Quantum field theory of photon – Diracfermion interacting system in graphene monolayer Bich Ha Nguyen1,2and Van Hieu Nguyen1,2 1 Advanced Center of Physics and Institute of Materials Scie
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Quantum field theory of photon–Dirac fermion interacting system in graphene monolayer
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2016 Adv Nat Sci: Nanosci Nanotechnol 7 025003
(http://iopscience.iop.org/2043-6262/7/2/025003)
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Trang 2Quantum field theory of photon – Dirac
fermion interacting system in graphene
monolayer
Bich Ha Nguyen1,2and Van Hieu Nguyen1,2
1
Advanced Center of Physics and Institute of Materials Science, Vietnam Academy of Science and
Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
2
University of Engineering and Technology, Vietnam National University, 144 Xuan Thuy, Cau Giay,
Hanoi, Vietnam
E-mail:nvhieu@iop.vast.ac.vn
Received 20 January 2016
Accepted for publication 22 February 2016
Published 30 March 2016
Abstract
The purpose of the present work is to elaborate quantumfield theory of interacting systems
comprising Dirac fermionfields in a graphene monolayer and the electromagnetic field Since the
Dirac fermions are confined in a two-dimensional plane, the interaction Hamiltonian of this
system contains the projection of the electromagneticfield operator onto the plane of a graphene
monolayer Following the quantization procedure in traditional quantum electrodynamics we
chose to work in the gauge determined by the weak Lorentz condition imposed on the state
vectors of all physical states of the system The explicit expression of the two-point Green
function of the projection onto a graphene monolayer of a free electromagneticfield is derived
This two-point Green function and the expression of the interaction Hamiltonian together with
the two-point Green functions of free Dirac fermionfields established in our previous work form
the basics of the perturbation theory of the above-mentioned interactingfield system As an
example, the perturbation theory is applied to the study of two-point Green functions of this
interacting system of quantumfields
Keywords: quantumfield, Dirac fermion, electromagnetic field, Green function, perturbation
theory
Classification numbers: 2.01, 3.00, 5.15
1 Introduction
After the discovery of graphene by Novoselov et al[1,2], a
new extremely promising interdisciplinary scientific area—
the physics, chemistry and technology of graphene and
similar two-dimensional hexagonal semiconductors—has
emerged and strongly developed as‘a rapidly rising star on
the horizon of materials science and condensed-matter
phy-sics, having already revealed a cornucopia of new physics
and potential applications’, as Geim et al stated [3] The
quantum motion of electrons as spinless point particles in
graphene is essentially governed by Dirac’s (relativistic) equations [4] in the (2+1)-dimensional Minkowski space-time
It is known that in the terminology of quantum field theory the spinless Dirac fermions in graphene monolayers are described by two spinor quantum fields y K(r,t) and
t
r, ,
K
y ¢ r={r r1, 2}={x y, }[5] The points K and K′ are the two nearest corners of the first Brillouin zone in the reciprocal lattice of the hexagonal graphene structure They are called Dirac points
Since the Dirac fermions are considered as the spinless fermions, the quantum fields y K(r,t) and y K¢(r,t) are the two-component spinors realizing the fundamental repre-sentation of the SU(2) group of rotations in some fictive three-dimensional Euclidean space Let us call them the
quasi-|Vietnam Academy of Science and Technology Advances in Natural Sciences: Nanoscience and Nanotechnology Adv Nat Sci.: Nanosci Nanotechnol 7 (2016) 025003 (7pp) doi:10.1088 /2043-6262/7/2/025003
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.
Trang 3spinors or pseudospinors in the analogy with the notion of
isospinor used in the theory of elementary particles[6–9]
Let us denoteτi, i=1, 2, 3 three generators of the SU(2)
group of rotations in thefictive three-dimensional Euclidean
space We call them the quasi-spin or pseudospin operators
acting on the quantum fields of Dirac fermions as
two-component spinors They are similar in the matrix form but
have a quite different physical meaning compared to the Pauli
matrices σi, i=1, 2, 3, representing conventional spin
operators of spin 1/2 fermions and being generators of the
SU(2) group of rotations in the physical three-dimensional
space In the unit system with =c=1 (c being the light
speed in the vacuum) and the approximation assuming the
linear dispersion law for the Dirac fermions, the Hamiltonian
of the system of free Dirac fermions in the graphene
mono-layer has the following expression[5]
1
m
( )
⁎
+
Let us chose the Cartesian coordinate system as follows:
the plane of a graphene monolayer is the coordinate plane
xOy and, therefore, the Oz-axis is perpendicular to this plane
The coordinate of a point in the three-dimensional physical
space is denoted{r, z}={x, y, z} In conventional quantum
electrodynamics it is known [6–11] that three components
Ai(r, z, t), i=1, 2, 3, of the vector potential field A(r, z, t)
together with the scalar potentialfield f(r, z, t)=A0(r, z, t)
form a four-component vectorfield Aμ(r, z, t), μ=1, 2, 3, 4,
A4(r, z, t)=iA0(r, z, t), in the (3+1)-dimensional
Min-kowski space-time In order to take into account the
interac-tion between Dirac fermionfieldsy K(r,t)andy K¢(r,t)with
the vector potential field A(r, z, t), we must perform the
substitution-i -i+eA r( , ,o t)in the Hamiltonian
(1), e being the absolute value of the electron charge [6–12]
Then we obtain the following expression of the Hamiltonian
of the interaction between the vector potentialfield A(r, z, t)
and Dirac fermionfieldy K(r,t)andy K¢(r,t)
2
( )
⁎
t
=
+
+
Dirac fermions interact also with the scalar potentialfield
f(r, z, t) The corresponding part of the interaction Hamiltonian
is
3
( )
ò y y
=
+
+
The interaction between the electromagnetic field and
Dirac fermionfields is completely described by the following
total interaction Hamiltonian
Hint=HintV +H intS ( )4
Its explicit expression contains only the projected vector potentialfield with two components
A i(r,t)def=.A i(r, ,o t),i=1, 2 ( )5 and projected scalar potential field
Let us denotef(r, t) as A0(r, t) and introduce the matrix
1 0
0 1
Then the set of three formulae(2)–(4) can be compactly rewritten as follows
7
m
K
m K
K
m K
int
0
2
( )
⁎
ò
=
=
+
The study of the interaction of the electromagnetic field with the Dirac fermionfield in a graphene monolayer requires the use of explicit formulae determining the projection Am(r,
t), m=0, 1, 2, of the electromagnetic field as well as the
projection D mn(r - ¢ r,t- ¢ m, nt) =0, 1, 2 of the
two-point Green function D mn(r - ¢ r,z- ¢z,t- ¢t) of the electromagnetic field onto the graphene plane These for-mulae are established in section2 Section3is devoted to the study of the interacting system comprising the Dirac fermion fields and electromagnetic field An application of the per-turbation theory is presented in section4 Section5 contains the conclusion and discussions For simplifying formulae we shall use the unit system with =c= 1
2 Projection of free electromagneticfield and its two-point Green function onto graphene monolayer The content of this section is a short presentation of the free electromagnetic field Aμ(x) and its two-point Green function onto the plane xOy of a graphene monolayer In the con-ventional relativistic quantum field theory [6–11] the electromagnetic field is described by a vector field Aμ(x), μ=1, 2, 3, 4, in the (3+1)-dimensional Minkowski space-time The coordinate vector x of each point in this space-time has four components xμ, μ=1, 2, 3, 4, x={x1, x2, x3,
x4}={x, y, z, it} The two-point Green function of the electromagneticfield Aμ(x) is defined as follows:
q q
where the symbol á⋅⋅⋅ñ denotes the average of the inserted expression(containing field operators) in the ground state G| ñ
Trang 4of the Dirac fermion gas
á⋅⋅⋅ñ = á ⋅ ⋅⋅ ñ⋅
This ground state G| ñ can be considered as the vacuum
state of the free electromagneticfield
Since the theory of the electromagneticfield is invariant
under a class of gauge transformations
¶
m
the vectorfield Aμ(x) is not uniquely determined In classical
electrodynamics [12] to simplify equations and calculations
the vector field Aμ(x) satisfying the following Lorentz
condition
A x
¶
m m
was frequently used
However, in quantum electrodynamics this condition
cannot hold for the quantum vector field A x m( ) Instead of
condition(11) it was reasonably proposed to assume another
similar but weaker condition imposed on the state vector of all
physical states of the electromagneticfield:
A x
m m
In the fundamental research works on quantum
electro-dynamics[10,11] it was demonstrated that due to condition
(12) the electromagnetic waves in the states with longitudinal
and scalar polarizations play no role in any physical
pro-cesses Therefore in the Hilbert space of state vectors of all
physical states of the electromagneticfield the vector potential
field A(x)=A(r, z, t) has the following effective Fourier
expansion formula
c
k
13
l
lz l t
l
k
kr k
k
3 2
1
( )
( ) {
}
( )
ò ò
å
p
x
=
´
W +
s
s
s s
=
+ -W
where ξσkl with σ=±1 are two three-component complex
unit vectors characterizing two transversely polarized states of
the electromagnetic plane waves with the wave vector{k, l}
Let us represent each vector ξσkl as a column with three
elements
14
l
l
l
l
k
k k k
1 2 3
( )
x
x x x
s
s s s
⎛
⎝
⎜
⎜⎜
⎞
⎠
⎟
⎟⎟
For two plane waves propagating along the direction of the Oz-axis we have
1 2
1 0
2
1 0
15
⎝
⎜ ⎞
⎠
⎝
⎠
⎟
It is straightforward to project the vectorfield (13) onto the graphene plane to obtain the vectorfield A||(r, t) with two components
c
k
1
e
16
i
3 2
1
( )
( ) [ ( ) ]
[ ( ) ]
ò ò
å
p
x x
=
´
W
s
=
-W
It looks like a linear combination of an innumerable set
of quantum fields Ai(r, t)l, each of them being labeled by a value of the index l
c
k
1
e
18
1
( )
[ ( ) ]
[ ( ) ]
ò
å
p
x x
=
´
W
s
=
-W
The field Ai(r,t)l with an index l≠0 looks like the conventional free vectorfield with transverse polarizations of
a massive particle with the mass|l| and the helicities σ=±1
in the(2+1)-dimensional Minkowski space-time
Note that the electromagnetic waves with the scalar polarization play no role in any physical processes Therefore the free scalar field f(r, z, t) effectively does not have the non-vanishing projection onto the graphene plane
Now we consider the projection of the two-point Green function
D mn( )0( )x =D mn( )0(k, ,z t)
of the free electromagneticfield onto the graphene monolayer
In the relativistic quantum electrodynamics [10, 11] it was
shown that D mn( )0( )x has following general expression
kx
0
4
( ) ( )
ò
p
=
k k
0
2
2
⎣⎢
⎤
⎦⎥
where k denotes a four-momentum vector with the components
k ,μ=1, 2, 3, 4, k4=ik0, in the Minkowski space-time,
k=(k, ,l ik0),
k2 k2 l2 k ,
0
Trang 5kx=kr+lz-k t0 ,
0
ò =ò ò ò
and d(k2) is a scalar function depending on the choice of the
gauge for the free electromagneticfield
Since the theory is invariant under gauge transformations
of the whole system of all interacting quantum fields, for
simplifying the calculations in certain cases one often chose
to work in such a gauge that
k
In this case formula(20) becomes
i k io
1
22 0
2
˜ ( )
( )
d
=
i.e
k
k
0
0 2
( )
d
=
The projection D mn( )0(r,t) of the Green function
D mn( )0(r, ,z t)onto a graphene monolayer is determined by the
following definition
D mn( )0(r,t) def=.D mn( )0(r, ,o t), (24)
where m, n=0, 1, 2 From the above presented formulae it is
easy to show that
25
0
0
0
( )
( ) ( ) = p ò ò ò ( - ) ( )
with
k
k
0
0 2
( )
d
=
Formula(25) shows that D mn( )0(r,t) is a linear
combina-tion
mn( )0( )= pò mn( )0( )l ( )
of an innumerable set of functions D mn( )0(r,t)l labeled by the
index l running all integer values from−∞ to +∞:
k
,
2
0
i
2 0 2
0
( )
( )
ò ò
d p
=
´
-Each function D mn( )0(r,t)lis the two-point Green function
of a massive relativistic particle with the mass l| | (in two
dimensions)
However, if we impose on the state vectors of all
phy-sical states of the system the weak Lorentz condition (12),
then D mn( )0( )x must satisfy the transversality condition
0 ( )
( ) ( )
¶
mn m
and instead of equation(21) we have the relation
k
meaning that the tensor D˜mn( )0(k, ,l k0)
has the following components:
k
k
k
, ,
1
i j
2
2
0
0
2 0 2
( )
d
-´
-⎛
⎝
⎠
⎟
k k
1
i
0 0
0
0 0
2
=
=
-with i, j=1, 2,
k
k
k
1
2
2
33 0
0
2 2 0
2 0 2
( )
-´
-⎛
⎝
⎠
⎟
lk
1
30 0
0
0 0
2
=
=
-and
k
k
k
1
35
2
2
00 0
0
2 0 2
( )
-´
⎛
⎝
⎠
⎟
3 Interacting Dirac fermionfields and electromagnetic field
In order to apply perturbation theory to the study of inter-acting system comprising the Dirac fermion fields and the electromagnetic field it is necessary to use explicit expres-sions of following physical quantities:
• Dirac fermion fieldsy K(r,t)andy K¢(r,t),
• Two-point Green functionsDK ab(r,t)( ) 0 andDab K¢(r,t)( ) 0
of free Dirac fermion fields,
• Projection A||(r, t) with the two-component Ai(r,t), i=1,
2, of the electromagnetic field onto the graphene monolayer,
• Projection D mn(r,t)( )0, m, n=0, 1, 2, of the two-point Green function of the free electromagneticfield onto the graphene monolayer, and
• Interaction Hamiltonian Hint(t) of the system
Trang 6The interaction Hamiltonian Hint(t) was determined by
formula (7) The projection A||(r, t) of the electromagnetic
field and the projection D mn(r,t)( )0, m, n=0, 1, 2, of the
two-point Green function of the free electromagnetic field
were investigated in the preceding section 2 It remains to
establish the explicit expressions of Dirac fermion fields
t
r, ,
K( )
y y K¢(r,t) and two-point Green functions
t
K ( )( ) 0
Dab DK ab¢(r,t)( ) 0 of free Dirac fermions
In our previous work [13] we derived explicit
expres-sions of two-point Green functionsDab K (r,t)( ) 0,DK ab¢(r,t)( ) 0
of free Dirac fermions in a free Dirac fermion gas at T=0
They depend on the value EF of the Dirac fermion gas For
simplicity let us consider the case with EF=0 The extension
to other cases is straightforward
In the simple case with EF=0 the Dirac fermion fields
t
r,
K( )
y andy K¢(r,t)have the following Fourier expansion
formula
k
36
E k t K K K K
kr
k kr
k
e h
( ) [ ( ) ]
[ ( ) ]
ò
y
p
=
+
where akK K, ¢ and bkK K, ¢ are the destruction operators of the
Dirac fermion and Dirac hole, respectively, with wave
functions being plane waves, k is the wave vector to be
considered also as the momentum of the Dirac fermion or
Dirac hole, akK K, ¢+ and bkK K, ¢+ are corresponding creation
operators, Ee(k) and Eh(k) are energies of the Dirac fermion
and Dirac hole, respectively, with momentumk,
,
e h, F
( )
=
vF is the speed of the relativistic Dirac fermion in the unit
system with c=1,
u
v
k
k
1 2
e
1 2
e e
38
K
K
k k k k
( ) ( )
( )
( ) ( ) ( ) ( )
h h
=
=
-q q q q
-⎛
⎝
⎛
⎝
and
u
v
k
k
1 2
e
1 2
e
39
K
K
k k k k
( ) ( )
( )
( ) ( ) ( ) ( )
h h
=
q q q q
¢
-¢
-⎛
⎝
⎠
⎟
⎛
⎝
⎠
⎟
arctg k k
2
η and η′ being two arbitrary phase factors| |h = ¢ = | |h 1
Two-point Green functions of Dirac fermions in free Dirac fermion gas at T=0 have the following definition
41
( )
( )
q q
Introducing their Fourier transformations
t
k
1 2
42
K K
t K K
kr
2
( )
( ) ( )
ò ò
ab
w ab
¢
we have
,
43
e
K K K K
h
( )
( )
⁎
⁎
w
w w
Thus the basics for elaborating the perturbation theory of
an interacting system comprising Dirac fermionfields and an electromagneticfield were established
4 Perturbation theory The most efficient tool for the theoretical study of interaction processes between quanta of any interacting system of quantum fields is the scattering matrix S, briefly called the S-matrix In the perturbation theory the S-matrix is expressed in terms of the interaction Hamiltonian Hint(t) of the system as follows
S=T{exp⎡-iòdt Hint( )t }, (44)
where the integration with respect to the time variable t is performed over the whole real axis from −∞ to +∞ By expanding the exponential function on the right-hand side of formula (44) into power series, we write the S-matrix in the form of a series
n n
1
( ) ( )
å
= +
=
¥
the term S( n)of nth order is
d d
46
n
n
int 1 int 2 int
( )
!
( )
ò
As an example of the application of perturbation theory let us study two-point Green functions of an interacting sys-tem comprising Dirac fermionfields and the projection of the electromagneticfield onto the graphene monolayer at T=0 They are expressed in terms of free Dirac fermion fields
Trang 7r,
K( )
y and y K¢(r,t), components Ai(r,t) of the projection
A||(r,t) of a free electromagnetic field onto the graphene
monolayer and S-matrix as follows:
S
ij
á ñ and
S
r
,
48
K K
,
( )
- ¢
¢
ab
¢
Using expansion formula (45) of the S-matrix, we write
each of the Green functions(47) and (48) in the form of a series:
n
0
2
=
¥
50
K K
n
,
0
( )
( )
å
=
¥
¢
n running all non-negative integers n=0, 1, 2 K
We have calculated D r ij( - ¢ r,t- ¢t)( )0 and
D r ij( - ¢ r,t- ¢t)( )2 let us consider matrix element
i
2
51
n m K
n K K
n K
K
m K K
m K
2
2
0 2
0 2
!
( )
!
( ) ( )
⁎
⁎
ò ò
ò ò ò ò å å
¢
¢
¢
-´ á
+
´
+
+
+
Similarly, in order to calculateDab K (r - ¢ r,t- ¢t),for
example, we consider matrix element
i
2
52
n m K
n K K
n K
K
m K K
m K
2
2
0 2
0 2
!
( )
!
( )
( )
⁎
⁎
ò ò
ò ò ò ò å å
¢
¢
¢
-´ á
+
´
+
+
+
+
The matrix elements on the right-hand side of equations(51) and (52) can be calculated by applying the Wick theorem in quantumfield theory They are expressed in terms of the two-point Green functionsDK K ab, ¢(r-r ¢,t- ¢t)( ) 0 of free Dirac fermionfieldsy ab K K, ¢(r,t)
and the projection D ij( ) 0(r,t)of two-point Green functions of the free electromagneticfield onto
a graphene monolayer
By using derived expressions of the above-mentioned matrix elements it is straightforward to calculate second-order terms in the series (49) and (50) We obtain the following result:
t t
53
K
K
2
1
( )
( )
( )
( )
ò ò ò ò
å
¢
¢
ab
aa
where
,
54
K
n m nm
2
( )
( )
( )
-aa
is the self-energy part of the Dirac fermionfieldy K(r,t),and
,
55
ij
n m in
2
( )
( )
( )
( )
ò ò ò ò
åå
¢
¢
where
,
56
n K
m K
n K
( )
( ) ( )
t t
-can be considered as the self-energy part of the projection of the electromagnetic field onto a graphene plane, DK(r,t)( ) 0 and DK¢(r,t)( ) 0 being 2×2 matrices with elements
t
r,
K ( )( ) 0
Dab andDK ab¢(r,t)( ) 0 All higher-order terms in the series(49) and (50) can be calculated analogously Summing them up, we obtain the Dyson equations for the whole Green functions(49) and (50)
of interacting quantum fields in the ladder approximation:
,
57
ij
n m in
( ) ( )
ò ò ò ò
åå
¢
¢
Trang 8t t
58
K K
K K
,
,
1
( )
( )
( )
ò ò ò ò
å
- ¢
ab
aa
¢
5 Conclusion and discussions
In the present work we have developed the quantum theory of an
interacting system comprising Dirac fermion fields and the
projection onto a graphene monolayer of an electromagnetic
field The explicit expressions of these fields, the interaction
Hamiltonian of the system and the two-point Green functions of
freefields as well as the integral equation determining the
two-point Green functions of interacting fields in the ladder
approximation were established
We have not yet investigated the electromagnetic
scattering processes taking place in the graphene monolayer In
our subsequent works the presented expressions and equations
will be applied to the study of various interaction processes
with the participation of photon and Dirac fermions In
particular, the application of the whole theoretical tool
elabo-rated in the present work is necessary and also sufficient
for the study of physical processes taking place
completely inside the graphene monolayer This would be
also useful for the study of electromagnetic properties of
gra-phene-based optoelectronic and photonic nanostructures and
nanocomposites
Acknowledgment The authors would like to express their deep gratitude to the Advanced Center of Physics and Institute of Materials Sci-ence, Vietnam Academy of Science and Technology for its support
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