Local density of states of electron-crystal phases in graphene in the quantum Hall regime
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Lo al density of states of ele tron- rystal phases in graphene in the quantum HallregimeO. Poplavskyy,1, 2, 3, ∗ M. O. Goerbig,2 and C. Morais Smith31Fa ulty of Mathemati s, Wilberfor e Road, University of Cambridge, CB3 0WA, United Kingdom2Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris Sud, F-91405 Orsay Cedex, Fran e3Institute for Theoreti al Physi s, University of Utre ht, Leuvenlaan 4, 3584 CE Utre ht, The Netherlands(Dated: O tober 25, 2018)We al ulate, within a self- onsistent Hartree-Fo k approximation, the lo al density of statesfor di(cid:27)erent ele tron rystals in graphene subje t to a strong magneti (cid:28)eld. We investigate boththe Wigner rystal and bubble rystals with M e ele trons per latti e site. The total density ofstates onsists of several pronoun ed peaks, the number of whi h in the negative energy range oin ides with the number of ele trons M e per latti e site, as for the ase of ele tron-solid phasesin the onventional two-dimensional ele tron gas. Analyzing the lo al density of states at the peakenergies, we (cid:28)nd parti ular s aling properties of the density patterns if one (cid:28)xes the ratio ν N /M e between the (cid:28)lling fa tor ν N of the last partially (cid:28)lled Landau level and the number of ele tronsper bubble. Although the total density pro(cid:28)le depends expli itly on M e , the lo al density of statesof the lowest peaks turns out to be identi al regardless the number of ele trons M e . Whereas theseele tron-solid phases are reminis ent of those expe ted in the onventional two-dimensional ele trongas in GaAs heterostru tures in the quantum Hall regime, the lo al density of states and the s alingrelations we highlight in this paper may be, in graphene, dire tly measured by spe tros opi means,su h as e.g. s anning tunneling mi ros opy.PACS numbers: 73.43.-f, 73.20.Qt, 73.21.-b, 68.37.-dKeywords: Wigner rystal; ele tron-bubble rystal; high magneti (cid:28)eld; lo al density of states; s anningtunneling spe tros opyI. INTRODUCTIONAs was shown by Wigner in 1934,1 the degenerateFermi gas is unstable towards the formation of a pe-riodi triangular latti e of lo alized ele trons (ele tron rystal), on e the Coulomb energy prevails over the ki-neti one. However, the riti al ele tron density at whi hthe transition to the Wigner rystal (WC) o urs is toolow for usual metals. Nevertheless, the situation is mu himproved if one applies a strong perpendi ular magneti (cid:28)eld to a two-dimensional (2D) ele tron gas (2DEG). Inthis ase, the single-parti le ontinuous energy spe trumis quantized into a sequen e of hugely degenerate Lan-dau levels (LL's). If one restri ts oneself to the ele tronswithin the last partially-(cid:28)lled LL, one (cid:28)nds that their ki-neti energy is quen hed, and the only energy s ale is theCoulomb energy, whi h favors the formation of an ele -tron rystal at small (cid:28)lling fa tors.2,3,4A quantum ele tron rystal in the presen e of adisorder potential is expe ted to be ome olle tivelypinned and to manifest itself as an insulator.5 Whileat small (cid:28)lling fa tors the 2D WC with triangular lat-ti e symmetry3,4 is expe ted to yield the global en-ergy minimum, it was predi ted that the phase dia-gram of the 2DEG in ludes also ele tron-bubble rys-tals (a periodi latti e with more than one ele tron persite), stripes,6,7,8 and even more exoti quantum Hallliquid- rystal phases.9 Unlike ele tron- rystal phases, theprominent quantum liquids, whi h display the fra tionalquantum Hall e(cid:27)e t in the two lowest LLs,10 are transla-tionally and rotationally invariant and remain ondu ting even in the presen e of disorder. Therefore, these di(cid:27)er-ent quantum phases may be distinguished experimentallywith respe t to the behavior in transport measurements.For instan e, a su ession of insulating and ondu tingphases yields a re-entrant integer quantum Hall e(cid:27)e t(IQHE) in the (cid:28)rst11 ( N =1) and se ond12 ( N = 2 ) ex- ited LLs and has been interpreted in terms of a ompeti-tion of su h ele tron-solid and quantum-liquid phases.13Further eviden e for ele tron rystals in LLs stems fromradio-frequen y spe tros opy14 and transport measure-ments under mi rowave irradiation,15,16,17,18,19 whi h ex- ites the olle tive pinning mode of the ele tron rystals.While all these experimental te hniques have been verysu essful in dis overing new insulating phases and have on(cid:28)rmed a number of theoreti al predi tions, they areindire t eviden e for high-(cid:28)eld ele tron rystals basedon transport measurements (cid:21) the 2DEG in GaAs het-erostru tures is buried deep inside the substrate, whi hrenders impossible a dire t opti al observation of a pe-riodi ele tron- rystal latti e, e.g., by means of s anningtunneling mi ros opy (STM).20 In ontrast to the on-ventional 2DEG, su h opti al studies might be ome pos-sible in graphene, a one-atom thi k sheet of graphite,with unique ele troni and me hani al properties.21,22 In-deed, graphene may be viewed as a parti ular 2DEG,where the ele trons behave as if they were massless par-ti les des ribed by the relativisti 2D Dira equation.In this Dira equation the Fermi velo ity v F plays therole of the speed of light c , although it is roughly 300times smaller than the latter in va uum. In addition, theBrillouin zone of graphene has two non-equivalent ornerpoints ( alled Dira points) whi h yield a twofold val-ley degenera y and whi h may formally be des ribed interms of an SU(2) pseudospin degree of freedom.In strong magneti (cid:28)elds, the energy of Dira fermionsin graphene is quantized into LL's, the stru ture of whi his di(cid:27)erent from that of non-relativisti ele trons in a onventional 2DEG. Apart from their un onventional(square root) magneti -(cid:28)eld dependen e, there exists aLL at exa tly zero energy, and ea h LL in the ondu tionband has a ounterpart in the valen e band. This parti -ular LL stru ture leads to the anomalous integral quan-tum Hall e(cid:27)e t observed in graphene.23,24 In addition,the energy gap between the subsequent LL's in grapheneis so large, that it is possible to observe the IQHE evenat room temperatures.25As the mobility of graphene samples is further im-proved, one may expe t to observe the fra tional quan-tum Hall e(cid:27)e t, whi h has been studied theoreti ally byseveral authors,26,27,28,29,30 and also olle tively-pinnedinsulating phases su h as Wigner- rystal and bubblephases, as predi ted in Refs. 31,32,33,34,35. In on-trast to GaAs heterostru tures, these ele troni phaseso ur at the surfa e of the graphene sheet and are, thus,dire tly a essible by spe tros opi means. Indeed STMhas been applied su essfully to probe the density distri-bution in exfoliated36 and epitaxial37 graphene, as wellas in graphene on a graphite substrate in a strong mag-neti (cid:28)eld.38 This ex iting prospe t motivated us to al- ulate theoreti ally physi al quantities of a 2D ele tron rystal whi h might be measured in an STM experiment:the (integrated) density of states (DOS) and the lo aldensity of states (LDOS). We should note that the quan-tum Hall regime is the only ase for whi h one may ex-pe t the formation of ele tron- rystal phases in graphene.Indeed, it is predi ted that the 2D Wigner rystalliza-tion is ompletely absent in graphene for any ele trondensity in the absen e of a magneti (cid:28)eld,39 due to thes ale invarian e of the dimensionless intera tion parame-ter r s = e / ~ ǫv F ≃ /ǫ for a 2D system with a linear dis-persion relation. Here, ǫ is the diele tri onstant whi hdepends on the environment where the graphene sheet isembedded.In this paper, we dis uss the DOS and the LDOSfor several ele tron rystals in the N = 2 LL within aHartree-Fo k approximation. We have performed similar al ulations for N = 1 , 3 and 4, but we on entrate inthe present paper on N = 2 for two reasons. First, theDOS and LDOS results for N = 2 are representative ofhigh-(cid:28)eld ele tron solids (cid:21) our al ulations yield indeedsimilar results for the other LL's. Se ond, for higher LL'sthere have been no lear indi ations so far for ele tron- rystal phases in GaAs in the quantum Hall regime. Forour numeri al al ulations, we have adopted the itera-tive s heme proposed by Cté and Ma Donald,40,41 whi hhas also been applied to al ulate the energies and thereal-spa e pro(cid:28)les of various ele tron- rystal phases ingraphene.31 As a test of the validity of our ode, we have orroborated the results obtained in Ref. 31 and then applied it for the al ulation of the DOS and the LDOS.Noti e that, despite the huge amount of Hartree-Fo kstudies of quantum Hall ele tron- rystal phases, none isdevoted to study the LDOS of these phases. We have al ulated the LDOS at energies where the integratedDOS has well-pronoun ed peaks, whi h fall into two dis-tin t lasses: bound states at negative energy with re-spe t to the hemi al potential and high-energy peaksabove. The number of negative-energy peaks is identi alwith the number of ele trons M e per bubble, in agree-ment with bubble rystals in the onventional 2DEG.8Furthermore, we (cid:28)nd that the sum of the LDOS at these M e negative-energy peaks reprodu es the real-spa e den-sity pro(cid:28)le of the M e -ele tron bubble rystal.This paper is organized as follows. In Se tion II, weoutline the basi steps of the Hartree-Fo k approximationto the 2DEG in graphene. In Se tion III, we present nu-meri al results for the DOS and the LDOS in graphene.Finally, we draw our on lusions in Se tion IV.II. HARTREE-FOCK HAMILTONIANFor a partially (cid:28)lled LL N , the low-energy ele troni properties are aptured within a model that takes intoa ount states only within this level. In this ase, thesingle-parti le kineti energy is the same for all of states,and thus only the intera tion term is relevant. Further-more, we omit the physi al spin whi h we onsider to be ompletely polarized, e.g. due to a su(cid:30) iently large Zee-man e(cid:27)e t. The derivation of the Hartree-Fo k Hamil-tonian for the 2DEG in GaAs has been extensively dis- ussed in the literature.40,41 In graphene, the intera tionHamiltonian for the 2DEG is similar to that in GaAs, al-beit with di(cid:27)erent form fa tors due to the spinorial formof the wave fun tions.26,42 This similarity allows one touse the same theoreti al methods whi h were used pre-viously to study the 2DEG in GaAs, with the importantdi(cid:27)eren e that we need to take into a ount the twofoldvalley degenera y in the form of an SU(2) pseudospindegree of freedom, β = ± . Provided that inter-LL tran-sitions are negle ted, we may write the intera tion partof the full Hamiltonian for the 2DEG of spinless ele tronsin graphene as26 ˆ H int = V C X β,β ′ , q | q | [ F N ( q )] ˆ ρ β,β ( − q )ˆ ρ β ′ ,β ′ ( q ) , (1)where V C ≡ e /l B ǫ is the Coulomb energy s ale, with l B = p ~ /eB the magneti length, B the magneti (cid:28)eld,and ǫ the diele tri sus eptibility of the medium, andq ≡ ( q x , q y ) is a 2D waveve tor. The (guiding enter)density operator in the Landau gauge reads ˆ ρ β ,β ( q ) = N − φ X X exp (cid:18) − iq x X − i l B q x q y (cid:19) × ˆ c † X,β ˆ c X + l B q y ,β . (2)Here, N φ = S/ πl B measures the LL degenera y, withthe square area S of the 2DEG sample, ˆ c X,β and ˆ c † X,β arethe ele tron's destru tion and reation operators, respe -tively, where X denotes single-parti le quantum stateswithin the N -th LL. Finally, in Eq. (1) the grapheneform fa tor F N ( q ) reads26,42 F N ( q ) = h L | N | (cid:16) q (cid:17) + L | N |− (cid:16) q (cid:17)i e − q , N = 0; e − q , N = 0 , (3)where q ≡ | q | , and L n ( x ) is the Laguerre polynomial oforder n . We note that the 2DEG form fa tor in GaAs isgiven by40 F N ( q ) = L N (cid:18) q (cid:19) e − q / . (4)It is apparent from Eqs. (3) and (4) that the graphene(relativisti 2DEG) form fa tor is simply a linear om-bination of form fa tors for adja ent LL's of the non-relativisti 2DEG in GaAs. This pe uliar fa t resultsfrom mixing the Dira parti le wavefun tions betweenthe sites of two sublatti es in graphene, and is also a onsequen e of the spinorial nature of these wavefun -tions. Apart from the di(cid:27)eren e in form fa tors given byEqs. (3) and (4), the 2DEG in GaAs and graphene is de-s ribed equivalently, as follows from the same analyti alstru ture of the Coulomb intera tion term given by Eq.(1).Finally, we note that the Hamiltonian in Eq. (1) isSU(2)-invariant with respe t to the valley pseudospin.In ontrast to the physi al ele tron spin, this SU(2) sym-metry is approximate. However, SU(2)-symmetry break-ing terms are suppressed linearly in a/l B ≪ where a = 0 . nm is the arbon- arbon distan e in grapheneand l B = 26 / p B [T] nm, i.e. at an energy s ale thatis well below the disorder broadening of the LL's.26,43This physi al model is similar to another two- omponentquantum Hall system (cid:21) if one repla es in Eq. (1) F N ( q ) by the non-relativisti form-fa tor F N ( q ) , one obtainsthe Hamiltonian for the non-relativisti 2DEG in ludingthe ele trons' spin in the absen e of a polarizing Zeemane(cid:27)e t. Alternatively, this model may des ribe a quantumHall bilayer in the theoreti al limit of zero layer separa-tion, where the two (cid:16)spin(cid:17) orientations denote the twodi(cid:27)erent layers.41 One may further simplify the model inEq. (1) by omitting the valley pseudospin degree of free-dom, in whi h ase one presupposes a omplete valleypolarization of the ele troni phases, whi h would maxi-mally pro(cid:28)t from the ex hange intera tion. This e(cid:27)e tiveU(1) model is des ribed by the intera tion term ˆ H int = V C X q | q | [ F N ( q )] ˆ ρ ( − q )ˆ ρ ( q ) , (5)where the density operator of spinless ele trons ˆ ρ ( q ) isobtained from Eq. (2) by negle ting the pseudospin in-di es. This simpli(cid:28)ed U(1) model of fully valley-polarized graphene whi h is des ribed by Eq. (5) is alled U(1)-graphene in the remainder of the paper. Now, if onesubstitutes into Eq. (5) the non-relativisti form-fa tor F N ( q ) , one obtains the usual single-layer quantum Hall2DEG for spin-polarized ele trons in GaAs.The Hartree-Fo k approximation applied to thegraphene intera tion term in Eq. (1) yields31,41 ˆ H (HF)int = N φ V C X β, Q n(cid:2) H ( Q ) − X ββ ( Q ) (cid:3) ˆ ρ β,β ( Q ) − X β ¯ β ( Q )ˆ ρ ¯ β,β ( Q ) o , (6)where ¯ β = − β , and Q's are the re ipro al waveve tors ofthe WC latti e. The Hartree and Fo k e(cid:27)e tive intera -tion potentials read, respe tively, H ( Q ) = e − Q / Q |F N ( Q ) | ρ ( − Q )(1 − δ Q , ) , (7) X ββ ′ ( Q ) = Z ∞ dxe − x |F N ( Q ) | J ( xQ ) ρ β,β ′ ( − Q ) , (8)where Q ≡ | Q | , J is a Bessel fun tion, and the densityaverages are ρ β,β ′ ( Q ) = h ˆ ρ β,β ′ ( Q ) i , ρ ( Q ) = P β ρ β,β ( Q ) .We assume a triangular ele tron latti e for the broken-symmetry state, with re ipro al latti e ve tors given byQ = Q n , n m √ ! , n, m ∈ Z . (9)Here Q is the length of the basis ve tor of the re ipro allatti e, Q = l − B (cid:18) πν N √ M e (cid:19) / , (10) ν N is the (cid:28)lling fa tor of the last partially (cid:28)lled LL,and M e is the number of ele trons per site ( M e = 1 orresponds to the WC, and M e ≥ to an ele tron-bubble rystal with M e ele trons per bubble). The single-parti le Green's fun tion in the imaginary-time Matsub-ara formalism44 reads G β ,β ( Q , iω n ) = − N − φ Z ~ /k B T dτ exp( iω n τ ) × X X exp (cid:20) − iQ x X + l B Q x Q y (cid:21) × hT τ ˆ c X − l B Q y ,β ( τ )ˆ c † X,β (0) i , (11)where T is the temperature, k B is the Boltzmann onstant, T τ denotes imaginary-time ordering, and ω n = π (2 n + 1) k B T / ~ are the Matsubara frequen ies. G β ,β ( Q , iω n ) may be determined self- onsistently fromthe quadrati Hamiltonian (6) by using the Heisenbergequations of motion within the iterative-solution methodproposed in Ref. 45 whi h we adopt in the present work.After analyti ontinuation to real frequen ies iω n → ω + i + , G β ,β ( Q , iω n ) yields the retarded Green's fun -tion whi h may be used to al ulate the DOS g ( ω ) , g ( ω ) = − N − φ π X β Im G β,β ( Q = 0 , iω n → ω + i + ) , (12)and the LDOS A ( r , ω ) , A ( r , ω ) = − N − φ π X β Im G β,β ( r , iω n → ω + i + ) , (13)where the Green's fun tion in real spa e reads G β ,β ( r , iω n ) = (2 πl B ) − X Q ,β exp( − i Q · r ) F N ( − Q ) × G β ,β ( Q , iω n ) . (14)III. RESULTS AND DISCUSSIONSIn this se tion, we dis uss the spe tros opi propertiesof the ele tron-solid phases for the LL N = 2 . As al-ready mentioned in the introdu tion, we on entrate onthis LL for illustration purposes and be ause they aremostly signi(cid:28) ant from the physi al point of view. Wehave obtained similar results for N = 1 , 3 and 4 (notdis ussed here).We have hosen three di(cid:27)erent ele tron-solid latti eswhi h have the same ratio ν N /M e = 0 . ≈ / , andhen e the same latti e period given by Eq. (10). Our hoi e for the ν N /M e ratio is rather arbitrary. We notethat the M e = 1 and M e = 2 states yield in graphene theglobal energy minima, while the M e = 3 does not. It hasbeen shown in Ref. 31 that the ground state of grapheneat ν N ≤ . is an anisotropi Wigner rystal whereas at . ≤ ν N ≤ . the ground state is the M e = 2 bubble rystal, and at ν N ≤ . the Wigner rystal yields thelowest energy ( M e = 1 ). Therefore, the M e = 3 phaseis not the lowest-energy state in graphene; nevertheless,it is useful to analyze on the same footing all three ases M e = 1 , , .For ompleteness, we mention that we have also al- ulated the ohesive energies of other types of ele tron- rystal phases (not only triangular bubble phases, butalso anisotropi Wigner rystals). Our results for theenergies oin ide with those of Ref. 31 with ex ellent a - ura y and, therefore, orroborate the DOS and LDOSresults dis ussed below.A. (Integrated) Density of statesOur results for the DOS in graphene at N = 2 arepresented in Fig. 1.We (cid:28)nd that the DOS onsists of two well-separated lasses of peaks: well-de(cid:28)ned low-energy peaks are foundbelow the hemi al potential µ , whi h is shifted to zero energy, whereas the large number of peaks above µ arenot that easily distinguished. We note here that ingraphene the number of low-energy peaks in all asesis equal to M e , the number of ele trons in a bubble.The same result has been obtained before in Hartree-Fo k studies of the simpler single-layer 2D quantumHall system.8 We he ked that the same property holdstrue also for U(1)-graphene, and in non-relativisti two- omponent quantum Hall systems, su h as a bilayer withzero layer separation.In the simpler single-layer 2D quantum Hall systemin GaAs, the DOS at M e = 1 exhibits the features ofthe Hofstadter butter(cid:29)y stru ture.8 It means the follow-ing: given that the (cid:28)lling fa tor may be represented bya ratio of two integers p, q without a ommon divisor, ν N = p/q , these integers p, q determine then the stru -ture of the single-parti le energy spe trum of the system;namely, there should exist p low-energy levels and q − p high-energy levels (Hofstadter butter(cid:29)y ounting rule).In the DOS, whi h is a fun tion of frequen y, these en-ergy levels are re ognized as smoothed peaks. The Hofs-tadter butter(cid:29)y ounting rule was on(cid:28)rmed for M e = 1 in the single-layer 2DEG in GaAs, while for M e ≥ it is laimed that ounting of the single-parti le levels is dif-ferent: the number of low-energy peaks is equal to M e ,whereas nothing is known about what is the pre ise rulefor ounting the number of high-energy peaks.8(A) (B) (C)Bound states −0.4 −0.2 0.0 ω Single-parti le ex itationsFIG. 1: Logarithmi plots for the density of states g ( ω ) ofgraphene in the N = 2 Landau level at (A) M e = 1 , ν N =0 . ; (B) M e = 2 , ν N = 0 . ; (C) M e = 3 , ν N = 0 . .The frequen y ω is given in units of the Coulomb s ale V C .The ω = 0 frequen y is the position of the hemi al potential(Fermi energy) µ . The in(cid:28)nitesimal imaginary frequen y shift ω → ω + i + is approximated by ω → ω + iδ ω , with δ ω = 10 − .Figures (A)-(C) in the (cid:28)rst row yield the DOS in the low-energy frequen y range (bound states of ele trons), while inthe se ond row the high-energy peaks of the DOS orrespondto single-ele tron ex itations above the ground state of thelatti e.What will be important in the following dis ussion isthe order of indexing of the DOS peaks. We will ountthe peaks in the DOS with respe t to in reasing the fre-quen y ω . In Fig. 1(A), the (cid:28)rst DOS peak is obvi-ously the lowest-energy one with energy ≈ − . V C . These ond peak in (A) is a higher-energy one with energy ≈ . V C . The numbering of peaks ontinues until werea h the utmost-right peak with energy ≈ . V C . Thesame indexing rule is applied to the ases (B) and (C).One should note that in all three ases the DOS peakswith the same index may have rather di(cid:27)erent energies:while in (A) the se ond DOS peak belongs already to thehigh-energy region, in (B) it still lies below the Fermilevel.In addition, one should have a pro edure of extra t-ing the energies of the DOS peaks from the smoothedDOS vs frequen y dependen e shown in Fig. 1. In thelow-energy regime it may be always done reliably. In thehigh-energy regime, however, there is a larger numberof losely-lo ated DOS peaks, the shapes, widths, andamplitudes of whi h depend sensitively on the imaginaryfrequen y shift δ ω . The latter is used for the analyti- al ontinuation into the upper omplex half-plane of theGreen's fun tion, iω n → ω + iδ ω . Physi ally, this imag-inary frequen y shift represents a level broadening due,e.g., to disorder. We have found that the best way to ex-tra t only those peaks whi h are physi al is to pla e a ut-o(cid:27) ∆ on the DOS peak amplitude, so that peaks with am-plitude less than g max ∆ are negle ted, with g max ∝ δ − ω the maximum peak amplitude. In our study, we have hosen δ ω = 10 − , and ∆ = 0 . . The number of shells ofre ipro al latti e ve tors Q's is N sh = 8 , so that the a -tual number of ve tors is N Q = 241 . Single-parti le ener-gies whi h are extra ted from the smoothed DOS, will beused below in the al ulation of the LDOS. In the U(1)-graphene and the single-layer ases at the same densities onsidered here, we are able to extra t DOS peaks; at ν N = 0 . , M e = 1 , there is one lowest-energy peak, and high-energy ones. These are exa tly the numbers ofsingle-parti le levels di tated by the Hofstadter butter(cid:29)y ounting rule.8In graphene and the quantum Hall bilayer, studiedwithin the two- omponent model, we obtain the num-ber of identi(cid:28)ed DOS peaks around [with deviationof not more than one wrongly identi(cid:28)ed peak℄. Due tothe additional SU(2)-symmetry in the latter two ases,it is natural to expe t that the number of single-parti lelevels is thus doubled.B. Real-spa e density pro(cid:28)leFor later omparison with our results for the LDOS,we al ulated the real-spa e ele tron density pro(cid:28)le n ( r ) = 12 πl B X Q exp( − i Q · r ) F N ( − Q ) ρ ( Q ) (15)for the same hoi es of ( ν N , M e ) as in Fig. 1. The results,whi h are shown in Fig. 2, agree with previous al ula- tions for graphene performed by Zhang and Joglekar.31(A) (B) (C) [0 . , .
1] [0 . , .
7] [0 . , . FIG. 2: (Color online) Real-spa e density pro(cid:28)le n ( r ) ingraphene in the N = 2 LL. Choi es (A)-(C) are the sameas in Fig. 1. Minima and maxima of n ( r ) written inside thesquare bra kets as [min, max℄ orrespond to the values [0.0,1.0℄ in the olour plots (blue and white olours, orrespond-ingly). C. Lo al density of statesOur results for the LDOS in graphene are presentedin Figs. 3, 4, and 5. The LDOS patterns is plotted forall three ases (A)-(C), as in Fig. 1, and at the ener-gies of all extra ted single-parti le DOS peaks situatedin in reasing order. We obtain that the res aled (to therange of [0 . , . ) real-spa e patterns of A ( r , ω ) , al u-lated at the (cid:28)rst four DOS peaks for all three hoi esof ( ν N , M e ) , oin ide among themselves. There is alsoan approximate mapping between the LDOS patterns atthe (cid:28)fth DOS peak, although less pronoun ed than forthe (cid:28)rst four ones (one sees orresponden e between thepositions of maxima and minima, but the olors deviateslightly in ea h ase). For larger values of the peak in-dex, we start to see onsiderable dis repan ies betweenthe LDOS patterns. Also the number of extra ted peaks N p is di(cid:27)erent for ea h ase. The latter property is dueto the very approximate nature of our extra tion pro e-dure: while low-energy peaks are always identi(cid:28)ed reli-ably, the high-energy peaks are determined only approx-imately. However, the a ura y is quite good. We alsonote a very interesting property of the LDOS at the lasttwo peaks for (B)-(C): the LDOS patterns are identi al,but their positions are swapped. We do not have anyphysi al argument why this should be the ase, but it ould be a feature that appears when M > N . However,this statement is a mere spe ulation and a more detailedinvestigation is required to larify this aspe t.D. Comparison with the real-spa e densityNow, to ompare the LDOS patterns shown in Figs.3, 4, 5 with the real-spa e density pro(cid:28)le n ( r ) de(cid:28)ned inEq. (15) and plotted in Fig. 2, we introdu e the resummed(A)1 2 3 4 56 7 8 9 1011 12 13FIG. 3: (Color online) LDOS A ( r , ω ) for graphene at ν N =0 . , M e = 1 [ ase (A)℄. Contour olours are graded in thesame way as de(cid:28)ned in Fig. 2. The ontour plots are orderedwith respe t to the index of DOS peaks [indi ated above theplots℄. The number of extra ted DOS peaks is N p =13.(B)1 2 3 4 56 7 8 9 1011 12 13 14 15FIG. 4: (Color online) LDOS A ( r , ω ) for graphene at ν N =0 . , M e = 2 [ ase (B)℄. N p = 15 .LDOS ˜ A ( r , ω ) , de(cid:28)ned for a (cid:28)xed single-parti le energy ω i as a sum of all LDOS patterns at smaller peak energies, ˜ A ( r , ω i ) = i X j =1 A ( r , ω j ) . (16)Given the ex ellent oin iden e of the LDOS patternsshown in Fig. 6 with the real-spa e densities in Fig. 2,one may empiri ally write n ( r , M e = i ) ↔ ˜ A ( r , ω i ) , i = 1 , , , (17) (C)1 2 3 4 56 7 8 9 1011 12 13 14FIG. 5: (Color online) LDOS A ( r , ω ) for graphene at ν N =0 . , M e = 3 [ ase (C)℄. N p =14.(A)1 2 3FIG. 6: (Color online) Resummed LDOS ˜ A ( r , ω ) for grapheneat the three (cid:28)rst DOS peaks for (A) [ ν N = 0 . , M e = 1 ℄.Contour olours are graded in the same way as de(cid:28)ned in Fig.2.where the sign ↔ means mapping between the res aledto the [0.0,1.0℄ interval quantities. This means that thereal-spa e density of the M e -ele tron bubble rystal isdetermined by the sum of the LDOS at the M e negative-energy peaks. More surprisingly, be ause of the orre-sponden e between the LDOS patterns of the low-energypeaks for all di(cid:27)erent M e bubble rystals, one may de-termine the real-spa e density pattern of the M e bubble rystal by summing the LDOS of the M e peaks of lowestenergy for any of the ele tron-solid phases (cid:21) the LDOSpatterns of the M e = 1 Wigner rystal, e.g., ontainsthus the information of the density of all other M e bub-ble rystals.E. U(1)-graphene: lo al density of statesIn Fig. 7 we present the LDOS for U(1)-graphene. Thenumber of extra ted DOS peaks for all three density hoi es (A)-(C) is N p = 7 . This is in a ordan e withthe Hofstadter butter(cid:29)y ounting rule. We also see anex ellent orresponden e of the LDOS for the (cid:28)rst fourDOS peaks, then, also a good oin iden e of the LDOS atthe peaks 6-7 for (A)-(B), whereas these two LDOS pat-terns for (C) inter hange pla es, as ompared with thepatterns 6-7 for (A)-(B). This inter hange phenomenonis the same as observed for graphene and is not yet un-derstood.In general, the U(1)-graphene results oin ide numeri- ally with those for graphene when one takes into a ountthe SU(2) symmetry for the valley pseudospin. This in-di ates that one may use the U(1)-model instead of themore omplex SU(2)-symmetri one for the dis ussion ofthe density patterns, hen e simplifying further al ula-tions on graphene. Moreover, it indi ates that in theele tron- rystal phases onsidered above the valley de-gree of freedom is fully polarized.(A)1 2 3 4 5 6 7(B)1 2 3 4 5 6 7(C)1 2 3 4 5 6 7FIG. 7: (Color online) LDOS A ( r , ω ) for U(1)-graphene. Thenumber of extra ted DOS peaks N p = 7 .IV. CONCLUSIONSThe aim of this work is to show how a high-(cid:28)eldele tron-solid phase in the 2DEG may be dete ted byopti al means in graphene. We have al ulated the DOSand the LDOS of ele tron-solid phases in the Hartree-Fo k approximation in the N = 2 LL.We show that the number of low-energy DOS peaks in graphene is given by the number of ele trons per site M e .This result is similar to the previous DOS al ulation inthe Hartree-Fo k approximation for GaAs.8We found that the res aled LDOS is identi al for di(cid:27)er-ent (cid:28)lling fa tors ν N , as long as the ratio ν N /M e , whi hdetermines the latti e spa ing of the M e -ele tron bubble rystal, is kept (cid:28)xed, and the LDOS frequen y is takenat the DOS peak with the same index (for the (cid:28)rst fourindi es). In parti ular, this result yields an unexpe ted on lusion that, e.g. by (cid:28)xing the (cid:28)lling fa tor ν N = 0 . in the N = 2 LL, and using STM, one ould observe inthe LDOS the whole su ession of ele tron- rystal den-sity patterns with M e = 1 , , by (cid:28)xing the appliedSTM voltage at the onse utive (cid:28)rst three single-parti leex itation energies, and summing up the LDOS to obtainthe resummed LDOS ˜ A ( r , ω ) .We believe that this LDOS orresponden e holds truefor all single-parti le ex itations resolved as individualDOS peaks so far (a ounting for inter hanging of thelast two peaks in the M e = 2 , ases) and for all LL's(similar on lusions follow from our al ulations in theLLs N = 1 , , ).We also obtained the same LDOS orresponden e forother models of the 2DEG: (i) in a single-layer GaAsheterostru ture; (ii) U(1)-graphene; (iii) bilayer. Thisimplies that the observed LDOS ν N /M e s aling is inde-pendent of the underlying intera tion potential and thenumber of inner dis rete degrees of freedom. The fa tthat the U(1)-graphene results oin ide numeri ally withthose for graphene indi ates that the ele tron rystals onsidered here are ompletely valley-pseudospin polar-ized. A knowledgmentsWe thank P. Lederer for fruitful dis ussions. O. P. a -knowledges (cid:28)nan ial support from the European Com-mission through the Marie-Curie Foundation ontra tMEST CT 2004-51-4307, and from the Gates CambridgeS holarship Trust. The work of C.M.S. was partially sup-ported by the Netherlands Organization for S ienti(cid:28) Re-sear h (NWO). ∗∗