You are an expert at summarizing long articles. Proceed to summarize the following text: 
a great deal of attention has recently been directed to graphene , a monolayer of carbon atoms , both experimentally  @xcite and theoretically  @xcite . graphene is marked with its novel charge carriers that behave like massless dirac fermions with effective speed of light @xmath2 m / s @xmath3 . it thus provides a special opportunity to study @xmath1relativistic " quantum dynamics in condensed - matter systems . experiments have revealed a number of exotic transport properties of graphene , such as the half - integer quantum hall ( qh ) effect and minimal conductivity . the dynamics of dirac fermions becomes particularly interesting in a magnetic field and leads to peculiar quantum phenomena , such as fermion number fractionalization  @xcite and spectral asymmetry ,  @xcite intimately tied to the chiral anomaly in 1 + 1 dimensions . actually the half - integer qh effect and the presence of the zero - energy landau levels observed  @xcite in graphene are a manifestation of fermion number fractionalization . it would be important to explore further possible signatures of relativistic quantum field theory in the low - energy physics of graphene . an interesting proposal  @xcite along this direction is to simulate the klein paradox  @xcite ( or tunneling ) in graphene . calculations  @xcite of the dielectric function also reveal that the electromagnetic response of graphene is substantially different from that of conventional two - dimensional systems . the difference becomes even prominent under a strong magnetic field . @xcite in particular , for graphene the vacuum state is a dielectric medium and carries an appreciable amount of electric and magnetic susceptibilities over all range of wavelengths ; this reflects the presence of the @xmath1dirac sea " . curiously the zero - energy landau levels , though distinct in hall conductance , hardly contribute to the susceptibilities . the purpose of this paper is to study further aspects of the response of graphene in a magnetic field at integer filling factors @xmath0 , with focus on the static structure factor @xmath4 , which is directly related to the cross section in inelastic light scattering by graphene . in particular , we point out that graphene has unusual characteristics of electronic correlations at short distances , which derive from the @xmath1relativistic " nature of massless quasiparticles . it is also shown that the zero - energy landau levels remain indistinguishable in spectral weight @xmath5 from the vacuum state .    in sec . ii we briefly review the low - energy effective theory of graphene in a magnetic field and derive , as a preliminary , the structure factor for conventional qh systems . in sec . iii we study the structure factor for graphene . section  iv is devoted to a summary and discussion . graphene has a honeycomb lattice consisting of two triangle sublattices of carbon atoms with one electron per site . it is a gapless semiconductor and its low - energy electronic transport is described by an effective hamiltonian of the form  @xcite @xmath6 , \nonumber\\ { \cal h}_{\pm}&= & v_{\rm f}\ ,   ( \sigma_{1}\pi_{1 }   + \sigma_{2}\pi_{2 } \pm m \sigma_{3 } )   - e a_{0 } ,   \label{hzero}\end{aligned}\ ] ] where @xmath7 [ @xmath8 or @xmath9 includes coupling to external electromagnetic potentials @xmath10 ; @xmath11 m / s is the fermi velocity . the two - component spinors @xmath12 and @xmath13 stand for the electron fields near the two inequivalent fermi points ( @xmath14 and @xmath15 ) where the spectrum becomes linear ; @xmath16 reside on the same sublattice and @xmath17 on another .    for generality we have introduced a tiny @xmath1mass " gap @xmath18 which spoils the valley su(2 ) symmetry of @xmath19 . actually the observed @xmath20 hall plateaus  @xcite suggest such a tiny mass gap . @xcite we keep @xmath21 to clarify the particle - hole character of the lowest landau levels , but practically set @xmath22 . we suppress the electron spin , which is treated as a global su(2 ) symmetry of @xmath19 by doubling the fields @xmath23 and @xmath24 . the zeeman splitting , though ignored for simplicity , is readily incorporated . the coulomb interaction is written as @xmath25 where @xmath26 is the fourier transform of the electron number density @xmath27 ; @xmath28 is the coulomb potential with the fine - structure constant @xmath29 and the substrate dielectric constant @xmath30 . we shall discuss the effect of @xmath31 later . let us place graphene in a strong magnetic field and study how the electrons in graphene respond to a weak potential @xmath32 . we set @xmath33 to supply a uniform magnetic field @xmath34 normal to the sample plane . when @xmath35 , the eigenmodes of @xmath19 are landau levels of @xmath23 and @xmath24 of energy @xmath36 labeled by integers @xmath37 , and @xmath38 ( or @xmath39 with the magnetic length @xmath40 ) ; @xmath41 is the basic cyclotron frequency . here @xmath42 specifies the sign of the energy @xmath43 .    for @xmath44 , @xmath23 and @xmath24 have the same spectrum symmetric about @xmath45 . the @xmath46 level of @xmath23 has negative energy @xmath47 while that of @xmath24 has positive energy @xmath48 ; these @xmath49 levels represent holes and electrons via quantization . with the electron spin taken into account , each landau level is thus four - fold degenerate , except for the doubly - degenerate @xmath50 levels . the @xmath50 eigenmodes have components only on each separate sublattice .    to make this landau - level structure explicit , it is useful to pass to the @xmath51 basis , with the expansion  @xcite @xmath52 . ( from now on , we shall only display the @xmath23 sector since the @xmath24 sector is obtained by reversing the sign of @xmath53 . ) the hamiltonian @xmath19 thereby is rewritten as @xmath54 and the charge density @xmath55 as  @xcite @xmath56 with the coefficient matrix @xmath57   \label{gpsikn}\end{aligned}\ ] ] and @xmath58 ; @xmath59 stands for the center coordinate with uncertainty @xmath60 = i\ell^{2}$ ] . here the coefficient functions @xmath61 are defined in terms of the harmonic oscillator eigenstates @xmath62 with @xmath63 and @xmath64=1 $ ] ; @xmath65 and @xmath66 . more explicitly , @xcite @xmath67 for @xmath68 , and @xmath69^{\dag}$ ] . actually @xmath70 are the coefficient functions that characterize the charge density for the `` nonrelativistic '' hall electrons @xmath71 expressed in terms of their eigenmodes @xmath72 with energy @xmath73 . let us first consider , as an exercise , the static structure factor or the spectral weight @xmath74 for a state @xmath75 of free `` nonrelativistic '' hall electrons with integer filling factor @xmath0 ( per spin ) . one may use eq . ( [ rhonr ] ) and note , in taking the matrix element @xmath76 , that @xmath77 for unoccupied levels @xmath78 so that the result is proportional to the number of electrons per occupied level @xmath79 . ( note here that @xmath80 with @xmath81 . ) this yields the spectral weight @xmath82 for @xmath83 , where @xmath84 denotes the total area . let us recall that the static structure factor  @xcite @xmath85 is defined by @xmath86 with its @xmath87 component isolated , where @xmath88 stands for the total electron number . accordingly , in general , @xmath89 owing to charge conservation . noting the formula @xmath90 allows one to cast eq . ( [ rhorhonr ] ) into the structure factor at filling factor @xmath0 @xmath91 which agrees with a known result . @xcite in particular , @xmath92    note first that @xmath93 since @xmath94 . note also that @xmath95 as @xmath96 for all @xmath0 . to see what this means let us recall the following : for a collection of classical particles , @xmath97 is written as  @xcite @xmath98 for @xmath99 , i.e. , as an average over the relative positions @xmath100 of particles surrounding a given particle . as a result , @xmath95 for @xmath96 if @xmath101 , e.g. , for particles formulated on a lattice . such behavior of @xmath97 thus implies the absence of particle correlations at short distances . in this section we study the case of graphene . in the present treatment the @xmath23 and @xmath24 sectors are independent and the spectral weight @xmath102 with @xmath103 is given by the sum @xmath104 for @xmath99 . for the @xmath23 sector one may simply replace , in eq . ( [ rhorhonr ] ) , @xmath105 by @xmath106 of eq . ( [ gpsikn ] ) and note that the level indices @xmath107 now run over all integers , @xmath108 . analogously , @xmath109 for the @xmath24 sector is obtained from @xmath106 by setting @xmath110 ( i.e. , @xmath111 ) , or equivalently @xmath112 let us denote @xmath113 and @xmath114 for short . they actually are functions of @xmath115 , are thus symmetric in @xmath107 , and have the property @xmath116 with @xmath117 . the spectral weight is now written as @xmath118 where @xmath119 ; @xmath120 stands for the spin degeneracy . one can write @xmath121 as @xmath122 when the @xmath23 landau levels are occupied up to the @xmath123th level @xmath124 ; analogously for @xmath125 . note that eq . ( [ gpsigchi ] ) relates @xmath121 and @xmath126 so that @xmath127 the equalities @xmath128 and @xmath129 then imply the following : ( i )  the @xmath23 and @xmath24 sectors contribute equally to the vacuum spectral weight , @xmath130 . ( ii )  the spectral weight @xmath5 is the same for the charge - conjugate states with @xmath131 . in view of this we shall focus on the case @xmath132 from now on .    in the limit @xmath22 of practical interest , distinctions between @xmath133 and @xmath134 disappear : @xmath135 for @xmath136 and @xmath137 for @xmath138 , as seen from eq . ( [ gpsikn ] ) . this yields @xmath139 in particular , @xmath140 shows that the graphene vacuum and the @xmath141 states , all of zero energy , have the same spectral weight @xmath142 . actually we earlier noted such degeneracy in electric susceptibility for the @xmath143 and @xmath141 states ,  @xcite and these states now turn out to be indistinguishable in both the real and imaginary parts of density response ; we discuss this point in more detail later .    to proceed further let us note some basic properties of the transition rates @xmath144 with @xmath145 . it is clear from eq . ( [ gammapsichi ] ) that the transition rates among the positive - energy states and those among the negative - energy states are essentially the same whereas they are different from the rates across the dirac sea ( @xmath146 with @xmath147 ) . this feature is seen from fig .  1  ( a ) , which illustrates typical profiles of @xmath148 for @xmath149 . in general , @xmath150 is significantly peaked at some value of @xmath151 and the peak position rises only gradually as the level gap @xmath152 increases . the structure factor @xmath97 for small @xmath153 is thus governed by virtual transitions to neighboring levels while its property at larger @xmath153 is determined by transitions across larger gaps . it is readily expected from this property that the sum over an infinite number of negative - energy levels would make the weight @xmath5 cutoff ( @xmath154 ) dependent for small @xmath155 , actually to @xmath156 . in particular , the vacuum weight @xmath157 vanishes at @xmath158 and grows rapidly with @xmath159 for fixed @xmath154 , whereas it diverges as @xmath160 for @xmath161 . see fig .  1  ( b ) . indeed , evaluating the @xmath162 terms in @xmath163 shows that the divergence is logarithmic in @xmath154 , @xmath164 with @xmath165 and for @xmath22 . this ultraviolet divergence is physical . the infinite depth of the dirac sea , of course , is an artifact of the continuum model  ( [ hzero ] ) and the cutoff scale @xmath166 is set by the energy scale above which the model loses its validity . let us recall here that for conventional qh systems the charge operator trivially annihilates the vacuum , @xmath167 , and the vacuum spectral weight vanishes . accordingly , the nonzero vacuum weight @xmath168 itself is a `` relativistic '' signature of graphene , @xmath169 which is a consequence of particle - hole pair creation or the presence of the dirac sea . it is perfectly legitimate to consider the vacuum weight with such a physical cutoff @xmath154 ( apart from its precise value ) . one can equally well extract cutoff - insensitive information out of it . one possible way is to consider a variation of @xmath168 for @xmath170 and @xmath171 . experimentally this means measuring the vacuum weight for @xmath170 and @xmath171 separately . for @xmath171 the spectral weight for the graphene vacuum @xmath172 with @xmath22 is written as @xmath173 which represents a collection of virtual transitions  @xcite from a negative - energy electron state with momentum @xmath174 to a positive - energy state with @xmath175 via the charge density @xmath176 ; here @xmath177 denotes the angle between @xmath174 and @xmath175 ; @xmath120 and @xmath178 denote the spin and valley degeneracy . this is again ultraviolet divergent . for regularization we cut off the @xmath174 integral at @xmath179 and choose the `` fermi momentum '' @xmath180 so that the dirac sea accommodates the same number of electrons as in the @xmath181 case , @xmath182 , i.e. , @xmath183 . a direct calculation yields @xmath184 the cutoff @xmath185 dependence thus correctly disappears from the difference @xmath186 , with the result @xmath187 as seen from fig .  1  ( b ) , the vacuum weight @xmath188 differs only slightly for @xmath181 and @xmath171 .    actually for all @xmath0 the spectral weight @xmath5 contains the vacuum fluctuations and the static structure factor @xmath97 is necessarily cutoff - dependent . a possible cutoff - independent measure in experiment is to compare the spectral weights for @xmath189 and @xmath143 ( with @xmath181 ) . correspondingly let us define by @xmath190 the structure factor with the vacuum contribution subtracted . for the @xmath191 states ( with @xmath192 ) , @xmath193 in terms of the cutoff - independent deviations @xmath194 and @xmath195 . for @xmath196 they are rewritten as @xmath197 analogously for @xmath198 . in reaching the second line , we have used the formulas ( valid for @xmath22 and @xmath160 ) : @xmath199\ ,   e^{z } -{1\over{4}}\ , { z^{|n| } \over{|n|!}},\ \ \ \ \   \label{sumgkn}\end{aligned}\ ] ] where @xmath145 and @xmath200 ; \end{aligned}\ ] ] @xmath201 and @xmath202 as @xmath203 ; the decrease is slower for larger @xmath204 , as depicted in fig .  2  ( a ) . the subtracted structure factor is now written as @xmath205\ ] ] for @xmath191 . in particular , @xmath206 and @xmath207.\ ] ]    in fig .  2 ( b ) and ( c ) we plot @xmath208 for @xmath209 and for higher filling @xmath210 . there are some notable features : ( 1 )  @xmath211 for @xmath212 . this is a consequence of the uniformity of charge at long wavelengths . ( 2 )  @xmath211 for @xmath213 . this shows that the large @xmath153 portion of @xmath214 is common to all @xmath0 and is governed by the vacuum weight @xmath168 . as a result , @xmath97 itself would rise with @xmath153 , in sharp contrast to the standard behavior @xmath215 as @xmath216 for `` nonrelativistic '' hall electrons in eq . ( [ spnr ] ) . ( 3 )  @xmath208 tends to saturate around 0.25 over a certain broad range of @xmath153 for higher filling factors @xmath0 . the subtracted structure factor @xmath208 , unlike @xmath97 , refers to quantum fluctuations of a finite number of filled levels above the vacuum and is governed by virtual transitions , newly allowed or suppressed by the presence of such levels , as seen from eq . ( [ dij ] ) . saturation is a consequence of a competition between the transitions among positive - energy states and the suppressed vacuum polarization effect .    in this connection , let us try to retain only the positive - energy states and transitions among them . in fig . 3  ( a ) we plot a projected structure factor @xmath217 calculated from the first term on the right - hand side of eq . ( [ dij ] ) . note that @xmath218 i.e. , even the positive - energy sector alone does not recover the @xmath215 behavior of conventional qh systems . while @xmath217 itself is not directly observable in experiment , this feature  ( [ splusp ] ) would indicate indirectly that @xmath208 is significantly smaller than 1/2 . some remarks on the influence of a mass gap are in order here . the relativistic treatment makes sense when the mass gap is tiny @xmath219 . in the `` nonrelativistic '' limit @xmath220 where the mass gap is large compared with the depth of the dirac sea , in contrast , the landau - level sum is limited to a finite interval @xmath221 and ceases to yield a divergence . the virtual transition rates across the mass gap scale like @xmath222 ( while @xmath223 for @xmath224 ) , and the vacuum spectral weight @xmath225 tends to zero like @xmath226 as @xmath227 . figure  3  ( b ) shows the manner how @xmath228 develops as the mass gap gets large for @xmath229 . the @xmath230 state is now distinguishable in @xmath97 from the vacuum . in the @xmath227 limit , @xmath231 recovers the standard behavior @xmath215 as @xmath216 for all integer @xmath0 .    to explore the origin of the unconventional behavior  ( [ splusp ] ) let us note the formula @xmath232 which follows from eq . ( [ sumgkn ] ) . actually this formula is valid for @xmath21 as well , since it is independently derived from the evaluation of the static weight @xmath233 for a hypothetical state @xmath75 with only one filled landau level @xmath234 and all other levels empty . it is thus a consequence of completeness of physical states .    in the `` nonrelativistic '' limit the dirac sea effectively disappears . then eq . ( [ completedirac ] ) is reduced to eq . ( [ completenr ] ) , which in turn leads to the behavior @xmath215 for @xmath216 . for @xmath22 the large-@xmath153 behavior of eq . ( [ completedirac ] ) is governed by the sums over @xmath235 so that @xmath236 as seen also from eq .  ( [ sumgkn ] ) . this precisely accounts for the behavior of @xmath217 in eq . ( [ splusp ] ) . it is now clear that both the masslessness ( or a tiny mass gap ) of the quasiparticles and the presence of the dirac sea are crucial for the unusual density correlations @xmath237 at short distances . in other words , correlations fail to vanish at short distances because of particle - hole pair creation ; the inability of localizing massless particles is a purely `` relativistic '' effect , underlying also the klein paradox . @xcite    equation  ( [ completedirac ] ) may suggest that the vacuum spectral weight @xmath168 would have a contribution of @xmath238 from the dirac sea , it actually rises like @xmath239 , as we have seen in eq . ( [ vacuumweight ] ) . had we defined the `` vacuum '' static structure factor @xmath97 with normalization by the total number of electrons in the dirac sea , it would vanish like @xmath240 . this shows again that , for massless particles and in the presence of the dirac sea , the standard ( classical ) picture of particle correlations does not necessarily make sense . finally we wish to comment on some peculiarities of the zero - energy @xmath50 levels , especially on how the coulomb interaction @xmath28 affects their response . in the random - phase approximation ( rpa ) the effect of @xmath31 of eq . ( [ hcoul ] ) is readily included  @xcite into the polarization function ( @xmath241 ) , @xmath242 once one knows the polarization function @xmath243 for noninteracting hall electrons . suppose now that we start with the vacuum state and fill up the @xmath244 level to reach the @xmath245 state . then @xmath243 will change by an amount proportional to @xmath246 which is easily seen to vanish for zero mass - gap @xmath22 . this shows that the @xmath143 and @xmath141 states remain indistinguishable in density response even at the rpa level , indicating the robustness of the zero modes against perturbations . in particular , the static structure factor @xmath247 and the dielectric function @xmath248 remain the same for @xmath143 and @xmath141 . in this paper we have studied the static structure factor for graphene in a strong magnetic field , with emphasis on revealing possible quantum signatures that distinguish graphene from conventional qh systems . in particular , for graphene the vacuum state is a dielectric medium with full of virtual particle - hole pairs . in other words , @xmath249 for graphene while @xmath250 for standard qh systems . correspondingly the graphene vacuum state has a nonzero density spectral weight @xmath168 , which actually diverges with the ( cutoff ) depth of the dirac sea and generally rises with @xmath153 significantly . experimentally the nonzero vacuum weight @xmath168 itself as well as its rise with @xmath153 , measured via inelastic light scattering , would be a clear signal of the quantum nature of the graphene vacuum state .    for graphene the static structure factor @xmath97 grows with @xmath153 for all @xmath0 , in sharp contrast to the behavior @xmath215 as @xmath251 of conventional qh systems ; this is because of pair creation at short distances . actually , even the transitions among positive - energy states fail to recover the standard behavior and lead to @xmath237 for @xmath216 , as noted in eq . ( [ splusp ] ) . this feature of @xmath217 itself , unfortunately , is not directly observable . nevertheless , for large filling factor @xmath0 , it competes with the vacuum polarization effect and would make the observable subtracted structure factor @xmath208 ( which is sensitive to quantum fluctuations of cutoff - independent longer wavelengths ) saturate around 1/4 over a certain broad range of @xmath153 ; this could be a possible signature of the underlying dynamics . such unusual features of particle correlations at short distances are a `` relativistic '' effect coming from massless dirac particles in graphene .    of special interest , in addition , are some peculiar features of the lowest ( @xmath50 ) landau levels of zero energy . remarkably they hardly affect the density response . the virtual transitions , both _ allowed _ and _ suppressed _ anew by the presence ( or absence ) of the @xmath50 levels , combine to leave the density response unchanged , and this compensation persists even when the coulomb interaction is taken into account in the rpa . the graphene vacuum ( @xmath143 ) state and the @xmath141 states would thus be indistinguishable in spectral weight @xmath252 as well as in electric susceptibility , although they are distinct in hall conductance . experimentally one would expect a definite nonzero signal for @xmath5 which largely remains the same over the range @xmath253 , except for some values of @xmath0 where nontrivial dynamics such as the fractional quantum hall effect may come into play . the author wishes to thank t. morinari for useful discussion . this work was supported in part by a grant - in - aid for scientific research from the ministry of education science sports and culture of japan ( grant no . 17540253 ) .
a close study is made of the static structure factor for graphene in a magnetic field at integer filling factors @xmath0 , with focus on revealing possible signatures of @xmath1relativistic `` quantum field theory in the low - energy physics of graphene . 
 it is pointed out , in particular , that for graphene even the vacuum state has a nonzero density spectral weight , which , together with the structure factor for all @xmath0 , grows significantly with increasing wave vector ; such unusual features of density correlations are a ' ' relativistic " effect deriving from massless dirac quasiparticles in graphene . 
 remarkably it turns out that the zero - energy landau levels of electrons or holes , characteristic to graphene , remain indistinguishable in density response from the vacuum state , although they are distinct in hall conductance .