Lattice gauge theory model for graphene
Alessandro Giuliani,1Vieri Mastropietro,2and Marcello Porta31Università di Roma Tre, L.go S. L. Murialdo 1, 00146 Roma, Italy 2Università di Roma Tor Vergata, Viale della Ricerca Scientifica, 00133 Roma, Italy
3Università di Roma La Sapienza, P.le A. Moro 2, 00185 Roma, Italy
共Received 31 August 2010; published 27 September 2010兲
The effects of the electromagnetic共em兲 electron-electron interactions in half-filled graphene are investigated in terms of a lattice gauge theory model. By using exact renormalization group methods and lattice Ward identities, we show that the em interactions amplify the responses to the excitonic pairings associated to a Kekulé distortion and to a charge-density wave. The effect of the electronic repulsion on the Peierls-Kekulé instability, usually neglected, is evaluated by deriving an exact non BCS gap equation, from which we find evidence that strong em interactions among electrons facilitate the spontaneous distortion of the lattice and the opening of a gap.
DOI:10.1103/PhysRevB.82.121418 PACS number共s兲: 71.10.Hf, 71.30.⫹h, 73.63.Bd, 64.60.ae
Graphene, a monocrystalline graphitic film that has been recently experimentally realized,1 has highly unusual elec-tronic properties, due to the quasirelativistic nature of its charge carriers.2In the presence of long-ranged interactions, graphene provides an ideal laboratory for simulating quan-tum field theory共QFT兲 models at low energies and to possi-bly observe phenomena such as spontaneous chiral symme-try breaking and mass generation. For a few years, there was essentially no experimental signature of electron-electron in-teractions in graphene, due to substrate-induced perturba-tions that obscured their effects. However, the realization of suspended graphene samples is allowing people to collect increasing evidence of interaction effects.3On the theoretical side, understanding the properties of a system of interacting fermions on the honeycomb lattice is a challenging problem, similar to quantum electrodynamics but with some peculiar differences that make its study new and nontrivial;4its com-prehension is essential for graphene and, at the same time, it has relevance for other planar condensed-matter systems, like high Tcsuperconductors, and even for basic questions in
QFT.
Most theoretical analyses deal with a simplified effective continuum model of massless Dirac fermions with static Coulomb interactions. Early results based on lowest-order perturbation theory predicted a growth of the effective Fermi velocity v共k兲 close to the Fermi points and excluded the spontaneous formation of a gap at weak coupling.5共a兲 The absence of a gap is a serious drawback for possible techno-logical applications of graphene. Therefore, people started to investigate possible mechanisms for its generation. One way to induce a gap is by adding an interaction with an external periodic field, which can be generated, e.g., by the presence of a substrate.6In the absence of a substrate, Ref.7proposed a mechanism similar to the one at the basis of spontaneous chiral symmetry breaking in strongly coupled QED3; the ap-plicability of these proposals to real graphene is a delicate issue, due to the uncontrolled approximations related to a large-N expansions. Another possible mechanism for gap generation is based on a Peierls-Kekulé distortion of the hon-eycomb lattice, which is a prerequisite for electron fractionalization.8,9One optimizes over the distortion pattern, by minimizing the corresponding electronic energy. In the
absence of interactions, a rather strong interaction with the classical phonon field is needed for the formation of a non-trivial distortion8while the effects of the electron interactions are still poorly understood.
In this Rapid Communication, we investigate the effects of electronic interactions in terms of a lattice gauge theory model. We show that the electronic interactions amplify the response functions associated to Kekulé共K兲 or charge den-sity wave 共CDW兲 pairings. Moreover, we derive the exact form of the Peierls-Kekulé gap equation in the presence of the electromagnetic共em兲 electron-electron interactions, from which we find evidence that strong em interactions enhance the Peierls-Kekulé instability 共despite the growth of the Fermi velocity,10 which apparently opposes this effect兲. The model is analyzed by the methods of constructive QFT, which have already proved effective in obtaining rigorous nonperturbative results in many similar problems;11 we rely neither on the effective Dirac description nor on a large-N expansion.
The charge carries in graphene are described by tight-binding electrons on a honeycomb lattice coupled to a three-dimensional 共3D兲 quantum em field. We introduce creation and annihilation fermionic operators xជ,⫾ =共a⫾ជ,x , bxជ+␦ជ1,
⫾ 兲
=兩B兩−1兰
kជ苸Bdkជkជ,⫾e⫾ikជxជ for electrons with spin index=↑↓
sitting at the sites of the two triangular sublattices⌳Aand⌳B
of a honeycomb lattice; we assume that ⌳A=⌳ has basis
vectors lជ1,2= 1
2共3, ⫾
冑
3兲 and that ⌳B=⌳A+␦ជj with␦ជ1=共1,0兲 and␦ជ2,3=1
2共−1, ⫾
冑
3兲 the nearest-neighbor vectors; B is the first Brillouin zone. The honeycomb lattice is embedded in R3 and belongs to the plane x3= 0. The grand-canonical Hamiltonian at half filling is H = H0+ HC+ HA, where
H0= − t
兺
xជ苸⌳A兺
j, ax+ជ,bxជ+␦ជj, − eie兰0 1␦ជ j·Aជ共xជ+s␦ជj,0兲ds+ c.c. 共1兲with t the hopping parameter and e the electric charge; the coupling with the em field is obtained via the Peierls substi-tution. Moreover, if nxជ=兺axជ, + axជ, − 共respectively, n xជ =兺bxជ,+ b−ជ,x 兲 for xជ苸⌳A 共respectively, xជ苸⌳B兲, PHYSICAL REVIEW B 82, 121418共R兲 共2010兲 RAPID COMMUNICATIONS
HC= e2
2xជ,yជ苸⌳
兺
A艛⌳B共nxជ− 1兲共xជ− yជ兲共nyជ− 1兲,
where ˆpជ is an ultraviolet regularized version of the static
Coulomb potential. Finally, HAis the energy共in the presence
of an ultraviolet cutoff兲 of the 3D photon field Aគ=共Aជ, A3兲 in the Coulomb gauge. We fix units so that the speed of light c = 1 and the free Fermi velocityv =32tⰆ1. If we allow dis-tortions of the honeycomb lattice, the hopping becomes a function of the bond lengthᐉxជ,jthat, for small deformations,
can be approximated by the linear function txជ,j= t +xជ,j with
xជ,j= g共ᐉxជ,j−ᐉ¯兲 and ᐉ¯ the equilibrium length of the bonds.
The Kekulé dimerization pattern corresponds to, see Fig.1,
xជ,j=0+⌬0cos关pជF
+共␦ជ
j−␦ជj0− xជ兲兴 共2兲
with j0苸兵1,2,3其. In order to investigate the effect of the em interactions and the Peierls-Kekulé instability, we use the following strategy: we first compute the ground-state energy and the correlations for xជ,j= 0 by exact renormalization
group 共RG兲 methods,12 finding, in agreement with previous analyses,5共b兲 that the quasiparticle weight vanishes at the Fermi points and the effective Fermi velocity tends to the speed of light as power laws with nonuniversal critical ex-ponents. In addition, the analysis of the response functions and of the corresponding exponents indicates a tendency to-ward excitonic pairing; the mass terms of K or CDW type are strongly amplified by the interactions. Next, we compute the electronic correlations in the presence of a nontrivial lat-tice distortion xជ,jand we show that a Kekulé dimerization
pattern Eq. 共2兲 is a stationary point of the total energy 共i.e., the sum of the elastic energy and the electronic energy in the Born-Oppenheimer approximation兲.
We start with xជ,j= 0; the analysis is very similar to the
one performed in the continuum Dirac approximation in Ref. 13共which we refer to for more details兲, the main difference being that the present lattice gauge theory model is automati-cally ultraviolet finite and gauge invariant: this avoids the need for an ultraviolet regularization, which can lead to well-known ambiguities.14 The n-points imaginary-time correla-tions can be obtained by the generating functional
eW共J,兲=
冕
P共d兲冕
P共dA兲eV共A+J,兲+共,兲, 共3兲 where:k⫾are Grassman variables, with k =共k0, kជ兲 and k0theMatsubara frequency , and P共d兲 is the fermionic Gaussian integration with inverse propagator
g−1共k兲 = −1 Z
冉
ik0 v⍀ⴱ共kជ兲 v⍀共kជ兲 ik0
冊
共4兲 with Z = 1 and ⍀共kជ兲=23兺j=1,2,3eikជ共␦
ជj−␦ជ1兲关note that g共k兲 is sin-gular only at the Fermi points k = kF⫾=共0,23,⫾32冑3兲兴; if = 0, 1, 2, A共p兲 are Gaussian variables with propagator w共p兲=␦兰dp3 共2兲共兩pជ兩 2+p 3 2兲 p2+p 3
2 , where is an ultraviolet cutoff
function; finallyV=eZ兰关j0A0+vjជAជ兴+h.o.t., where h.o.t. indi-cates higher order interaction terms in A produced by the Taylor expansion of the exponential in H0and jis the bare lattice current.15We compute W共J,兲 via a rigorous Wilso-nian RG scheme, writing the fields , A as sums of fields
共k兲, A共k兲, living on momentum scales 兩k−k
F
⫾兩,兩p兩⯝Mkwith kⱕ0 a scale label and M ⬎1 a scaling parameter; the itera-tive integration of the fields on scales h⬍kⱕ0 leads to an effective theory similar to Eq. 共3兲 with an ultraviolet cutoff around the Fermi points of width Mh and with effective
scale-dependent wave-function renormalization Zh, Fermi
velocity vh, and effective charge eh. This approach works
only if ehdoes not flow to strong coupling; the boundedness
of eh follows from an exact Ward identity共WI兲, derived by
the lattice phase transformation x,⫾→e⫾ie␣xx,⫾ in Wh共0,兲16
p⌳共h兲共k,p兲 = e关Sk+p−1 ⌫0共kជ, pជ兲 − ⌫0共kជ, pជ兲Sk
−1兴, 共5兲 where Sk is the interacting propagator, ⌫0共kជ, pជ兲=共
−i 0 0 −ie−ipជ␦ជ1兲
and ⌳共h兲共k,p兲 is the vertex function17 that, if computed at external momenta兩k−kF⫾兩⬃Mhand兩p兩 ⰆMh, is proportional
to eh共if= 0兲 or ehvh共if= 1 , 2兲. Using Eq. 共5兲, we find that eh→e−⬁= e + e3F共e兲 with F共e兲 a series in e2 with bounded
coefficients, i.e., the effective charge tends to a line of fixed points; F共e兲 is vanishing at lowest order, see Fig.2, but the WI does not exclude that F共e兲 is nonzero at higher orders. A similar WI implies that the photon remain massless and, as an outcome of the above procedure, we get an expansion of the Schwinger functions in the effective couplings ehthat is finite at all orders, see Ref.13 for the proof; this is in con-trast with the naive perturbation theory, which is plagued by logarithmic divergencies. The boundedness of ehmakes such
expansion meaningful and it allows one to control the flow of Zh andvh. One finds that:共i兲 limh→−⬁vh= 1, i.e., Lorentz
invariance spontaneously emerges; 共ii兲 both Zh
−1
and 1 −vh
vanish with two anomalous power laws, see Ref.13. There-fore, ifxជ,j= 0, the dressed propagator has a form similar to
FIG. 1. The Kekulé pattern; the hopping parameter is t +⌬0and
t −⌬0/2 on the double and single bonds, respectively.
FIG. 2. The second-order graphs contributing to the dressed charge e−⬁, given by the contribution of the vertex part minus the
graphs coming from the wave-function renormalization共if=0兲 or the velocity renormalization共=1,2兲; their sum is exactly vanish-ing, in agreement with the WI Eq.共5兲.
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Eq.共4兲 with Z and v replaced by Z共k−kF⫾兲 and v共k−kF⫾兲; if k is far from the Fermi points, Z共k−kF⫾兲 and v共k−kF⫾兲 are
close to their unperturbed values, namely, 1 and 32t. On the contrary, if 兩k
⬘
兩Ⰶ1, Z共k⬘
兲⬃兩k⬘
兩− with =12e22+¯ ananomalous critical exponent that is finite at all orders in e2, and v共k
⬘
兲 tends to the speed of light. Moreover, 1−v共k⬘
兲 ⬃共1−v兲兩k⬘
兩˜with˜ =2e2
52+¯ another anomalous critical
ex-ponent.
The above analysis confirms, at all orders and in the pres-ence of a lattice cutoff, the results found long ago in Ref. 5共b兲, where graphene was described by an effective con-tinuum Dirac model with an ultraviolet dimensional regular-ization; the exponents agree at lowest order. Gauge invari-ance implies that the dressed Fermi velocity has a universal value 共the speed of light兲 and Lorentz invariance emerges; this is what happens both in Ref. 5共b兲 共thanks to the use of dimensional regularization兲 and in the present more realistic lattice model, thanks to the exact lattice WI Eq.共5兲. On the other hand, we expect that as soon as gauge invariance is broken the limiting Fermi velocity is smaller than the speed of light; this is indeed what happens in Ref.13, where gauge invariance is broken by the momentum cutoff.
In order to understand which instabilities are likely to occur in the system, we compute response functions or gen-eralized susceptibilities 共which apparently have never been systematically computed before not even in the Dirac ap-proximation兲; more precisely, for xជ,j= 0, we compute R共␣兲共x, j;y, j
⬘
兲=具x,j共␣兲共␣兲y,j⬘典 with, e.g., x,j共K兲 =兺共ax,+bx+共0,␦ជ j兲, − eie兰01␦ជj·Aជ(x+s共0,␦ជj兲,0)ds+ c.c.兲 or x,j 共CDW兲 =兺共ax,+ax,−− bx+共0,␦ជ j兲, + bx+共0,␦ជ j兲, −兲. The two latter operators describe, respectively, internode and intranode excitonic pairings of K and CDW type; this is because the possible presence of a condensate in the k = kF⫾channel forx,j共K兲 or in the k = 0 channel for x,j共CDW兲 would signal the emergence of long range order 共LRO兲 of K-type 共see Fig.1兲 or of CDW type 共a period-2 alternation of excess/deficit of electrons in the sites of the A/B lattice兲. Other relevant bilinears are the Cooper pairings, i.e., linear combinations of terms of the form ax,+ax,−+ or ax,+bx+共0,␦ជ
j兲,⬘
+
. The large distances asymptotic behavior of the response functions is
R共␣兲共x, j;0, j兲 ⬃ G1共␣兲共x兲 + cos共pជF
+
· xជ兲G2共␣兲共x兲 with兩Gi共␣兲共x兲兩⬃共const兲兩x兩−i
共␣兲
two scaling invariant functions 共similar formulas are valid for j⫽ j
⬘
兲. In the absence of in-teractions, i共␣兲= 4, for all ␣ and i; there are no preferredinstabilities. The presence of the interaction with the em field removes the degeneracy in the decay exponents: some re-sponses are enhanced and some other depressed. It turns out that
1共CDW兲= 4 − 4e2/共32兲 + ¯ , 2共K兲= 4 − 4e2/共32兲 + ¯ . On the contrary, 2共CDW兲and1共K兲are vanishing at second or-der while all the Cooper pairs responses decay faster than 兩x兩−4. The conclusion is that the responses to excitonic pair-ing of K or CDW type are amplified by the em interaction: in
this sense, we can say that the em interaction induces quasi LRO of K and CDW type.
Correspondingly, possible small distortions or inhomoge-neities of the K or CDW type are dramatically enhanced by the interactions. For instance, let us choosexជ,jas in Eq.共2兲.
The RG analysis can be repeated in the presence of a Kekulé mass term⌬0兺xជ,jcos关pF
+
·共xជ−␦ជj+␦ជj0兲兴xជ,j共K兲in the Hamiltonian,
which produces a new relevant coupling constant, the effec-tive Kekulé mass. Therefore, the interaction produces an ef-fective momentum-dependent gap⌬共k兲 that increases with a power law with exponent Kfrom the value⌬0up to
⌬共kF⫾兲 = ⌬01/共1+
K兲
, K= 2e2/共32兲 + ¯ . 共6兲 Note that the ratio of the dressed and bare gaps diverges as ⌬0→0. The enhancement of the dressed gap is related to the phenomenon of gap generation in Ref.7but it is found here avoiding any unrealistic large-N expansion. A similar en-hancement is found for the gap due to a CDW modulation.
Finally, let us discuss a possible mechanism for the spon-taneous distortion of the lattice and the opening of a gap 共Peierls-Kekulé instability兲. We use a variational argument, which shows that a Kekulé dimerization pattern of the form Eq. 共2兲 is a stationary point of the total energy 2g2兺xជ,jxជ,j
2
+ E0共兵xជ,j其兲, where the first term is the elastic energy and E0共兵xជ,j兲其 is the electronic ground-state energy in the
Born-Oppenheimer approximation. The extremality condition for the energy isxជ,j= g2具共K兲xជ,j典, where具·典is the ground-state
average in the presence of the distortion pattern兵xជ,j其.
Com-puting 具xជ,j共K兲典by RG with the multiscale analysis explained
above, we find that Eq. 共2兲 is a stationary point of the total energy, provided that 0= c0g2/+¯ for a suitable constant c0and that⌬0satisfies the following non BCS gap equation:
⌬0⯝g2
冕
⌬ⱗ兩k⬘兩ⱗ1dk⬘
Z−1共k
⬘
兲⌬共k⬘
兲兩⍀共kជ⬘
兲兩2 k02+v2共k⬘
兲兩⍀共kជ⬘
+ pជF兲兩2+兩⌬共k⬘
兲兩2,
where ⌬=⌬01/共1+K兲 and, for ⌬ⱗ兩k
⬘
兩Ⰶ1, Z共k⬘
兲⬃兩k⬘
兩−, 1 −v共k⬘
兲⬃共1−v兲兩k⬘
兩˜, and ⌬共k⬘
兲⯝⌬0兩k⬘
兩−K. In the absence of interactions, Z共k⬘
兲=1, v共k⬘
兲=v, and the above equation reduces to the free one in Ref.8. Our gap equation is quali-tatively equivalent to the simpler expression1 = g2
冕
⌬ 1 d −K 1 −共1 − v兲˜ 共7兲 from which its main features can be easily inferred.At weak em coupling, the integral in the right-hand side 共r.h.s.兲 is infrared convergent, which implies that a nontrivial solution is found only for g larger than a critical coupling gc;
remarkably, gc⬃
冑
v with v the free Fermi velocity, eventhough the effective Fermi velocity flows to the speed of light. Therefore, at weak coupling, the prediction for gc is
qualitatively the same as in the free case;8this can be easily checked by noting that the denominator in the r.h.s. of Eq. 共7兲 is sensibly different from v only if is exponentially small inv/˜ .
On the other hand, if one trusts our gap equation also at
LATTICE GAUGE THEORY MODEL FOR GRAPHENE PHYSICAL REVIEW B 82, 121418共R兲 共2010兲
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strong em coupling and if in such a regime K−= 7e2
122
+¯ exceeds one, then the r.h.s. of Eq. 共7兲 diverges as ⌬ →0, which guarantees the existence of a nontrivial solution for arbitrarily small g; this can be easily checked by rewrit-ing Eq.共7兲, up to smaller corrections, as 1⯝K−g2−1⌬1+−
K , that is ⌬0⯝g2共1+K兲/共K−−1兲: note the non BCS form of the gap, similar to the one appearing in certain Luttinger superconductors.18The existence of a nontrivial solution for arbitrarily small g suggests that strong em interactions be-tween fermions enforce the Peierls-Kekulé mechanism and facilitate the spontaneous distortion of the lattice and the gap generation, by lowering the critical phonon coupling gc; this
is in agreement with the one-dimensional case, where the Peierls instability is enhanced by the electronic repulsion, see Ref.19. Note the crucial role in the above discussion played by the momentum dependence of the gap term 关leading to the factor−Kin Eq.共7兲兴; on the contrary, the growth of the
Fermi velocity plays a minor role. A similar analysis can be repeated for the gap generated by a CDW instability.
In conclusion, we considered a lattice gauge theory model for graphene and we predicted that the electron repulsion enhances dramatically, with a nonuniversal power law, the gaps due to the Kekulé distortion or to a density asymmetry between the two sublattices, as well as the responses to the corresponding excitonic pairings. Moreover, we derived an exact non BCS gap equation for the Peierls-Kekulé instabil-ity from which we find evidence that strong em interactions facilitate the spontaneous distortion of the lattice and the gap generation, by lowering the critical phonon coupling.
A.G. and V.M. gratefully acknowledge financial support from the ERC under Grant No. CoMBoS-239694. We thank D. Haldane and M. Vozmediano for many valuable discus-sions.
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=兰2兩B兩dk 共2兲dp3k+p
+ ⌫共k,p兲
k
−
where⌫are matrices which can be deduced from Eq. 共1兲 with ⌫0共kF, 0兲=−iI, ⌫1共kF, 0兲=−2,
⌫2共kF, 0兲=−1. 16W
his defined in the same way as W with an extra infrared cutoff
suppressing momenta smaller than Mh. 17More precisely, p关S
k+p⌳共h兲共k,p兲Sk兴ij=␣p
2
k,j−k+p,i+
Wh共␣,兲兩␣==0.
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Carl-son, D. Orgad, S. A. KivelCarl-son, and V. J. Emery, Phys. Rev. B
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