Graphene and quantum electrodynamics
A single atomic layer of carbon, graphene, has the low-energy "relativistic- like" gapless quasiparticle excitations which in the continuum approximation are described by quantum electrodynamics in 2+1 dimensions. The Dirac- like character of charge carriers in graphene leads to several un...
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nasplib_isofts_kiev_ua-123456789-1118692025-02-09T13:42:15Z Graphene and quantum electrodynamics Графен i квантова електродинамiка Графен и квантовая электродинамика Gusynin, V.P. Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина A single atomic layer of carbon, graphene, has the low-energy "relativistic- like" gapless quasiparticle excitations which in the continuum approximation are described by quantum electrodynamics in 2+1 dimensions. The Dirac- like character of charge carriers in graphene leads to several unique electronic properties which are important for applications in electronic devices. We study the gap opening in graphene following the ideas put forward by P. I. Fomin for investigation of chiral symmetry breaking and particle mass generation in quantum field theory. Одноатомний шар в углецю, графен, має низькоенергетичнi безщiлиннi квазiчастинковi збудження, якi в континуальному наближеннi описуються квантовою електродинамiкою в розмiрностi 2+1. Дiракiвський характер заряджених носiї в у графенi призводить до багатьох унiкальних електронних властивостей, що мають важливе значення для застосувань в електронних пристроях. Ми дослiджуємо вiдкриття щiлини в графенi, слiдуючи iдеям, запропонованим П.I.Фомiним для дослiдження порушення киральної симетрiїi генерацiї мас частинок у квантовiй теорiї поля. Одноатомный слой углерода, графен, обладает низкоэнергетическими безщелевыми квазичастичными возбуждениями, которые в континуальном приближении описываются квантовой электродинамикой в размерности 2+1. Дираковский характер заряженных носителей в графене приводит ко многим уникальным электронным свойствам, имеющим важное значение для применений в электронных устройствах. Мы исследуем открытие щели в графене, следуя идеям, предложенным П.И.Фоминым для исследования нарушения киральной симметриии генерации масс частиц в квантовой теории поля. I have benefited a lot from inspiring common work with my teacher Petr Ivanovich Fomin, the pioneer of the theory of the dynamical origin of masses of elementary particles. This work was supported in part by the European FP7 program Grant No. SIMTECH 246937, the joint Ukrainian-Russian SFFR-RFBR Grant F53.2/028, and the Grant STCU #5716-T ”Development of Graphene Technologies and Investigation of Graphene-based Nanostructures for Nanoelectronics and Optoelectronics”. The author acknowledges also a collaborative grant from the Swedish Institute. 2013 Article Graphene and quantum electrodynamics / V.P. Gusynin // Вопросы атомной науки и техники. — 2013. — № 3. — С. 29-34. — Бібліогр.: 27 назв. — англ. 1562-6016 PACS: 81.05.uW, 03.65.Pm https://nasplib.isofts.kiev.ua/handle/123456789/111869 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина |
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Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина Gusynin, V.P. Graphene and quantum electrodynamics Вопросы атомной науки и техники |
| description |
A single atomic layer of carbon, graphene, has the low-energy "relativistic- like" gapless quasiparticle excitations which in the continuum approximation are described by quantum electrodynamics in 2+1 dimensions. The Dirac- like character of charge carriers in graphene leads to several unique electronic properties which are important for applications in electronic devices. We study the gap opening in graphene following the ideas put forward by P. I. Fomin for investigation of chiral symmetry breaking and particle mass generation in quantum field theory. |
| format |
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| author |
Gusynin, V.P. |
| author_facet |
Gusynin, V.P. |
| author_sort |
Gusynin, V.P. |
| title |
Graphene and quantum electrodynamics |
| title_short |
Graphene and quantum electrodynamics |
| title_full |
Graphene and quantum electrodynamics |
| title_fullStr |
Graphene and quantum electrodynamics |
| title_full_unstemmed |
Graphene and quantum electrodynamics |
| title_sort |
graphene and quantum electrodynamics |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2013 |
| topic_facet |
Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина |
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https://nasplib.isofts.kiev.ua/handle/123456789/111869 |
| citation_txt |
Graphene and quantum electrodynamics / V.P. Gusynin // Вопросы атомной науки и техники. — 2013. — № 3. — С. 29-34. — Бібліогр.: 27 назв. — англ. |
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Вопросы атомной науки и техники |
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2025-11-26T08:10:09Z |
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2025-11-26T08:10:09Z |
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1849839706283442176 |
| fulltext |
GRAPHENE AND QUANTUM ELECTRODYNAMICS
V.P.Gusynin∗
Bogolyubov Institute for Theoretical Physics NAS of Ukraine, 03680, Kiev, Ukraine
(Received January 1, 2013)
A single atomic layer of carbon, graphene, has the low-energy “relativistic- like” gapless quasiparticle excitations
which in the continuum approximation are described by quantum electrodynamics in 2+1 dimensions. The Dirac-
like character of charge carriers in graphene leads to several unique electronic properties which are important for
applications in electronic devices. We study the gap opening in graphene following the ideas put forward by P. I. Fomin
for investigation of chiral symmetry breaking and particle mass generation in quantum field theory.
PACS: 81.05.uW, 03.65.Pm
1. INTRODUCTION
Graphene, a one-atom-thick sheet of crystalline car-
bon with atoms packed in the honeycomb lattice, is
a remarkable system with many unusual properties
that was fabricated for the first time 8 years ago [1].
Graphene is the building block for many forms of car-
bon allotropes, for example, graphite is obtained by
the stacking of graphene layers that is stabilized by
weak van der Waals forces, carbon nanotubes are
formed by graphene wrapping, and fullerenes can
also be obtained from graphene by replacing several
hexagons into pentagonsand heptagons.
The ability to sustain huge (> 108A/cm2) elec-
tric currents and a very high electron mobility, µ ∼
2·105cm2/Vs for suspended graphene, make graphene
a promising candidate for applications in electronic
devices such as nanoscale field effect transistors. Its
thermal conductivity ∼ 5 · 103W/mK is larger than
for carbon nanotubes or diamond. Graphene is op-
tically transparent: it absorbs πα ≈ 2.3% of white
light (α = 1/137 – fine structure constant) that can
be important for making liquid crystal displays. It
is the strongest material ever tested with stiffness
340 N/m. Because of this graphene remains stable
and conductive at extremely small scales of order sev-
eral nanometers.
Theoretically, it was shown long time ago [2] that
quasiparticle excitations in graphene have a linear
dispersion at low energies and are described by the
massless Dirac equation in 2+1 dimensions. The ob-
servation of anomalous integer quantum Hall effect
in graphene [3, 4] is in perfect agreement with the
theoretical predictions [5]–[7] and became a direct
experimental proof of the existence of gapless Dirac
quasiparticles in graphene. In the continuum limit,
graphene model on a honeycomb lattice, with both
on-site and nearest-neighbor repulsions,maps onto a
2+1-dimensional field theory of Dirac fermions in-
teracting through the Coulomb potential plus, in
general, some residual short-range interactions rep-
resented by local four-fermion terms.
The vanishing density of states at the Dirac points
ensures that the Coulomb interaction between the
electrons in graphene retains its long-range character
due to vanishing of the static polarization function
when the wave vector ~q → 0. The large value of the
“fine-structure” coupling constant αg = e2/~vF ∼ 1
(vF ≈ 106 m/s is the Fermi velocity) means that a
strong attraction takes place between electrons and
holes in graphene at the Dirac points. For graphene
on a substrate with the effective coupling αg/κ ¿ 1,
κ being a dielectric constant, the system is in a weak-
coupling regime and exhibits semimetallic properties
due to the absence of a gap in the electronic spec-
trum. All the currently proposed applications of
graphene are based on that fact. Much less is known
about suspended graphene where the coupling con-
stant is large.In fact, suspended graphene provides a
condensed-matter analog of strongly coupled quan-
tum electrodynamics (QED) intensively studied by
P.I. Fomin and collaborators in the 1970s and 1980s.
[8, 9]. The dynamics of the vacuum in QED leads
to many peculiar effects not yet observed in nature.
Some QED-like effects such as zitterbewegung (trem-
bling motion), Klein tunneling, Schwinger pair pro-
duction, supercritical atomic collapse and symmetry
broken phase with a gap at strong coupling have a
chance to be tested in graphene. In fact, graphene
could be used as a bench-top particle-physics labora-
tory,allowing us to investigate the fundamental inter-
actions of matter(recently, the Klein tunneling and
supercritical atomic collapse in graphene were ob-
served experimentally, see papers [10] and [11], re-
spectively).
∗Corresponding author E-mail address: vgusynin@bitp.kiev.ua
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2013, N3(85).
Series: Nuclear Physics Investigations (60), p.29-34.
29
2. DYNAMICAL MASS GENERATION IN
QUANTUM FIELD THEORY
Proposals that electron-electron interactions in
graphene could generate an electronic gap were in-
vestigated and actually preceded the discovery of
graphene [12, 13]. The gap opening in graphene is
an analogue of the phenomenon of dynamical mass
generation which is studied in quantum field theory
since the Nambu-Jona-Lasinio (NJL) paper in 1961
[14] The Lagrangian of the NJL model//
L = Ψ̄iγµ∂µΨ +
G
2
[
(Ψ̄Ψ)2 + (Ψ̄iγ5Ψ)2
]
(1)
is invariant under ordinary phase and chiral transfor-
mations,
Ψ → eiαΨ, Ψ̄ → Ψ̄e−iα,
Ψ → eiαγ5Ψ, Ψ̄ → Ψ̄e−iαγ5 ,
(2)
and the chiral invariance forbids the mass generation
in perturbation theory. The self-consistent Hartree-
Fock equation for a mass (Λ is a cutoff),
4π2m
GΛ2
= m− m3
Λ
ln
(
1 +
m2
Λ2
)
(3)
has a nontrivial solution m 6= 0 if the coupling con-
stant exceeds some critical value, G > Gc = 4π2
Λ2 ,
leading to a nontrivial chiral condensate < Ψ̄Ψ > 6= 0.
In QED the problem of a mass generation starts
since the work by V. Weisskopf [15] who calculated
for the first time the one-loop correction to the bare
electron mass m0,
m = m0 +
3α
2π
m0 ln
(
Λ
m
)
(4)
The total electron mass m vanishes for m0 = 0. On
the other hand, solving the self-consistent equation
for the mass,
m = m0 +
3α
2π
m ln
(
Λ
m
)
(5)
one gets a nontrivial solution,
m = Λ exp
(
−3π
2α
)
, (6)
even for zero bare electron mass m = 0. This so-
lution, called the superconducting-type solution be-
cause of its nonanalytical dependence on the coupling
α, was thoroughly studied in the works by P. I. Fomin
(see review [16] and the references therein) using the
Schwinger-Dyson equations for the electron and pho-
ton propagators and for the vertex function. The
solution (6) should be compared to the solution for
a quasiparticle gap in the Bardeen-Cooper-Schrieffer
(BCS) theory of superconductivity,
∆ = ωD exp(− 2
gνF
), (7)
where ωD is the Debye frequency, g the electron-
phonon coupling constant, and νF is the density of
states on the Fermi surface. In the BCS theory the
gap in the quasiparticle spectrum is generated for any
value of the coupling constant. The physical reason
for zero value of the critical coupling constant in the
BCS theory is connected with the presence of the
Fermi surface.
More refined study of the Schwinger-Dyson equa-
tion for the fermion propagator in QED (Fig. 1), leads
in the ladder approximation to the solution [17],
m ' Λ exp
(
− const√
α− αc
)
, (8)
where the critical coupling constant is of order one,
αc ∼ 1 Note the essentially nonanalytical dependence
of the mass on the coupling constant α. Such a behav-
ior of a mass remains valid in more refined approx-
imation when fermions loops are neglected but all
diagrams with crossed photon lines are taken into ac-
count (the so-called quenched approximation). Tak-
ing into account the vacuum polarization [18] changes
the result to
m ' Λ(α− αc)β , α & αc, β . 1
2
. (9)
Fig.1.The Schwinger-Dyson equation for the
fermion propagator
The solution (8) for the dynamical electron mass
combines features of a superconducting gap(non-
analytical dependence on the coupling constant) and
of the NJL gap (the presence of a critical coupling).
The presence of a magnetic field makes the situa-
tion even more interesting. It was shown in Ref. [19]
that a magnetic field catalyzes the gap generation
for gapless fermions in relativistic-like systems, and
even the weakest attraction leads to the formation
of a symmetry-breaking condensate and a gap gen-
eration. For example, in the four-dimensional NJL
model in an external magnetic field the gap (mass) is
generated at any coupling constant [20]:
∆ '
√
|eB| exp(− 1
2ν0G
) , (10)
where B is the strength of a magnetic field and
ν0 = |eB|/4π2 is the density of states at the low-
est Landau level (compare with a superconducting
gap).This phenomenon is called magnetic catalysis
and its main features are model independent. The
essence of the magnetic catalysis is that the dynam-
ics of the electrons in a magnetic field, B,corresponds
effectively to a theory with spatial dimension reduced
by two units (note a close similarity with the role of
30
the Fermi surface in the BCS theory) if their energy
is much less than the Landau gap
√
|eB|. The zero-
energy Landau level, which plays, in fact, the role
of the Fermi surface, has a finite density of states
and this is the key ingredient of magnetic catalysis.
The magnetic catalysis plays an important role in
quantum Hall effect studies in graphene where it is
responsible for lifting the degeneracy of the Landau
levels.
3. LOW-ENERGY EFFECTIVE THEORY
OF GRAPHENE
The electronic band structure of graphene close to
the Fermi level forms the basis of the low-energy ef-
fective theory of graphene. This band structure is a
reflection of the hexagonal arrangement of the car-
bon atoms. As was shown by P. Wallace [2], the
two-dimensional graphene honeycomb lattice has two
atoms per unit cell and its tight-binding band struc-
ture consists of conduction and valence bands. The
energy spectrum of electrons as a function of a wave
vector has the form
E(~k) = ±t
√
1 + 4 cos
√
3kxa
2
cos
kya
2
+ 4 cos2
kya
2
,
(11)
where t ≈ 2.8 eV is nearest-neighbor hopping energy
(hopping between different sublattices), in the tight-
binding lattice Hamiltonian, a = 2.46 Å is the lattice
constant. The two bands touch each other and cross
the Fermi level, corresponding to zero chemical po-
tential µ = 0, in six K points located at the corners
of the hexagonal 2D Brillouin zone, but only two of
them, say K, K ′ = (± 4π
3α , 0), are nonequivalent. Low-
energy excitations near two nonequivalent K-points
have a linear dispersion, E(~k) = ±~vF |~k| where
~ = h/2π is the Planck constant, vF =
√
3ta/2~ ≈
106 m/s is the Fermi velocity which is only 300 times
less the velocity of light. Electron states at each K-
point are described by the spinor,
ΨKσ =
(
ΨKAσ
ΨKBσ
)
,
which consists of electron wave functions on atoms A,
B of two sublattices, σ = ± describes the real spin,
and satisfies the massless Dirac equation,
(
~γ0i∂t + ~vF (iγ1∂x + iγ2∂y)
)
ΨKσ(x, y) = 0. (12)
Two-dimensional Dirac gamma matrices are given in
terms of the Pauli matrices γµ = (τ3, iτ2,−iτ3), µ =
0, 1, 2. Spinors at two K-points can be combined into
one four-dimensional Dirac spinor
Ψσ = (ΨKAσ,ΨKBσ, ΨK′Aσ, ΨK′Bσ)T ,
which satisfies the Dirac equation (12) but with 4x4
gamma matrices γµ = τ̃3 ⊗ (τ3, iτ2,−iτ3) (the Pauli
matrix τ̃3 acts in the space of two K-points). The
four-component spinor structure accounts for quasi-
particle excitations of sublattices A and B around the
two Dirac points (valleys) in the band structure.
Taking into account the Coulomb interaction be-
tween quasiparticles we come at the effective low-
energy theory described by the QED-like action (the
spin index is omitted)
S =
∫
dtd3r
{
1
2
(∂iA0)2 −A0j0+
[
Ψ̄(t, r)(~γ0i∂t + ~vF iγi∂i −∆0)Ψ(t, r)
]
δ(z)
}
,
(13)
where j0 = eΨ̄(t, r)γ0Ψ(t, r)δ(z) is the charge den-
sity. Note that the electric field, described by the
scalar potential A0 and responsible for interparticle
interaction, propagates in a three-dimensional space,
while fermion fields, describing electron- and hole-
type quasiparticles, are localized on two-dimensional
planes. This is a typical example of the so-called
brane world models studied last years in high en-
ergy physics.Integration over z-coordinate leads to
the standard nonlocal Coulomb interaction for qua-
siparticles in a plane. In Eq. (13) we included also
a bare gap ∆0 which can be generated due to the
interaction with a substrate [21]. In the presence of
this term the dispersion law for quasiparticles takes
the form E(~k) = ±
√
(~vF
~k)2 + ∆2
0 and two zones are
separated by 2∆0. For ∆0 = 0 the continuum effec-
tive theory described by the action (13) possesses the
large “flavor” U(4) symmetry in the spin-valley space.
In the next section we will study the dynamical sym-
metry breaking of this symmetry and generation of
a quasiparticle gap due to the Coulomb interaction
when the bare gap is zero.
4. GAP GENERATION AND
SEMIMETAL-INSULATOR PHASE
TRANSITION IN GRAPHENE
In pristinegraphenean analog of the fine-structure
constant αg & 1. Given the strong attraction,
one may expect an instability in the excitonic chan-
nel (electron-hole pairing) with subsequent quantum
phase transition to a phase with gapped quasipar-
ticles that may turn graphene into an insulator.
In graphene sheets deposited on a substrate,such a
transition is effectively inhibited due to the screen-
ing of the Coulomb interaction by the dielectric
(αg → αg/κ). This semimetal-insulator transition
in graphene is widely discussed now in the literature
since the first study of the problem in Refs. [12, 13].
As is well known, in the BCS theory the Bethe-
Salpeter (BS) equation for an electron-electron bound
state in the normal state of metal has a solution with
imaginary energy, i.e., a tachyon. This means that
normal state is unstable and a phase transition to
the superconducting state takes place.
In order to analyze excitonic instability in
graphene, we consider the BS equation for an
electron-hole bound state. In the random phase ap-
proximation, the BS equation for an amputed bound-
31
state wave function reads
χαβ(q, P ) =
iαg
(2π)2
∫
d3kD(k0, |~q − ~k|)×
×
[
γ0S(q +
P
2
χ(k, P )S(k − P
2
)γ0
]
αβ
, (14)
where q = (q0, ~q), P = (P0, ~P ) are relative and total
energies-momenta of bound electron and hole. The
Coulomb propagator has the form
D(ω, ~q) =
1
|~q|+ Π(ω, ~q)
, (15)
and the one-loop polarization function is
Π(ω, ~q) =
πe2
2κ
~q2
√
(~vF ~q)2 − ω2
, (16)
which in the instantaneous approximation reduces
to Π(ω, ~q) = παg|~q|/2κ. We consider the following
ansatz for the matrix structure of the wave function
of an excitonic bound state,
χ(q, P ) = χ5(q, P )γ5 + χ05(q, P )~q~γγ0γ5 . (17)
In the random phase approximation with static polar-
ization function, i.e., Π(ω = 0, ~q), we find an analyti-
cal solution with pure imaginary energy (tachyon) if
the coupling constant αg exceeds some critical value
αgc (for details, see Ref. [22]):
P 2
0 = −4Λ2 exp
(
− 4πn√
4λ− 1
+ δ
)
, δ ≈ 7.3, (18)
λ > λc =
1
4
, λ =
αg
2 + παg
, n = 1, 2, ...
In the instantaneous approximation αgc = 2.33, while
more accurate numerical treatment gives αgc = 1.62.
In the supercritical regime the wave function χ5(~q) as
a function of a relative momentum behaves asymp-
totically as
χ5(|~q|) ∼ |~q|−1/2 cos
(√
λ− 1
4
ln
( |~q|
|P0|
)
+ const
)
.
Such oscillatory behavior is typical for the phenom-
enon known in quantum mechanics as the “fall into
the center” (collapse): in this case the energy of a sys-
tem is unbounded from below and there is no ground
state. Nodes of the wave function of the bound state
signify the existence of the tachyon states with imag-
inary energy P0.
To find a stable ground state we solved a gap
equation which in the random phase approximation
with static polarization has the form [23],
∆(P ) = λ
∫ Λ
0
q∆(q)K(p, q)√
q2 + (∆(q)/vF )2
.dq (19)
Here the symmetric kernel of the equation is
K =
2
π
1
p + q
K
(
2
√
pq
p + q
)
,
and K(x) is the complete elliptic integral of the first
kind. The nontrivial solution for the momentum
dependent gap function ∆(p) exists if the coupling
λ > 1/4 (αgc > 1.62 and its value at p = 0 (gap
itself) is given by
∆(p = 0) = ΛvF exp
(
− π√
λ− 1/4
)
. (20)
The essentially nonanalytical dependence of the gap
(20) on the coupling constant corresponds to a contin-
uous phase transition of infinite order. Such a behav-
ior is inherent for the Berezinskii-Kosterlitz-Thouless
phase transition, or the conformal phase transition,
and is related to the scale invariance of the problem
under consideration. Note, however, that taking into
account the finite size of graphene samples should
turn this phase transition into a second-order one.
In the case of frequency dependent polarization
function (dynamical screening) the critical constant is
lower, αgc = 0.92. A dynamical gap is generated only
if αg > αgc. Since for suspended clean graphene the
fine-structure constant αg = 2.19 is supercritical, the
dynamical gap will be generated making graphene an
insulator. Note that for graphene on SiO2 substrate,
the dielectric constant κ=2.8 and αg = 0.78, i.e., the
system is in the subcritical regime. The value of αgc
is rather large that implies that a weak-coupling ap-
proach might be quantitatively inadequate for the
problem of the gap generation in suspended clean
graphene. Therefore, it is instructive to compare our
analytical results to lattice MonteCarlo studies, [24]
where αgc = 1.08 ± 0.05 that is rather close to our
analytical findings.
Additional short-range four-fermion interactions
were also included in the continuum model to account
for the lattice simulation results [23]. In spite of being
small, the induced local interactions can play a signif-
icant role in the critical behavior observed in lattice
simulations.Instead of a critical point, we obtained
the critical line in the plane of electromagnetic, α,
and four-fermion, g̃, coupling constants (Fig. 2), and
found a second-order phase transition separating zero
gap and gapped phases with critical exponents close
to those found in lattice calculations.
We expect that the form of the critical curve in
graphene can be checked in further lattice simula-
tions.
Fig.2. Phase diagram
32
5. CONCLUSIONS
We studied in this paper an intriguing possibility
that spontaneous formation of excitons, electron-hole
bound states, and the concomitant formation of exci-
tonic condensate which breaks chiral symmetry, may
turn graphene into a Mott insulator. The insulator
phase of graphene with gapped quasiparticles is in
line with the strong coupling phase of QED stud-
ied by P.I. Fomin and his collaborators in the 70s
and 80s. The strong coupling phase of QED was
searched in heavy ion collisions in GSI-Darmstadt in
the 80s. Various sharp lines were observed in energy
spectra of electron-positron and photon-photon emit-
ted back-to-back with kinetic energies near 350 keV.
Suggested explanation was that the correlated sig-
nals corresponded to the decay of a neutral particle
produced in the collision, and that particle is natu-
rally present in strong coupling phase of QED (tightly
bound electron-positron pairs). [25] However, later
on the Darmstadt experiments have faded.
The spontaneous gap generation and insulator
phase in graphene(strong coupling phase of graphene)
have been predicted [12, 13] before graphene was fab-
ricated in the laboratory. So far, however experi-
mentally there has been no conclusive evidence that
semimetal-insulator transition of suspended mono-
layer graphene occurs at low temperature. [26] In
this respect, it has been already suggested that the
growth of the Fermi velocity,when approaching the
charge neutrality point [23, 27], makes the effective
interaction strength smaller than a critical value, thus
preventing a gap generation.Certainly, this problem
worth of further studying, both theoretically and ex-
perimentally.
ACKNOWLEDGEMENTS
I have benefited a lot from inspiring common work
with my teacher Petr Ivanovich Fomin, the pioneer
of the theory of the dynamical origin of masses of
elementary particles.
This work was supported in part by the Euro-
pean FP7 program Grant No. SIMTECH 246937,
the joint Ukrainian-Russian SFFR-RFBR Grant
F53.2/028, and the Grant STCU #5716-T ”Devel-
opment of Graphene Technologies and Investigation
of Graphene-based Nanostructures for Nanoelectron-
ics and Optoelectronics”. The author acknowledges
also a collaborative grant from the Swedish Institute.
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ГРАФЕН И КВАНТОВАЯ ЭЛЕКТРОДИНАМИКА
В.П. Гусынин
Одноатомный слой углерода, графен, обладает низкоэнергетическими безщелевыми квазичастичными
возбуждениями, которые в континуальном приближении описываются квантовой электродинамикой в
размерности 2 + 1. Дираковский характер заряженных носителей в графене приводит ко многим уни-
кальным электронным свойствам, имеющим важное значение для применений в электронных устрой-
ствах. Мы исследуем открытие щели в графене, следуя идеям, предложенным П.И. Фоминым для
исследования нарушения киральной симметрии и генерации масс частиц в квантовой теории поля.
ГРАФЕН I КВАНТОВА ЕЛЕКТРОДИНАМIКА
В.П. Гусинiн
Одноатомний шар вуглецю, графен, має низькоенергетичнi безщiлиннi квазiчастинковi збудження,
якi в континуальному наближеннi описуються квантовою електродинамiкою в розмiрностi 2+1. Дiра-
кiвський характер заряджених носiїв у графенi призводить до багатьох унiкальних електронних вла-
стивостей, що мають важливе значення для застосувань в електронних пристроях. Ми дослiджуємо
вiдкриття щiлини в графенi, слiдуючи iдеям, запропонованим П.I. Фомiним для дослiдження пору-
шення киральної симетрiї i генерацiї мас частинок у квантовiй теорiї поля.
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