations (7) and (8) both have two solutions

corresp-onding to two eigenvalues

= 



,

F 1 2 2 2

( )

and two eigenfunctions

 ¢

 r e u k , 10

E k

k

kr

, , ( )( ) i , ( ) ( )

h

=



q q



-⎛

⎝

⎜ ⎞⎠⎟

2

e

k

i 2

i 2 ( )

( ) ( )

h

=

q q

¢

-⎛

⎝

⎜ ⎞⎠⎟

2

e

k

i 2

i 2 ( )

( ) ( )

where

⎝

⎜ ⎞

⎠

⎟

k k

2

η and η′ are two arbitrary phase factors h| |= ¢ = 1.| |h

The quantum field of free Dirac fermions in the hexagonal

graphene lattice has the expression

Y(r,t)=eiKrYK(r,t)+eiK¢rYK¢(r,t) (14)

with the following expansion of YK K, ¢(r,t):

å å

¢

=

- n ¢ ¢

t

K K

c

E k t K K K K

k

kr

k

( ) [ ( ) ] ( ) ( )

whereakK K n, ¢is the destruction operator of the Dirac fermion

with the wave function being the plane wave whose wave

vectork satisfies the periodic boundary condition for a very

large square graphene lattice containing Ncelementary cells

Note that the role of the electron spin was omitted and

electrons are considered as the spinless fermions

Two-component 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

( )

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, quasi-spinors or pseudo-quasi-spinors It is worth investigating the symmetry with respect to the rotations in the above-mentionedfictive 3D Euclidean space

The Hamiltonian of the quantumfield of free Dirac fer-mions is

t 

+

¢ + ¢

0 F { ( ) ( ) ( )

From the expansion formula (15) and the canonical anticommutation relations between destruction and creation operatorsakK K n, ¢and a K K n ¢ +

k

,

( ) it follows that Dirac equations

t

¶Y

t

r

r

K

K

F

( )

t 

¶Y

¢

¢

t

r

r

K

K

F

( )

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

¶Y

¢

¢

t

r

r

K K

K K

,

0 ,

( )

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|Gñthe 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|Gñis expressed in terms of the Dirac fermion creation operators and the state vector|0ñof the vacuum

 

å å

¢

+

¢ ¢ ¢ +

n n

F F

( ) ( )

With respect to the ground state |Gñ the destruction/ creation operator a K K n ¢ a K K n ¢ +

, ( , ) of the Dirac fermion with energy less than EFbecomes 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

Adv Nat Sci.: Nanosci Nanotechnol 7 (2016) 015013 V H Nguyen et al

These are three different cases depending on the position

of the Fermi level EF(figure4)

Case 1: EF=0 (figure4(a))

All levels with energies E k+( ) are empty and all those

with energiesE k-( ) are occupied We set

e h

and obtain

å

- +

t

v b

k

c

E k t

E k t

k

kr

k kr

k

i

i h

e

[ ( ) ] [ ( ) ]

Case 2: EF>0 (figure4(b))

All states with energies E k+( )>EF are empty and for

them we set

+ + +

,

F e

( ) ( )

All states with energiesE k+( )<EFare occupied and for

them we set

-+ + - + +

,

h

F 1

( ) ( ) ( )

All states with energiesE k-( ) are occupied and for them

we set

- +

,

h

F 2

( ) ( ) ( )

In this case we obtain

å q

q

Y

-+ - - +

t

r

k k k

1

e e

E t

c

E k t

E k t

E k t

k

kr

k

kr

k kr

k

i

F i

i 2 2

h h

F

e

1 2

( )

[ ( ) ] [ ( ) ] ( ) ( ) [ ( ) ] ( ) ( )

( ) ( )

Case 3: EF<0 (figure4(c)) All states with energiesE k+( ) are empty and for them we set

+

,

h

F 2

( ) ( )

( ) ( ) ( )

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

,

F e1

( ) ( )

( ) ( ) ( )

All states with energiesE k-( )<EFare occupied and for them we set

-

,

h

F

( ) ( )

In this case we obtain

å

q q

- - +

t

k k

e e

23

E t

c

E k t

E k t

E k t

k

kr

k kr

k kr

k

F i 1 1

F i h

e1

( )

[ ( ) ] ( ) ( )

[ ( ) ] ( ) ( ) [ ( ) ]

( )

( )

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

Yˆ (r,t)=eiE tFY(r,t)⋅ (24)

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

¶Y

¶ t = - ¢ Y

r

r

where

å

H E e k a a E h k b b , 26

k

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

Adv Nat Sci.: Nanosci Nanotechnol 7 (2016) 015013 V H Nguyen et al

in the case 1 with EF=0,

å q

q

-+

+

E E k E k b b

h h

k

k k

F 1 1 1

2 2 2

{ [ ( ) ] ( )

( ) ( ) ( ) ( ) ( ) ( )

in the case 2 with EF>0, and

å

q q

¢ =

-+

- +

E k E E k a a

k

k k

0 e

2 2 2

F e

1 1 1 F

{ ( ) [ ( ) ] ( )

( ) ( ) ( )

( ) ( ) ( )

in the case 3 with EF<0

4 Green functions of Dirac fermions in the free Dirac

fermion gas atT=0

Green functions of Dirac fermions in the free Dirac fermion

gas at T=0 are defined by the following formulae

Dab r- ¢r,t- ¢ = - át i G T Ya r,t Yb r¢ ¢,t + Gñ,

29

K ( ) | [ ˆ (K ) ˆ (K ) ]|

( ) and

Dab¢ r- ¢r,t- ¢ = - át i G T Ya¢r,t Yb¢ r¢ ¢,t + Gñ

30

K ( ) | [ ˆK( ) ˆK( ) ]|

( )

Using the Heisenberg quantum equation of motion(25)

as well as the equal-time canonical anticommutation relations

between the quantum field operators Ya K K, ¢(r,t) and

Yb K K, ¢(r¢ ¢,t) we derive the following inhomogeneous differ-,

ential equations for these Green functions

t



¶

ab

t t

r r

F

and

t



¶

ab

¢

t t

r r

32

F

( )

⁎

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 EFthere exist three

different cases In the first case with EF=0 the operator

Yˆ (ar,t) is expressed in terms of the components u a( ) andk

a

v k( ) by means of formula (21), in the second case with

EF>0 it is expressed in terms of the components u a( )k ,

a

v( )1( )k andv a( )2( )k by means of formula(22), while in the third

case with EF<0 it is expressed in terms of the components

a

u( )2( )k,u a( )1( )k andv k a( ) by means of formula(23)

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

ò

å p w w

N

c k

kr

i

It is straightforward to derive the expressions of

w

D˜ab(k, ) in all three cases In thefirst case with EF=0 we obtain

w

E k

E k

e

( )

( ) ( )

In the second case with EF>0 we have

*

*

*

w q

w w

+ +

E k

E k

E k

,

i0

i0

h

h

F

e F

1 1 1

2 2 2

( )

( )

( ) ( ) ( ) ( ) ( )

( )

while in the third case with EF<0

*

*

*

w w q

w q

w

E k

E k

E k

,

i0

i0

h

2 2 e 2 F

1 1 e 1 F

( )

( )

( ) ( ) ( )

( ) ( ) ( )

5 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 Ttemp Instead of for-mulae(29) and (30) now we have the following definition of Green functions of Dirac fermions:

ab

b

b

- ¢

t t

Tr

r r

,

K

H

T

T

0

0

{ [ ˆ ( ) ˆ ( ) ]}

( ) and

ab

b

b

¢

- ¢ ¢ ¢ +

- ¢

t t

Tr

r r

,

K

H

T

T

0

0

{ [ ˆ ( ) ˆ ( ) ]}

where

b =

k T

1

T

B temp

( )

Adv Nat Sci.: Nanosci Nanotechnol 7 (2016) 015013 V H Nguyen et al

and kBis the Boltzmann constant Note that formula(38) can

be obtained from formula(37) by means of the replacement

 ¢

K K Field operators Yˆa K K, ¢(r,t) and Yˆa K K, ¢(r,t)+ are

obtained from the corresponding operators at t=0 by means

of the action of the time translation operator eitH0,namely

t

and

t

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-dependentfield operators (40)

and(41) we introduce corresponding τ-dependent ones

t

Ya K K, ¢(r, )M=et H0¢Ya K K, ¢(r, 0 e)+ -t H0¢ (42)

and

t

¢ ¢ ¢ + - ¢

K K

They obey the Heisenberg quantum equation of motion

t

t

¶Y

¶Y

a

a a

a

¢

¢

¢

¢

H

H

r

r r

r

,

,

K K

M

K K

M

,

0 , ,

0 ,

( )

From this common form it is easy to derive concrete

forms of the differential equations for different fields

t

Ya K(r, )M,Ya K¢(r,t) andM Y¯ (a K r,t)M,Y¯a K¢(r,t)M.We obtain

t

¶Y

a

r

r

,

K

M

( )

t

t

t 

¶Y

-´ Y

a

g

¢

¢

r

r

,

i

K

M

K M

( )

⁎

t

t

t 

¶Y

-´ Y

a

g

r

r

,

i

K

M

K M

¯ ( )

⁎

t

¶Y

a

¢

¢

r

r

,

48

K

M

¯ ( )

( )

The Matsubara temperature Green functions of Dirac

fermions are defined by the following formula

t

b

b

- ¢

- ¢

T

Tr

e

,

49

K

K M

M K M H

T

T

0

0

( ) 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

t

t

´ ¢ ¢ Y

a

t t

K M K M K

M K M

K

[ ( ) ¯ ( ) ]

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

t t

- ¢ - ¢

t t

t t

t

ab

gb ab

r r

r r

r r

,

i ,

K

M

K

M

t t

t t

t 

ab

gb ab

¢

¢

r r

r r

r r

,

i ,

K

M

K

M

⁎

Now let us derive the explicit expressions of the Green functions Dab K (r- ¢r,t)M and Dab K¢(r- ¢r,t)M Since the 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:

*

*

å

t

-+

ab

- ¢

- - +

-⎧

⎨

⎩

⎫

⎬

⎭

N

r r

1 e

1 e

52

M c

E k

E k

k

k r r

i

T T

T h h

T h

e

( ) ( )

( ) ( )

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

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

å

t

-+

´

-+

ab

b

b

- ¢

+

- - +

-+

-⎧

⎨

⎩

⎫

⎬

⎭

N

E k E

E E k

r r

1 e

1 e

1 e

53

M c

E k

E k

E k

k

k r r

i

F

F

1 1

2 2

T T

T h h

T h

T h h

T h

e

1

2

( ) ( )

( )

( ) ( ) ( ) ( )

( )

⁎

( ) ( ) ( )

( ) ( ) ( ) ⁎

( ) ( ) ( )

( )

( ) ( ) ⁎

( ) ( ) ( )

( ) ( ) ( )

Adv Nat Sci.: Nanosci Nanotechnol 7 (2016) 015013 V H Nguyen et al

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

*

*

*

å

t

-+

´

-+

´

-+

ab

b

b

b

- ¢

-⎧

⎨

⎩

N

E k E

E E k

r r

1 e

1 e

1 e

54

M

c

E k

E k

E k

k

k r r

i

2 2

F

1 1

F

T T

T T

T h h

T h

e2 e2

e2

e1 e1

e1

( ) ( )

( ) ( )

( ) ( ) }

( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

It is easy to verify that Green functions(52)–(54) satisfy

the following condition of antiperiodicity

Dab(r- ¢r, + T M) = -Dab(r- ¢r, )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

ò

å

t

b

ab

b

e t ab

- ¢

N

r r

k

M

n M n

k

k r r

i

0 i

T n

( )

Inverting the expansion formulae of the functions(52)–

(54), we obtain

e

E k

E k

n M

( )

( ) ( )

( ) ( )

in the case 1 with EF=0,

*

*

*

e q

e e

-+

+ +

E k

E k

E k

,

i

i

n M

n

F

e F

1 1

1

2 2

2

( )

( )

( ) ( )

( ) ( ) ( )

( )

in the case 2 with EF>0, and

e

e q

e q

e

+

-+

E k

E k

E k

,

i

i

n M

n

n

2 2 e F

1 1 e F

( )

[ ( ) ] ( ) ( )

( )

( ) ( ) ⁎ ( )

( ) ( ) ⁎ ( )

⁎

in the case 3 with EF<0

6 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 func-tions In the definition of Green functions of complex time-dependent field operators it was proposed to define the

‘extended chronological ordering’ TC of two complex vari-ables 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]:

b

b

- ¢

- ¢

Tr

e

60

H

T

T

0

0

( ) and

b

b

- ¢ ¢ ¢

- ¢

Tr

61

K

K C H

K

C H

T

T

0

0

( ) where complex time-dependent field operators Ya K K, ¢(r,z)C

and Y¯a K K, ¢(r,z) have the formC

¢ ¢ ¢ + - ¢

¢ ¢ ¢ + - ¢

⎪

⎪

⎧

⎨

⎩

z z

62

,

C i , i ,

C i , i

They satisfy the Heisenberg quantum equation of motion

¶Y

¶Y

a

a

a

a

¢

¢

¢

¢

z

z

r

r r

r

K K

K K

K K

K K

, C

0 , C ,

C

0 , C

( )

From this common form it is easy to derive concrete forms of the differential equations for different fields

Ya K(r,z)C,Ya K¢(r,z) and YC ¯ (a K r,z)C,Y¯a K¢(r,z)C.We obtain

d

t

¶Y

a

z

r

r

K

K

C

( )

t 

¶Y

a

¢

¢

z

r

r

K

K

C

( )

t 

¶Y

a

z

r

r

K

K

C

¯ ( )

Adv Nat Sci.: Nanosci Nanotechnol 7 (2016) 015013 V H Nguyen et al

t

¶Y

a

¢

¢

z

r

r

K

K

C

¯ ( )

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 C2 C3 C ,4

C1being 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

b

-t0 i0,t0 i T

[ ] and +¥ +[ i0,+¥ -i0](figure5) 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, C2and 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

field operators Ya K K, ¢(r,t) and Y¯a K K, ¢(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

q q

´ Y ¢ ¢ + ñ

b

t

r

, i0

K

11

q q

-´ Y ¢ ¢ - ñ

b

t

r

, i0

69

K

22

( )

-´ Y ¢ ¢ + ñ

b

t

r

K

21

Dab r- ¢r,t- ¢t =i áYb r¢ ¢ -,t i0 Ya r,t+i0 ñ

71

12

( )

They satisfy following differential equations:

d

t

ab

gb ab

t t

t t

t t

r r

r r

r r

,

K

K

11

11

d

t

ab

gb ab

t t

t t

t t

r r

r r

r r

,

K

K

22

22

d

t

ab

gb

t t

t t

r r

r r

K

K

21

21

d

t

ab

gb

t t

t t

r r

r r

K

K

12

12

For the set of four functions Dab K¢(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 fol-lowing result:

In the case 1 with EF=0

*

*

å

q

q

=

ab

- ¢ - - ¢

- ¢

t t

r r

k

k

, i

e

c

E k t t

E k t t

h

k

k r r

11

i i

e

i h

e

[ ( ) ( )]

( ) ( )

( ) ( )( )

( )( )

Figure 5.Contour C consists of four parts

Adv Nat Sci.: Nanosci Nanotechnol 7 (2016) 015013 V H Nguyen et al

*

å

q

q

=

ab

- ¢ - - ¢

- ¢

t t

r r

k

,

i

77

c

E k t t

E k t t

h

k

k r r

22

i i

e

i h

e

( )

( ) ( )( )

( )( )

*

*

å

=

-ab

- ¢ - - ¢

- ¢

t t

r r

,

i

c

E k t t

E k t t

h

k

k r r

12

i i

e

i h

e

( ) ( )( )

( )( )

*

*

å

=

-ab

- ¢ - - ¢

- ¢

t t

n

r r

k

, i

1

c

E k t t

E k t t

h

k

k r r

21

i i

e

i h

e

[ ( )]

( ) ( )( )

( )( )

where

= +

b b

1 e

E k

E k

e

T T

e e

( )

( ) ( )

and

b b

1 e

h

E k

E k

T h

T h

( )

( ) ( )

In the case 2 with EF>0 we have

*

*

*

q q

q q

-ab

- ¢

+ - - ¢

+ - ¢

- ¢

t t

r r

k

k

,

i

e

e

c

E k t t

E k t t

h

E k t t

h

k

k r r

11 i

F i e

F i 1 1

1

i 2 2

2

h

h

e

1

2

( ) ( ) [ ( ) ( )]

( ) ( )( )

( )( ) ( ) ( ) ( )

( )( ) ( ) ( )

( )

( )

( )

*

*

*

q q

q q

ab

- ¢

+ - - ¢

+ - ¢

- ¢

t t

r r

k

k

,

i

e

e

c

E k t t

E k t t

h

E k t t

h

k

k r r

22 i

F i e

F i 1 1

1

i 2 2

2

h

h

e

1

2

( ) ( ) [ ( ) ( )]

( ) ( )( )

( )( ) ( ) ( ) ( )

( )( ) ( ) ( )

( )

( )

( )

*

*

*

q

-´

-ab

- ¢

+ - - ¢

+ - ¢

- ¢

t t

n

n

r r

k

k

, i

e 1

e

c

E k t t

E k t t

h

E k t t

h

k

k r r

12 i

F i e

F i 1 1 1

i 2 2 2

h h

e

1

2

( ) ( ) ( )

( ) ( )( )

( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

( )

( )

*

*

*

q

ab

- ¢

+ - - ¢

+ - ¢

- ¢

t t

n

r r

k

, i

1 e e

83

c

E k t t

E k t t

h

E k t t

h

k

k r r

21 i

F i e

F i 1 1 1

i 2 2 2

h h

e

1

2

( ) ( ) [ ( )]

( )

( )

( ) ( )( )

( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( )

( )

( )

In the case 3 with EF<0 the result is

*

*

*

å

q q q q

q

=

-ab

- ¢ - - ¢

- - - ¢

- - ¢

t t

r r

k

k

k

, i

e

e

c

E k t t

E k t t

E k t t

h

k

k r r

11

i i 2 2

e 2

F i 1 1 e

1

F i h

e2

e1

( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( )

( )( )

( )

( )

*

*

*

å

q q q q

q

=

ab

- ¢ - - ¢

- - - ¢

- - ¢

t t

r r

k

k

k

, i

e

e

c

E k t t

E k t t

E k t t

h

k

k r r

22

i i 2 2 e

2

F i 1 1 e

1

F i h

e2

e1

( ) ( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

( )( )

( )

( )

*

*

*

å

q q

=

-ab

- ¢ - - ¢

- - - ¢

- - ¢

t t

r r

, i

e

86

c

E k t t

E k t t

E k t t

h

k

k r r

12

i i 2 2

e

F i 1 1 e

1

F i h

e2

e1

( )

( ) ( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( )( )

( )

( )

Adv Nat Sci.: Nanosci Nanotechnol 7 (2016) 015013 V H Nguyen et al