You will be presented with the full text of a science research paper. Please write an abstract for this paper. Your response should include the abstract and no additional text.

Paper text:
graphene is a two - dimensional ( 2d ) allotrope of carbon on a honeycomb lattice with one electron per @xmath3 orbital ( half - filling ) .
its bare electronic spectrum is described in terms of a linearly dispersing , massless , chiral dirac field ( @xmath4 ) . since its isolation
a few years ago @xcite it was realized that graphene displays a number of unique properties that are at odds with the standard theory of metals .
@xcite one of the most important unresolved questions in graphene is the role of electron - electron interactions . @xcite
even though , due to the vanishing of the density of states at the fermi energy , the electron - electron interactions are expected to remain unscreened and strong , it is not clear what is their influence in the properties of graphene .    in the present work we study the influence of the electron - electron interactions on the static dielectric function of graphene at half filling .
we perform calculations to one order beyond the conventional random phase approximation ( rpa ) vacuum polarization bubble , by including self - energy and vertex corrections into the polarization loop .
our main finding is that vertex contributions become important in the coupling regime @xmath2 , which in turn means that other non - rpa contributions should also be included .
we were mainly motivated by the question whether the interactions can significantly affect the screening properties .
this issue is particularly relevant in graphene for two reasons : ( 1 . )
the effective coupling constant @xmath0 ( see the precise definition below ) in graphene is large @xmath2 , and thus interactions are expected to be generically important , and ( 2 . ) despite of the above , to the best of our knowledge , no clear signatures of interaction effects have been observed so far in graphene .
for example measurements of the compressibility @xcite have not detected electron correlation effects .
in addition , screening of external charged impurities introduced in graphene is also expected to be sensitive to interaction effects , at least on theoretical level , @xcite and could be relevant for interpretation of recent experiments on charged impurity scattering . @xcite it is thus generally important to investigate the problem of how the correlations affect the effective charge of the carriers in graphene , which is determined by the vacuum polarization .
we will assume that graphene at half filling ( i.e. when the chemical potential crosses the dirac point ) remains a homogeneous gas of quasiparticles , which in itself is not necessarily an innocent assumption due to the possibility of puddles , ripples , etc .
@xcite however we assume that the system is homogeneous as the importance of the above effects is still unsettled .
our starting point is the low - energy hamiltonian of graphene which can be written as ( we use units such that @xmath5 ) , @xmath6 where @xmath7 is the fermi velocity , @xmath8 is the chemical potential away from half - filling , @xmath9 is the 2@xmath102 identity matrix , @xmath11 , and @xmath12 are pauli matrices .
the first term in the hamiltonian ( [ ham1 ] ) reflects the effective lorentz invariance that exists in the non - interacting problem at low energies and gives rise to bizarre electronic behavior analogous to the one found in quantum electrodynamics ( qed ) .
@xcite    in eq.([ham1 ] ) , @xmath13 is the electron - electron interaction , @xmath14 where @xmath15 is the fourier transform of the coulomb potential in 2d .
the relative strength of the coulomb interactions to the kinetic energy is determined by graphene s  fine structure constant " @xmath16 . unlike qed
, the dirac fermion velocity is much smaller than the speed of light , @xmath17 , and hence the coulomb field can be treated as instantaneous ( @xmath18 ) . as a result
, the coulomb interaction breaks the lorentz invariance of the problem leading to fundamental differences between the graphene problem and qed . from now on we absorb the dielectric constant of the medium @xmath19 into the definition of the effective charge @xmath20 .
for example in the typical case of graphene on a sio@xmath21 substrate with dielectric constant @xmath22 , we have @xmath23 , where @xmath24 is the charge of the electron .
keeping in mind that @xmath25 , one then finds the coupling constant @xmath26 .
@xcite nevertheless , even in this situation when the relation @xmath27 is not strictly satisfied , perturbation theory is expected to give a good indication for the behavior of physical quantities .
the rest of the paper is organized as follows . section ii deals with corrections to the polarization loop arising from the dressing of the electron propagators . in the section
iii the true interaction ( correlation ) insertion , the vertex correction , is examined .
section iv contains our conclusions .
we concentrate on the most interesting case of zero fermi energy ( @xmath28 ) , when the low - energy physics is controlled by the proximity to the dirac point .
the free dirac green s function is @xmath29 the interaction effects lead to the dressed green s function @xmath30 , where the self - energy @xmath31 is a sum of two terms with different matrix structure : @xmath32 , @xmath33 . at hartree - fock
( hf ) level ( first order in @xmath0 ) a divergent contribution appears , due to the long - range nature of the coulomb interaction @xcite where @xmath34 is an ultraviolet cutoff ( @xmath35 is the lattice spacing ) .
one finds @xmath36 which implies that the effective velocity changes @xmath37 , and grows without bound at low energies @xmath38 .
this should in principle lead to anomalies in thermodynamic and spectral properties of graphene .
@xcite from theoretical viewpoint , most importantly , the single logarithmic behavior was found to persist to second order of perturbation theory as well , @xcite and consequently this is expected to be the case to all orders , reflecting the fairly simple ( at least at weak - coupling ) renormalization structure of the theory .
we now turn to the calculation of the static polarization , @xmath39 .
the frequency variable in @xmath40 is omitted from now on . the bare polarization bubble ( without any interaction lines in the loop )
is known to be @xmath41 from now on the trace stands for summation over spin ( s ) , valley ( v ) and pseudospin ( pauli matrix @xmath42 ) indices , i.e. @xmath43 the momentum sums are performed as @xmath44 .
next , we calculate the bubble dressing due to the electron - electron interactions to first order in @xmath0 .
the two diagrams at this order are shown in fig .
we write the total polarization as @xmath45 where @xmath46 and @xmath47 stand for the contributions of fig .
[ fig1](a ) and fig .
[ fig1](b ) , respectively .        the self - energy dressing of fig .
[ fig1](a ) can be written as ( the factor of @xmath48 originates from the two possible insertions ) @xmath49 at lowest order , the self - energy is simply the hartree - fock one , meaning that in ( [ pol1 ] ) we replace @xmath50 the large logarithm present in @xmath51 at low momenta , eq .
( [ hf ] ) , is expected to appear also in some form in @xmath52 .
let us define the following quantities which appear in our results from now on @xmath53 @xmath54 after performing the frequency , and then momentum integrations in ( [ pol1 ] ) , and using the self - energy from ( [ se ] ) , we obtain @xmath55^{2 } }   \nonumber \\ & &   = \frac{\alpha}{16 } \
\frac{|{\bf q}|}{v } \
\ln(\lambda/|{\bf q}| ) , \ \
\lambda/|{\bf q}| \gg 1.\end{aligned}\ ] ] this result means that the large logarithm in @xmath52 simply reflects the renormalization of the fermi velocity , i.e. this divergence is not independent , but can be simply reabsorbed into the velocity by replacing @xmath56 in the one - loop result @xmath57 . due to the simple logarithmic structure of the theory this
is expected to hold to all orders of perturbation theory , i.e. all self - energy corrections lead to a replacement of the coupling @xmath0 in all final expressions with the  running " coupling @xmath58 , accounting for the velocity renormalization .
we therefore assume that the velocity renormalization procedure is performed in all higher order diagrams . at finite small chemical potential , @xmath59 , which is the case in any realistic experimental situation , the divergence is cut - off , @xmath60 . due to the slow variation of the logarithmic function or possible other factors ( such as strong dielectric screening ) ,
no significant variation of the velocity has been found in experiment . @xcite    an interesting effect , related to the interaction contribution @xmath52 , eq .  ( [ pol10 ] )
, was recently discussed in ref .   within the rg approach .
our calculation , leading to eq .
( [ pol10 ] ) , provides an explicit perturbative confirmation of the rg results .
if we imagine an external coulomb impurity with charge @xmath61 , probing the polarization of the vacuum , then the induced charge density , in momentum space , is @xmath62 . here
while the first term leads to induced charge @xmath64 , localized in real space at the impurity site and with a screening sign , the interaction term @xmath65 has an opposite sign and decays as an inverse power law ( @xmath66 , with logarithmic corrections ) .
this peculiar behavior simply reflects , however , the renormalization ( increase ) of the fermi velocity at low momenta , which leads to suppression of screening at large distances .    , given by eq .
( [ vertex ] ) . ]
we proceed with the calculation of the vertex correction in fig .
[ fig1](b ) . before evaluating this expression , it is useful to examine the ( possible ) singularity structure separately in the vertex operator , shown in fig .
for simplicity we use the notation @xmath67 which stands for the more conventionally used form @xmath68 , where the variables denote both frequency and momenta ( @xmath69 is the bosonic momentum / frequency ) .
since the coulomb interaction is non - retarded , we have the simple expression @xmath70 after evaluating the frequency integral , the result is a sum of an off - diagonal and diagonal parts ( with respect to the pauli matrix indexes ) , @xmath71 . more explicitly , @xmath72 we are interested only in the zero frequency limit ( and only in the real part of @xmath73 , since the imaginary part does not contribute to the polarization ) .
in this case the off - diagonal piece vanishes identically , @xmath74 .
this is expected to be the case since the coulomb interaction is diagonal and thus the vertex can not generate a static contribution with a different matrix structure . on the other hand
the diagonal part is finite in the same limit @xmath75 an explicit evaluation shows that @xmath76 does not have any divergent contributions .
for example at @xmath77 , @xmath78 while in the opposite limit @xmath79 for our purposes the exact formulas are not important ( we also do not show the dependence on the angle between @xmath80 ) ; our main conclusion at this stage is that the vertex does not have any divergent parts .
we have also examined diagrams of higher order , such as  ladder " and  crossed " ladder vertex corrections , and have found that all of them are finite . therefore the vertex insertions into the polarization function are expected to give a finite contribution to that quantity , and below we evaluate the lowest order vertex correction numerically .
it is clear that a ward identity relating divergent contributions in the self - energy and in the vertex does not hold here , unlike conventional qed where lorentz ( and gauge ) invariance guarantees cancellation between vertex and self - energy corrections , @xcite and charge is renormalized only through simple polarization loops in the photon propagator . on the other hand in graphene , where the only non - trivially renormalized quantity is the velocity @xmath7
, both the polarization operator and the vertex operator do not show any independent divergencies .    the diagram of fig .
[ fig1](b ) now reads @xmath81 where the full expression for @xmath73 from eq .
( [ vertex ] ) should be used .
an explicit calculation , starting by evaluation of the energy integrations , leads to the result @xmath82 taking into account : @xmath83 where @xmath84 stands for a vector product , we then arrive at the final formula @xmath85 it is clear on dimensional grounds that @xmath86 is linear in @xmath87 .
this is in fact the case for polarization diagrams in all orders of perturbation theory .
the four - dimensional integrals , appearing in ( [ pol20 ] ) , can not be evaluated analytically .
we have found , as expected in light of our previous discussion of the vertex function , that the expressions converge in the ultraviolet limit .
after computing the integrals numerically , we obtain the following result for the combination @xmath88 , which appears in the dielectric function , @xmath89 adding also the one - loop rpa result , we have finally ( where @xmath90 is the static dielectric constant , defined by the formula below ) @xmath91 @xmath92    we conclude that , at @xmath2 , the vertex correction is more than @xmath1 of the one - loop result .
it also has a screening sign , i.e. it reduces the effective charge .
one also expects that finite contributions will appear to all orders in @xmath0 .
however , resummation of perturbation theory by simple means seems impossible , as the contributions in question are finite and accumulate over a wide range of momenta in the corresponding diagrams ( rather than within a specific integration window , from where divergent parts typically originate , and thus can be easily collected ) . even though the vertex contribution is a sizable one , two remarks
are in order : ( 1 . ) it does not change drastically the structure of the theory , apart from contributing towards further screening of the interactions .
the fact that perturbation theory is used with the intention of being applied at a rather strong coupling is in itself questionable .
nevertheless , perturbation theory provides a clear indication that a significant contribution to screening exists beyond the conventional one - loop rpa result . on the other hand in the weak - coupling regime , @xmath27 ,
rpa is parametrically well justified as far as the static polarization properties are concerned ( although the rpa is not justified for the self - energy .
it is also useful to compare our results to the situation in ordinary metals with a finite fermi surface .
certain approximations are typically used to account for vertex corrections , such as the hubbard form of the dielectric function . when extrapolated to low momentum
, the vertex contribution tends to decrease the screening length , @xcite i.e. it reduces further the range of the interactions .
naturally in graphene , where the screening length is infinite ( for the case of zero chemical potential considered here ) , the vertex correction affects directly the effective charge , without changing the shape of the coulomb potential .
finally we mention two recent related works , discussing interaction effects , that appeared while the present manuscript was being prepared . in ref .
, the effect of self - energy and vertex corrections to lowest order ( @xmath0 ) on the minimal conductivity in graphene was discussed , with the conclusion that the corrections is of order @xmath93 .
dynamical polarization properties were studied in ref .  , where the vertex diagrams were found to have logarithmically singular contributions near the threshold @xmath94 , leading to the possibility of a plasmon mode .    in summary , we have shown that vertex corrections can have sizable effect in the static vacuum polarization diagrams in the regime of strong coupling , while for small coupling their importance diminishes parametrically .
the self - energy corrections are naturally absorbed into the renormalization of the fermi velocity .
the non - rpa vertex diagram at lowest order of perturbation theory was found to decrease the effective charge , meaning that in principle correlation effects at higher order must also be taken into account .
thus the ultimate asymptotic behavior of the static polarization function for @xmath2 remains an open problem .
we discuss the effect of electron - electron interactions on the static polarization properties of graphene beyond rpa . 
 divergent self - energy corrections are naturally absorbed into the renormalized coupling constant @xmath0 . 
 we find that the lowest order vertex correction , which is the first non - trivial correlation contribution , is finite , and about @xmath1 of the rpa result at strong coupling @xmath2 . 
 the vertex correction leads to further reduction of the effective charge . 
 finite contributions to dielectric screening are expected in all orders of perturbation theory .