Toward theory of quantum Hall effect in graphene
We analyze a gap equation for the propagator of Dirac quasiparticles and conclude that in graphene in a magnetic field, the order parameters connected with the quantum Hall ferromagnetism dynamics and those connected with the magnetic catalysis dynamics necessarily coexist (the latter have the for...
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irk-123456789-1175562017-05-25T03:03:43Z Toward theory of quantum Hall effect in graphene Gorbar, E.V. Gusynin, V.P. Miransky, V.A. Graphene and graphite multilayers We analyze a gap equation for the propagator of Dirac quasiparticles and conclude that in graphene in a magnetic field, the order parameters connected with the quantum Hall ferromagnetism dynamics and those connected with the magnetic catalysis dynamics necessarily coexist (the latter have the form of Dirac masses and correspond to excitonic condensates). This feature of graphene could lead to important consequences, in particular, for the existence of gapless edge states. Solutions of the gap equation corresponding to recently experimentally discovered novel plateaus in graphene in strong magnetic fields are described. 2008 Article Toward theory of quantum Hall effect in graphene / E.V. Gorbar, V.P. Gusynin, V.A. Miransky // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1007-1011. — Бібліогр.: 35 назв. — англ. 0132-6414 PACS: 73.43.Cd;71.70.Di;81.05.Uw http://dspace.nbuv.gov.ua/handle/123456789/117556 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Graphene and graphite multilayers Graphene and graphite multilayers |
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Graphene and graphite multilayers Graphene and graphite multilayers Gorbar, E.V. Gusynin, V.P. Miransky, V.A. Toward theory of quantum Hall effect in graphene Физика низких температур |
| description |
We analyze a gap equation for the propagator of Dirac quasiparticles and conclude that in graphene in a
magnetic field, the order parameters connected with the quantum Hall ferromagnetism dynamics and those
connected with the magnetic catalysis dynamics necessarily coexist (the latter have the form of Dirac masses
and correspond to excitonic condensates). This feature of graphene could lead to important consequences, in
particular, for the existence of gapless edge states. Solutions of the gap equation corresponding to recently
experimentally discovered novel plateaus in graphene in strong magnetic fields are described. |
| format |
Article |
| author |
Gorbar, E.V. Gusynin, V.P. Miransky, V.A. |
| author_facet |
Gorbar, E.V. Gusynin, V.P. Miransky, V.A. |
| author_sort |
Gorbar, E.V. |
| title |
Toward theory of quantum Hall effect in graphene |
| title_short |
Toward theory of quantum Hall effect in graphene |
| title_full |
Toward theory of quantum Hall effect in graphene |
| title_fullStr |
Toward theory of quantum Hall effect in graphene |
| title_full_unstemmed |
Toward theory of quantum Hall effect in graphene |
| title_sort |
toward theory of quantum hall effect in graphene |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2008 |
| topic_facet |
Graphene and graphite multilayers |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117556 |
| citation_txt |
Toward theory of quantum Hall effect in graphene / E.V. Gorbar, V.P. Gusynin, V.A. Miransky // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1007-1011. — Бібліогр.: 35 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT gorbarev towardtheoryofquantumhalleffectingraphene AT gusyninvp towardtheoryofquantumhalleffectingraphene AT miranskyva towardtheoryofquantumhalleffectingraphene |
| first_indexed |
2025-07-08T12:27:46Z |
| last_indexed |
2025-07-08T12:27:46Z |
| _version_ |
1837081734870466560 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10, p. 1007–1011
Toward theory of quantum Hall effect in graphene
E.V. Gorbar and V.P. Gusynin
Bogolyubov Institute for Theoretical Physics, 03680, Kiev, Ukraine
E-mail: vgusynin@bitp.kiev.ua
V.A. Miransky*
Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada
Received March 25, 2008
We analyze a gap equation for the propagator of Dirac quasiparticles and conclude that in graphene in a
magnetic field, the order parameters connected with the quantum Hall ferromagnetism dynamics and those
connected with the magnetic catalysis dynamics necessarily coexist (the latter have the form of Dirac masses
and correspond to excitonic condensates). This feature of graphene could lead to important consequences, in
particular, for the existence of gapless edge states. Solutions of the gap equation corresponding to recently
experimentally discovered novel plateaus in graphene in strong magnetic fields are described.
PACS: 73.43.Cd Theory and modeling;
71.70.Di Landau levels;
81.05.Uw Carbon, diamond, graphite.
Keywords: graphene, quantum Hall effect, Dirac quasiparticle.
The properties of graphene, a single atomic layer of
graphite [1], have attracted great interest, especially after
the experimental discovery [2,3] and (made indepen-
dently) theoretical prediction [4–6] of an anomalous
quantization in the quantum Hall (QH) effect. In this case,
the filling factors are ν = ± +4 1 2(| | )n / , where n is the
Landau level index. For each QH state, a four-fold (spin
and sublattice-valley) degeneracy takes place. These pro-
perties of the QH effect are intimately connected with
relativistic like features in the graphene dynamics [7–9].
In recent experiments [10,11], it has been observed
that in a strong enough magnetic field, B � 20 T, the new
QH plateaus, ν = ±0 1, and ± 4, occur, that was attributed
to the magnetic field induced splitting of the n = 0 and
n = ± 1Landau levels (LLs). It is noticeable that while the
degeneracy of the lowest LL (LLL), n = 0, is completely
lifted, only the spin degeneracy of the n = ± 1 LL is
removed.
On theoretical side, there are now two leading sce-
narios for the description of these plateaus. One of them is
the QH ferromagnetism (QHF) [12–15] (the dynamics of
a Zeeman spin splitting enhancement considered in
Ref. 16 is intimately connected with the QHF). The se-
cond one is the magnetic catalysis (MC) scenario in
which excitonic condensates (Dirac masses) are spon-
taneously produced [17–20]. For a brief review of these
two scenarios, see Ref. 21.
While the QHF scenario is based on the dynamical
framework developed for bilayer QH systems [22], the MC
scenario is based on the phenomenon of an enhancement of
the density of states in a strong magnetic field, which
catalyzes electron-hole pairing (leading to excitonic con-
densates) in relativistic like systems. The essence of this
effect is the dimensional reduction D D→ − 2 in the elect-
ron-hole pairing dynamics and the presence of the LLL
with energy E = 0 (containing both electron and hole sta-
tes) in relativistic systems in a magnetic field. This uni-
versal phenomenon was revealed in Ref. 23 and was first
considered in graphite in Refs. 24, 25.
On technical side, the difference between these two
scenarios is in utilizing different order parameters in
breaking the spin-sublattice-valley U ( )4 symmetry of the
noninteracting Hamiltonian of graphene. While the QHF
order parameters are described by densities of the con-
© E.V. Gorbar, V.P. Gusynin, and V.A. Miransky, 2008
* On leave from Bogolyubov Institute for Theoretical Physics, 03680, Kiev, Ukraine
served charges connected with diagonal generators of the
nonabelian subgroup SU U( ) ( )4 4⊂ , the order parameters
in the MC scenario are Dirac mass like terms. Note that
while the latter are bifermion operators which are in-
variant under 2 + 1 dimensional Lorentz transformations
(with the Fermi velocity v /F � 10 6 m s playing the role of
light velocity), the QHF charge densities are time like
components of the corresponding conserved currents
which transform as vectors under the Lorentz trans-
formations.
One may think that the QHF and MC order parameters
should compete with each other. However, as will be
shown in this paper, the situation is quite different: These
two sets of the order parameters necessarily coexist,
which implies that they have the same dynamical origin.
The physics underlying their coexistence is specific for
relativistic like dynamics that makes the QH dynamics of
the U ( )4 breakdown in graphene to be quite different
from that in bilayer QH systems [22] whose dynamics
have no relativistic like features.
Our approach is based on studying the gap equation for
the propagator of Dirac quasiparticles. For the description
of the dynamics in graphene, we will use the same model as
in Refs. 24, 25, in which while quasiparticles are confined to
a two-dimensional plane, the electromagnetic (Coulomb)
interaction between them is three-dimensional in nature.
The dynamics will be treated in the Hartree–Fock (mean
field) approximation, which is conventional and appropriate
in this case [12,13,17,24,25]. Then, at zero temperature and
in the clean limit (no impurities), the gap equation takes the
form:
G x y S x y i G x y x y U C
− −= + − − −1 1 0 0
0 0( , ) ( , ) ( , ) ( ) ( )�γ γ δ x y
− −i G x x x y U
C
F
�γ γ δ0 0 3 0tr [ ( , )] ( ) ( )
( )
. (1)
Here x x≡ ( , )0 x , with x t0 ≡ being time coordinate,U C ( )x
is the Coulomb potential in a magnetic field, given in Eq.
(46) in Ref. 25, U
C
F( )
( )0 is its Fourier transform at k = 0,
G x y−1( , ) is the full inverse quasiparticle propagator, and
S x y−1( , ) is the bare inverse quasiparticle propagator,
iS x y i B v x yt B F
− = ∂ + − − −1
0
3 0 3( , ) [( ) ] ( )� μ μ σ γ δ�� ,
(2)
where μ 0 is the electron chemical potential, � = − +i e /c�� A
is the canonical momentum, and μ γ σB B 0 3 is the Zeeman
term (the vector potential A corresponds to the magnetic
field B, B ≡ | |B , μ B is the Bohr magneton, and the Pauli
matrix σ 3 acts on spin indices) [26].
For Dirac matrices γ 0, �, we use the same represen-
tation as in Refs. 17, 25 (xy plane is chosen for graphene).
Note that while the second term on the right hand side of
Eq. (1) describes exchange interactions, the third one is the
Hartree term describing annihilation interactions.
The analysis of gap equation (1) beyond the LLL
approximation is a very formidable problem. Because of
that, we will utilize the following approximation: the
Coulomb potential U C ( )x in the gap equation will be
replaced by the contact interaction G int ( )δ 2
x :
G x y S x y i G G x x x y− −= + − −1 1 0 0 3( , ) ( , ) ( , ) ( )int� γ γ δ
− −i G G x x x y� int [ ( , )] ( )γ γ δ0 0 3tr , (3)
where G int is a dimensional coupling constant. Such an
approximation is common in quantum chromodynamics
(QCD), where long range gluon interactions are replaced
by contact (Nambu–Jona–Lasinio) ones. This leads to a
good description of nonperturbative dynamics in low-
energy region in QCD (for a review, see, for example,
Ref. 27). Because of the universality of the MC phe-
nomenon and because the symmetric and kinematic
structures of the gap equations (1) and (3) are the same, we
expect that approximate gap equation (3) should be at least
qualitatively reliable for the description of the LLL and
first few LLs, say, n = ± 1 LL [28]. This in turn implies
that in the analysis of this gap equation, one should use an
ultraviolet cutoff Λ of the order of the Landau scale
L B eB v /c BF( ) | | [ ] [ ]≡ ⊥ ⊥�
2 300� T K ( i n K e l v i n ) ,
where B⊥ is the component of B orthogonal to the
graphene plane measured in Tesla. The dimensional
coupl ing constant G int should be taken then as
G / eBint ~ 1 ⊥ (see below).
Because of the Zeeman term, the U ( )4 symmetry is bro-
ken down to the «flavor» symmetryU U( ) ( )2 2+ −× , where
the subscript ± corresponds to up and down spin states,
respectively. The generators of the U s( )2 , with s = ± , are
I Ps⊗ , − ⊗i Psγ 3 , γ 5 ⊗ Ps , and γ γ3 5 ⊗ Ps (here I is the
4 4× unit matrix, γ γ γ γ γ5 0 1 2 3= i , and P /± = ±( )1 23σ are
projectors on spin up and spin down states) [17].
Our goal is searching for solutions of Eq. (3) both with
spontaneously broken and unbroken SU s( )2 , where
SU s( )2 is the largest nonabelian subgroup of the U s( )2 .
The Dirac mass term
~ ~ †Δ Δs s s sP Pψ ψ ψ γ ψ≡ 0 , where
~
Δ s is
a Dirac gap (mass), is assigned to the triplet repre-
sentation of the SU s( )2 , and the generation of such a mass
would lead to spontaneous flavor SU s( )2 symmetry brea-
king down to the
~
( )U s1 with the generator γ γ3 5 ⊗ Ps
[17,24,25]. There is also a Dirac mass term of the form
Δ s sPψγ γ ψ3 5 that is a singlet with respect to SU s( )2 , and
therefore its generation would not break this symmetry.
On the other hand, while the triplet mass term is even
under time reversal � , the singlet mass term is � -odd (for
a recent review of the transformation properties of dif-
ferent mass terms in graphene, see Ref. 29). It is noti-
ceable that consequences of the presence of the mass Δ in
graphite were discussed long ago in Ref. 8.
1008 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
E.V. Gorbar, V.P. Gusynin, and V.A. Miransky
The analysis of gap equation (3) that we use is closely
connected with that in Ref. 23 and based on the de-
composition of the quasiparticle propagator over the LL
poles with the residues expressed through the generalized
Laguerre polynomials. A detailed description of the ana-
lysis will be presented elsewhere. Here we will describe
its main results. It was found that, for a fixed spin, the full
inverse quasiparticle propagator takes the following ge-
neral form (compare with Eq. (2)):
iG x y is t s s
− = ∂ + + −1 3 5 0( , ) [( ~ )� μ μ γ γ γ
− + −v x yF s s�� − γ γ δ
~
] ( )Δ Δ 3 5 3 , (4)
where the parameters μ s , ~μs , and
~
Δ s are determined from
gap equation (3). Note that the chemical potential μ ±
inc ludes the Zeeman energy �Z, wi th Z BB= =μ
= 0 67. [ ] [ ]B T K , and the chemical potential ~μs is related to
the density of the conserved pseudospin charge
ψ γ γ ψ† 3 5Ps , which is assigned to the triplet representation
of the SU s( )2 . Therefore, while the masses Δ s and
~
Δ s are
related to the MC order parameters 〈 〉ψγ γ ψ3 5Ps and
〈 〉ψ ψPs , the chemical potentials μ μ μ3 2≡ −+ −( ) / and ~μs
are related to the conventional QHF ones: the spin density
〈 〉ψ σ ψ† 3 and the pseudospin density 〈 〉ψ γ γ ψ† 3 5Ps ,
respectively. Note that while the triplet Dirac mass term
describes the charge density imbalance between the two
graphene sublattices [17,24], the pseudospin density des-
cribes the charge density imbalance between the two valley
points in the Brillouin zone.
The dispersion relations for higher LLs (| |n ≥ 1) fol-
lowing from Eq. (4) are
Ens s s
( ) ~σ μ σμ= − + +
+ + +⊥sign( ) | | / (
~
) ,n neB v cF s s2 2 2
� Δ Δσ (5)
where σ = ± 1 are connected with eigenvalues of the
pseudospin matrix γ γ3 5. The case of the LLL is special,
and its dispersion relation is
E eB eBs s s s s
( )
[~ ( )
~
] ( )
σ μ σ μ= − + + +⊥ ⊥sign signΔ Δ .
(6)
One can see from Eqs. (5), (6) that at a fixed spin, the
terms with σ are responsible for splitting of LLs.
In fact, for each value of spin, our analysis revealed
the following three types of solutions: a) a singlet solu-
tion with a nonzero singlet mass Δ and with no triplet
parameters
~
Δ and ~μ , b) a triplet solution with nonzero
~
Δ
and ~μ, and with the singlet mass Δ being zero, and c) a
mixed solution with Δ,
~
Δ, and ~μ being nonzero. The latter
is realized only in higher LLs. In order to find the most
stable solution among them, we compare the free energy
density Ω of the corresponding ground states. In the mean
field approximation that we use, Ω takes the following
form on solutions of the gap equation [30]
ΩVT i G S G= + −⎡
⎣⎢
⎤
⎦⎥
− −Tr Ln 1 11
2
1( ) , (7)
where VT is the space-time volume, the trace, the loga-
rithm, and the product S G−1 are taken in the functional
sense, and G G G= + −diag( , ).
The process of filling the LLs is described by varying
the electron chemical potential μ 0. We will consider po-
sitive μ 0 (dynamics with negative μ 0 is related by elect-
ron-hole symmetry and will not be discussed separately).
In this paper we will mostly consider the LLL dynamics
(results for the n = 1 LL will be briefly described at the
end of the paper).
For the case when only the LLL is doped, which cor-
responds to the condition | ~ | ( )μ μs s L B± << , we arrive at
the following results:
i) A solution with singlet Dirac masses both for spin up
and spin down is the most favorable for 0 20≤ < +μ A Z,
where A G eB / c≡ ⊥int | | 8π� [31]. It is:
~ ~ , , ( )Δ Δ± ± ± ± ± ⊥= = = = ±μ μ μ0 � A M eBsign (8)
with μ μ± ≡ 0 � Z and M A/≡ −( )1 λ , A L B /= λ π 2 2( ) Λ
where the dimensionless coupling constant λ is
λ π≡ G / vFint
/( )Λ 4 3 2 2 2
� [32]. From dispersion relation
(6), we find that E+ > 0 and E− < 0, i.e., the LLL is half
filled (the energy spectrum in this solution is σ inde-
pendent). Therefore the spin gap ΔE E E0 = −+ − corres-
ponds to the ν = 0 plateau. The value of the gap is
ΔE M Z A0 2 2= + +( ). It is instructive to compare ΔE0
with the spin gap in Ref. 16. The latter contains an en-
hanced Zeeman spin splitting, which corresponds to the
second term 2( )Z A+ in ΔE0. However, besides this term,
there is also the large contribution 2M in ΔE0 in the
present solution, which is connected with a dynamical
singlet Dirac mass for quasiparticles. The presence of this
mass could have important consequences for gapless edge
states whose relevance for the physics of the ν = 0 plateau
was pointed out in Ref. 33. Generalizing the analysis in
Ref. 33, we have found that such states exist only when
the full Zeeman splitting Z A+ is larger than the Dirac gap
M A/= −( )1 λ . This leads to the constraint Z A/> −λ λ( )1 .
Let us consider the case with B B= ⊥ . Then, since Z B~ ⊥
and A B~ ⊥ (see below), this constraint leads to a lower
limit B⊥
( )cr
for the values of B⊥ at which gapless edge
states exist. On the other hand, since Z depends on total B
while A depends only on B⊥ , adding a longitudinal B ||
will decrease the lower limit for B⊥ . It would be inte-
resting to check experimentally this point. Also, these
features could be relevant for the interpretation of the
recent experiments [34], in which no gapless edge states
were detected for B B= ≤⊥ 14 T. We shall return to this
issue below.
Toward theory of quantum Hall effect in graphene
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1009
ii) A hybrid solution, with a triplet Dirac mass for spin
up and a singlet Dirac mass for spin down, is the most
favorable for 2 60A Z A Z+ ≤ < +μ . It is
~
, ~ ( ) , ,Δ Δ ++ + ⊥ + += = = − =M A eB Aμ μ μsign 4 0 ,
~ ~ , , ( )Δ Δ− − − − − ⊥= = = − = −μ μ μ0 3A M eBsign . (9)
As follows from Eq. (6), while E+
+ >( )1
0 , the energies
E +
−( )1
and E E−
+
−
−=( ) ( )1 1 are negative. Consequently, the
LLL is now three-quarter filled and, therefore, the gap
ΔE E E M A1
1 1
2= − = ++
+
+
−( ) ( )
( ) corresponds to the ν = 1
plateau. The latter, unlike the ν = 0 plateau, is directly
related to spontaneous SU ( )2 + flavor symmetry break-
ing.
iii) A solution with equal singlet Dirac masses for spin
up and spin down states is the most favorable for
μ 0 6> +A Z. It is
~ ~ , , ( )Δ Δ± ± ± ± ± ⊥= = = − = −μ μ μ0 7A M eBsign
(10)
(compare with Eq. (8)). It is easy to check from (6) that
both E+ and E− are negative in this case, i.e., the LLL is
completely filled. Therefore, this solution corresponds to
the ν = 2 plateau related to the energy gap ΔE L B2 2� ( )
between the LLL and the n = 1 LL.
This analysis leads us to the picture for the LLL
plateaus which qualitatively agrees with that in experi-
ments [10,11]. In particular, taking the dimensionless
coupling λ to be a free parameter and choosing cutoff Λ to
be of the order of the Landau scale L B( ), we arrive at the
scaling relations, A eB~ | |⊥ , M eB~ | |⊥ , and, there-
fore, ΔE A M eB1 2= + ⊥( ) ~ | | for the gap related to the
ν = 1 plateau. One can check that the experimental value
ΔE1 100~ K for B⊥ = 30 T [11] corresponds to λ ~ .0 02.
However, because interactions with impurities are igno-
red in the clean limit used in the present model, it would
be more reasonable to consider λ, say, in interval
0.02–0.2. Then, for these values of λ, we find from the
constraint Z A/> −λ λ( )1 in the solution i) above that the
gapless edge states exist for | |
( )
B B⊥ ⊥> cr
, where 0 01. T�
B⊥
( )cr
� 200 T. One can see that B⊥
( )cr
is sensitive to the
choice of λ. Therefore in order to fix the critical value
B⊥
( )cr
more accurately, one should utilize a more realistic
and constrained model.
As to the n = 1 LL, we found that there are the gaps
Δ ΔE E A3 5 2= � and ΔE Z A4 2� ( )+ corresponding to
the plateaus ν = 3 5, and ν = 4, respectively (the contri-
butions of Dirac masses are suppressed at least by factor
M /L B2 2( ) there). Note that ΔE3 5, and ΔE4 are essentially
smaller than the LLL gaps ΔE1 and ΔE0, respectively
(ΔE3 5, � ΔE /1 2). On the other hand, the experimental data
yield ΔE Z4 2� , and no gaps Δ ΔE E3 5, have been ob-
served [10,11]. We believe that a probable explanation of
this point is that, unlike Z, the value of the dynamically
generated parameter A corresponding to the | |n > 1 LLs
will be essentially reduced if a considerable broadening
of higher LLs in a magnetic field is taken into account
[17]. If so, the gap ΔE4 will be reduced to 2Z and the gaps
Δ ΔE E3 5, will become unobservable.
Recently, in Ref. 35, a large width Γ1 of 400 K was
determined for the n = 1 LL. The plateaus ν = 3 5, could
become observable if the gaps Δ ΔE E A3 5 2= � calcu-
lated in the clean limit are at least of order Γ1 or larger
[17]. The LLL gap ΔE1 100� K at | |B⊥ = 30 T corres-
ponds to ΔE3 5, � 50 K. Then, taking a conservative es-
timate Γ1 100= K and using A eB~ | |⊥ , we conclude that
to observe the ν = 3 5, plateaus, the magnetic fields should
be at least as large as B ~ 100 T.
In conclusion, we have shown that the QHF and MC
order parameters in graphene are two sides of the same
coin and they necessarily coexist. This feature could have
important dynamical consequences for low energy ex-
citations, in particular, for gapless edge states. It would be
desirable to extend the present analysis to a more realistic
model setup, including the genuine Coulomb inter-
actions, LLs impurity scattering rates, and temperature.
Useful discussions with S.G. Sharapov and I.A. Shov-
kovy are acknowledged. The work of E.V.G. and V.P.G. was
supported by the SCOPES-project IB 7320-110848 of the
Swiss NSF, the grant 10/07-H «Nanostructure systems, na-
nomaterials, nanotechnologies», and by the Program of
Fundamental Research of the Physics and Astronomy Divi-
sion of the National Academy of Ukraine. V.A.M. acknow-
ledges the support of the Natural Sciences and Engineering
Research Council of Canada.
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Toward theory of quantum Hall effect in graphene
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