Fermi wave vector for the non-fully spin polarized composite-fermion Fermi sea
FFermi wave vector for the non-fully spin polarized composite-fermion Fermi sea
Ajit C. Balram and J. K. Jain Niels Bohr International Academy and the Center for Quantum Devices,Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark and Department of Physics, 104 Davey Lab, Pennsylvania State University, University Park, Pennsylvania 16802, USA (Dated: December 4, 2017)The fully spin polarized composite fermion (CF) Fermi sea at half filled lowest Landau levelhas a Fermi wave vector k ∗ F = √ πρ e , where ρ e is the density of electrons or composite fermions,supporting the notion that the interaction between composite fermions can be treated perturbatively.Away from ν = 1 /
2, the area is seen to be consistent with k ∗ F = √ πρ e for ν < / k ∗ F = √ πρ h for ν > /
2, where ρ h is the density of holes in the lowest Landau level. This result is consistentwith particle-hole symmetry in the lowest Landau level. We investigate in this article the Fermiwave vector of the spin-singlet CF Fermi sea (CFFS) at ν = 1 /
2, for which particle-hole symmetryis not a consideration. Using the microscopic CF theory, we find that for the spin-singlet CFFS theFermi wave vectors for up and down spin CFFSs at ν = 1 / k ∗↑ , ↓ F = (cid:113) πρ ↑ , ↓ e ,where ρ ↑ e = ρ ↓ e = ρ e /
2, which implies that the residual interactions between composite fermionsdo not cause a non-perturbative correction for non-fully spin polarized CFFS either. Our resultssuggest the natural conjecture that for arbitrary spin polarization the CF Fermi wave vectors aregiven by k ∗↑ F = (cid:113) πρ ↑ e and k ∗↓ F = (cid:113) πρ ↓ e . PACS numbers: 73.43.-f
I. INTRODUCTION
The emergence of a Fermi sea at half filled Landau level(LL) is remarkable given that the original Hamilto-nian has no kinetic energy. Its appearance is a conse-quence of the formation of composite fermions , whichexperience, on average, no magnetic field at a filling fac-tor ν = 1 /
2. In recent years, accurate determination ofthe Fermi wave vector of composite fermions from com-mensurability oscillations at and near half filled Landaulevel (LL) has shed important new light into thephysics of the composite fermion (CF) Fermi sea. At ν = 1 /
2, the measured Fermi wave vector of compositefermions, denoted by k ∗ F , is found to be k ∗ F = √ πρ e , con-sistent with a fully spin polarized Fermi sea of compositefermions with density ρ e . Slightly away from ν = 1 / , i.e., k ∗ F = √ πρ e for ν < / k ∗ F = √ πρ h for ν > /
2, where ρ h is the density of holes in the LLL.The dimensionless quantity k ∗ F (cid:96) , where (cid:96) = (cid:112) (cid:126) c/eB isthe magnetic length, is given by k ∗ F (cid:96) = √ ν for ν < / k ∗ F (cid:96) = (cid:112) − ν ) for ν > / ρ e . Al-ternatively, the state can be described in terms of com-posite fermions formed from binding of vortices to holesin the LLL, which have the density ρ h . Are these two dif-ferent states of matter? This point of view has been takenin Ref. , which investigates the consequences of a spon- taneous breaking of the particle-hole (PH) symmetry inthe LLL. However, numerical calculations strongly sug-gest that, in spite of the seemingly different physics, thestates formed from composite fermions made of electronsand composite fermions made of holes are ultimately dualdescriptions of the same state. For example, the CFFSat ν = 1 / . As an-other example, the states at ν = ( n + 1) / (2 n + 1) canbe constructed either as ν ∗ = n of composite fermionsformed from holes or ν ∗ = n + 1 of composite fermionsmade of electrons (in a negative magnetic field); bothof these descriptions produce identical quantum numbersfor the ground and low energy excited states, and, indeed,their actual wave functions have close to 100% overlap forfinite but not very small systems . The PH symmetryof composite fermions has motivated a Dirac CF theorythat builds PH symmetry in a manifest fashion . TheChern-Simons field theory approach is not projectedinto the LLL, and thus does not allow a considerationof composite fermions formed from binding of vortices tothe holes of the LLL, but has nonetheless been shown toproduce results consistent with PH symmetry to certainnontrivial orders in perturbation theory. For experimen-tal parameters, the positions of the commensurability os-cillations minima predicted from the Dirac CF theory andthe Chern-Simons field theory agree to a high degree .Even if one assumes PH symmetry, the question “Isthe k ∗ F determined by ρ e , by ρ h , or by something else?”still remains. At ν = 1 /
2, where ρ e = ρ h , one wouldexpect k ∗ F = √ πρ e , if one assumes that the residualinteractions between composite fermions can be treatedperturbatively. However, away from ν = 1 / a r X i v : . [ c ond - m a t . s t r- e l ] D ec swer is not obvious. In Ref. 26, we addressed this ques-tion in an unbiased fashion starting from the microscopicwave functions of composite fermions, which are knownto be very close to the exact solutions of the Coulombproblem . Following an earlier work , we used theFriedel oscillations in the pair correlation function g ( r ),which is equal to the normalized probability of findingtwo particles a distance r apart from each other in theground state, to determine the CF Fermi wave vector at ν = 1 / k ∗ F into better agreement with the value as-certained from the minority carrier rule. We also obtainthe static structure factor for fully spin polarized sys-tems and compare it with the predictions from topolog-ical field theories and the Dirac composite fermionapproach .) For this purpose, we fitted the oscilla-tions in g ( r ) at an intermediate range of r , because forsmall r the short distance correlations dominate, whereasfor large r the pair correlation function exponentially ap-proaches unity due to the gap. We stress that no Fermiwave vector was built into the initial wave function. It isnotable that well defined oscillations occur at intermedi-ate distances at ν = n/ (2 n ±
1) even for moderate valuesof n . We also showed that the result was indepen-dent of whether composite fermions made of electrons orholes are used.The goal of this work is to extend the work of Ref. 26to include the spin degree of freedom and study a non-fully spin polarized CFFS at ν = 1 /
2. The reason is thatPH symmetry is not relevant to a CFFS that is not fullyspin polarized, because, once the spin degree of freedomis active, particle-hole symmetry relates states at ν and2 − ν . The apparent dichotomy that exists in the CFtheory for fully polarized states around ν = 1 / electrons see a zeroeffective magnetic field at filling factor 1 /
2. From thisperspective, one might argue that provided a Fermi sea isformed, its area must be determined by the density of upand down spin electrons, and not up and down spin holes.Nonetheless, it is in principle possible that the Luttingertheorem is violated for this state. Furthermore, as wemove away from ν = 1 /
2, there is no longer a symmetrythat relates the k ∗ F (cid:96) ’s on the two sides.Additionally, the question has experimental rele-vance. It is well known, both theoretically andexperimentally , that the CFFS is partially spin po-larized at relatively small but experimentally attainableZeeman energies. This is expected from the fact that thefractional quantum Hall (FQH) states at ν = n/ (2 n ±
1) have, in general, several spin polarizations ,which have been qualitatively and quantitatively ex-plained by the CF theory in terms of partially spin polar-ized states of composite fermions . Even a spin-singlet CFFS can be obtained by reducing the Land´e g factor to zero by application of pressure or by goingto a two-valley system with zero valley splitting .These questions have motivated us to consider theFermi wave vector for a non-fully spin polarized CFFS.For technical reasons, it is convenient for us to study aspin-singlet CFFS, but our results are straightforwardlygeneralizable to CFFS with arbitrary spin polarization.Since the technical details of this work are very simi-lar to those of Ref. 26, it would suffice to give an out-line. We evaluate the CF Fermi wave vector for spin-singlet states from Friedel oscillations within the CFframework and find that it is consistent with the den-sity of electrons. For this purpose, we consider boththe spin-singlet CFFS at ν = 1 / ν = n/ (2 n ±
1) for large n (only evenvalues of n produce spin-singlet states) where the den-sities of spin up and spin down composite fermions aregiven by ρ ↑ e = ρ ↓ e = ρ e /
2. We use the projected Jainwave functions for the calculation, which are known todescribe the physics of the LLL very accurately .At ν = 1 /
2, our calculated CF Fermi wave vector forboth up and down spin composite fermions is consistentwith k ∗ F = √ πρ e . Given that the model of noninteract-ing composite fermions is valid for both fully spin polar-ized and spin singlet states, we expect the Fermi wavevectors for partially spin polarized CFFS to be given by k ∗↑ F = (cid:113) πρ ↑ e and k ∗↓ F = (cid:113) πρ ↓ e , where ρ ↑ e and ρ ↓ e aredensities of up- and down-spin electrons or compositefermions. The situation is less clear away from ν = 1 / k ∗ F ,approaching k ∗ F = √ πρ e sufficiently close to ν = 1 / II. BACKGROUND
As a background, the FQH effect of electrons at fillingfactors along the Jain sequence ν = n/ (2 pn ±
1) is de-scribed as the integer quantum Hall (IQH) state of com-posite fermions carrying 2 p vortices (denoted as p CFs)with n filled Λ levels (ΛLs) . (The term ΛLs refers toemergent Landau-like levels of composite fermions, whichreside entirely within the LLL of electrons.) We shallspecialize to 2 p = 2 below. For spinful electrons, wewrite the CF filling as n = n ↑ + n ↓ , where n ↑ and n ↓ are the number of filled spin-up and spin-down ΛLsrespectively. The Jain wave function for this state isgiven by:Ψ n n ± = P LLL Φ ± n J = P LLL Φ ± n ↑ Φ ± n ↓ J (1)where J = (cid:81) ≤ j 2. For all our calculations we shall usethe Jain-Kamilla projection method, details of which canbe found in the literature . For n → ∞ the se-quence n/ (2 n ± 1) approaches the filling factor ν = 1 / atzero Zeeman energy. Throughout this work we use thespherical geometry in which N electrons reside onthe surface of a sphere and see a radial magnetic flux of2 Qhc/e emanated from a Dirac monopole sitting at thecenter of the sphere. All states considered in this workhave a uniform density on the sphere i.e., have total or-bital angular momentum L = 0, and have a total spin S = 0.Note that unlike for the fully polarized states around ν = 1 / 2, the CF theory gives a unique description for thestates at ν = n/ (2 n ± 1) involving spins. These statesare understood as IQH states of composite fermionsformed from attaching vortices to electrons ; these com-posite fermions see a positive effective magnetic field at ν = n/ (2 n + 1) and negative effective magnetic field at ν = n/ (2 n − III. CALCULATIONS AND RESULTS We extract the Fermi wave vector for compositefermions from the Fermi-sea like Friedel oscillations seenin the pair-correlation function of FQH states .The pair-correlation function for a homogeneous non-interacting Fermi gas in zero magnetic field with equalnumber of up and down spins in two dimensions is givenby : g σ,σ (cid:48) ( r ) = 1 − δ σ,σ (cid:48) (cid:32) J ( rk F ) rk F (cid:33) (2)where the Fermi wave vector k F = √ πρ , ρ is the totalfermion density and J ( x ) is the Bessel function of or-der one of the first kind. Clearly, we have g ↑ , ↓ ( r ) = 1 = g ↓ , ↑ ( r ) for all r as there are no Pauli correlations betweennon-interacting fermions of opposite spins. The oscilla-tory part of g σ,σ ( r ) for large rk F goes as ( rk F ) sin(2 k F r ),which lead to the well-known 2 k F Friedel oscillations.This motivates us to define the Fermi wave vector forFQH states through the the pair correlation function, forwhich we assume the form : g σσ ( r ) = 1 + A ( r (cid:112) πρ e ) − α sin(2 k ∗ F r + θ ) (3)where ρ e is the electron density and A , α , k ∗ F and θ arefitting parameters.We have evaluated the same-spin pair-correlation func-tion g σ,σ ( r ) for ν = n/ (2 n + 1) for up to n = 14 using theMonte Carlo method in the spherical geometry choos-ing r as the chord or the arc distance. The g σ,σ ( r ) forthe largest systems considered in this work are shown inFig. 1(a). In the same figure we also show the fits forthe pair-correlation function obtained using Eq. (3). Inthis fitting we have discarded small values of rk F , whereshort distance physics is important and large values of rk F where curvature effects become significant and oscil-lations decay exponentially due to the gap. For states inthe sequence n/ (2 n + 1), it is possible to perform calcula-tions for very large spin-singlet systems at filling factorsslightly below ν = 1 / 2. We have carried out these cal-culations for up to N = 350 electrons and have extrap-olated k ∗ F (cid:96) to the thermodynamic limit from our finitesystem results. The largest system available at ν = 1 / k ∗ F at ν = 1 / n and N than for fully spin polarizedcomposite fermions. The limitation in this respect is setby the ΛL index; it becomes increasingly more compu-tationally time consuming to fill higher and higher ΛLs.Since we need to fill only n/ n .For ν = 1 / . This state can also be viewed asthe n → ∞ limit of the n/ (2 n ± 1) sequence.We obtain thermodynamic extrapolation for the valueof k ∗ F (cid:96) by fitting our pair correlation function to both arcand chord distances, and the results are shown separatelyin Fig. 1(b). Both should give the same result in thethermodynamic limit, but seen from the error bars, thefits from the arc distance are more accurate for the finitesystems accessible to our calculations.An important result of our calculations is that the ex-trapolated value of the Fermi wave vector at ν = 1 / k ∗ F = √ πρ e (i.e. k ∗ F (cid:96) = √ ν ), as ex-pected in a model that assumes composite fermions tobe noninteracting. It is stressed that we do not makeany assumption regarding the interaction between com-posite fermions; in fact, the wave functions representinga strongly correlated liquid of electrons include all ef-fects of interaction and are very accurate representationsof the exact states. We also note, parenthetically, thatwe have also tried to calculate the Fermi wave vector at ν = 1 / / 2, the calculated Fermiwave vector appears to approach the value k ∗ F (cid:96) = (cid:112) / k ∗ F (cid:96) devi-ates from k ∗ F (cid:96) = √ ν . The presumably more accurate arcresults suggest that k ∗ F (cid:96) has a tent-like shape, but thisresult is not fully corroborated by the chord extrapola-tions. Furthermore, as we go far from ν = 1 / g ( r ) hasvery few oscillations, and it is unclear how meaningfulthe concept of Fermi wave vector remains.A technical comment is in order. To observe the samenumber of oscillations as for fully spin polarized states,the system size needs to be doubled, which is not alwaysfeasible. For this reason, our calculation on spinful sys-tems is more sensitive to finite size effects than the resultsof Ref. 26. The problem is especially severe for stateswith reverse flux attachment, where the LLL projectionsuffers from numerical precision issues . This requiresus to store all quantities to a high precision, which con-siderably slows down the calculation for large N . Due tothis technical issue, when we approach the CFFS fromabove, our extrapolations are based on smaller systems(and hence are less reliable) than as we approach it frombelow. Due to this reason we do not have enough accu-racy for ν > / 2. Nonetheless, the results are consistentwith k ∗ F = √ πρ e . IV. CONCLUDING REMARKS We end the article with several observations.Given that the prediction from a model of non-interacting composite fermions is valid for the spin-singlet and fully spin polarized CFFSs, it is natural toexpect that it remains valid for partially spin polarizedCFFS as well. As noted above, a partially spin polarizedCFFS will likely produce two different Fermi wave vec-tors, given by k ∗↑ F = (cid:113) πρ ↑ e and k ∗↓ F = (cid:113) πρ ↓ e . There ispreliminary experimental evidence for the reduction ofthe Fermi wave vector from its fully spin polarized value.Ref. 26 showed that the Fermi wave vectors of fullyspin polarized states related by particle-hole symmetry,when measured in units of the inverse magnetic length,are identical. In other words, we have ( k ∗ F (cid:96) ) − ν = ( k ∗ F (cid:96) ) ν .By the same token, for partially spin polarized CFFS,we have ( k ∗ F (cid:96) ) − ν = ( k ∗ F (cid:96) ) ν , and our results apply to theFermi wave vector of partially spin polarized states in thevicinity of ν = 3 / . That is beyond thescope of this work. To gain some qualitative insight, wehave calculated the Fermi wave vector from the unpro-jected Jain wave functions:Ψ un n n ± = Φ ± n J = Φ ± n ↑ Φ ± n ↓ J (4)These wave functions have a small amplitude in higherLLs and are likely adiabatically connected to theprojected wave functions . For these wave functions, the pair correlation function at filling factors ν = n/ (2 n − ν = n/ (2 n + 1) are the same for a given N whenplotted in units of the the radius of the sphere. For afinite spin-singlet system this gives the relation( k ∗ unF (cid:96) ) n n − ( k ∗ unF (cid:96) ) n n +1 = (cid:115) N − Q ∗ N − − Q ∗ , Q ∗ = 2 N − n n which in the thermodynamic limit implies:( k ∗ unF (cid:96) ) n n − = (cid:18) n + 12 n − (cid:19) / ( k ∗ unF (cid:96) ) n n +1 (5)The estimated values of k ∗ unF (cid:96) for the unprojected statesare shown in Fig. 2(b). These numbers show that tothe extent it can be defined, the CFFS area away from ν = 1 / : Ψ CS − MF n n ± = Φ ± n ( B ∗ ) J | J | (6)where B ∗ = B − ρφ . Since the absolute value of thesewave functions is the same as that of the IQH stateΦ ± n ( B ∗ ), we have k ∗ MFF = √ πρ e for all ν = n/ (2 n ± ρ e is the density of electrons.For completeness, we have also calculated the Coulombinteraction energies of the spin singlet FQH states. Theseare tabulated in Appendix B.In summary, we have shown that the Fermi wave vec-tor for composite fermions for a spin singlet Fermi sea at ν = 1 / ν = 1 / 2, our calcu-lations admit the possibility of a tent-like structure for k ∗ F (cid:96) . ACKNOWLEDGMENTS We thank M. Mulligan, T. Senthil and D. Son for use-ful communications, and C. T¨oke for help with computercalculations and useful discussions. ACB was supportedin part by the European Research Council (ERC) underthe European Union Horizon 2020 Research and Inno-vation Programme, Grant Agreement No. 678862; bythe Villum Foundation; and by The Center for QuantumDevices funded by the Danish National Research Foun-dation. JKJ was supported by the U. S. National Sci-ence Foundation Grant no. DMR-1401636. Some calcu-lations were performed with Advanced CyberInfrastruc-ture computational resources provided by The Institutefor CyberScience at The Pennsylvania State University. 10 15 20 250.951.001.05 r /ℓ, r is the arc distance g σσ ( r ) N = , ν = / = , ν = / = , ν = / = , ν = / = , ν = / FIG. 1. (a) Pair correlation function g ( r ) as a function of r/(cid:96) , where r is the arc distance on the sphere, obtained usingthe projected wave functions of Eq. (1). The solid lines arefits using Eq. (3) for oscillations in an intermediate range of r .The curves (except for 14/29) have been shifted up or down bymultiples of 0.02 to avoid clutter. (b) Thermodynamic valueof k ∗ F (cid:96) as a function of ν from arc fits (solid vertical bars) andchord fits (dashed vertical bars), slightly shifted horizontallyfor clarity. The mean-field value k ∗ MFF (cid:96) = √ ν is shown forreference. Appendix A EXTRAPOLATION OF THE FERMIWAVE VECTOR IN THE SPHERICALGEOMETRY In this appendix we show the thermodynamic extrap-olation of the Fermi wave vector obtained from finitesize systems in the spherical geometry for spin singletstates. The extrapolations of the Fermi wave vectors forthe projected and unprojected states are shown in Fig. 3and 4 respectively. The thermodynamic values of theFermi wave vector obtained from these extrapolations areshown in Fig. 1(b) and Fig. 2(b) where the range shownis obtained from linear fitting in 1 /N of the Fermi wavevector obtained from the arc and chord distance results.Some remarks regarding the extrapolation are in or-der. At filling factors where we have a large number ofsystems, the thermodynamic values of the Fermi wavevector given by a linear fit to the arc and chord distancedata approach one another. We have also tested thata quadratic fit gives a value within 0.05 of the linearfit. At filling factors n/ (2 n + 1) we have considered verylarge systems, so we expect corrections to be small. For ν = 1 / ν = n/ (2 n − 1) we only have a few systems 10 15 20 250.951.001.05 r /ℓ, r is the arc distance g σσ ( r ) N = , ν = / = , ν = / = , ν = / = , ν = / FIG. 2. Same as in Fig. 1 but for the unprojected wavefunctions. Also shown for reference is the mean field value k ∗ MFF (cid:96) = √ ν corresponding to composite fermions made fromelectrons. and, given the scatter, a quadratic fit is not appropriatefor them. Appendix B GROUND STATE COULOMBINTERACTION ENERGIES OF SPIN-SINGLETSTATES For completeness, we list the Coulomb energies of theJain wave functions for various spin-singlet states alongthe sequence n/ (2 n ± using standard Monte Carlomethods, assuming zero thickness and no LL mixing.The density for a finite system in the spherical geom-etry depends on the number of electrons N and is dif-ferent from its thermodynamic value. To eliminate thiseffect we use the so-called “density-corrected” energy E (cid:48) N = ( QνN ) / E N for extrapolation to the thermody-namic limit N → ∞ . All energies quoted here are theper particle density-corrected energies E (cid:48) N /N .Fig. 5 shows the thermodynamic extrapolation of theprojected and unprojected ground state Coulomb ener-gies for the various spin-singlet FQH states along thesequence n/ (2 n ± 1) as well as for the limiting case ofthe ν = 1 / k ∗ F ℓ ν = / k ∗ F ℓ ν = / k ∗ F ℓ ν = / k ∗ F ℓ ν = / k ∗ F ℓ ν = / FIG. 3. Thermodynamic extrapolation of the Fermi wave vector k ∗ F (cid:96) for the projected spin-singlet Jain wave function at variousfilling factors along the sequence n/ (2 n ± 1) and the ν = 1 / those of the projected states , because the unprojectedstates have better correlations at short distance. We notethat the ground state energies for the unprojected statesalong the sequence n/ (2 n − 1) is related to the energies of n/ (2 n + 1) FQH states. This is because | Ψ un | evaluatedon the unit sphere for the unprojected states at fillingfactors ν = n/ (2 n − 1) and ν = n/ (2 n + 1) is the samefor a given N . The Coulomb energies scale inversely withthe radius of the sphere ( √ Q(cid:96) ), thus we have the relation: E un n n − E un n n +1 = (cid:115) N − Q ∗ N − − Q ∗ , Q ∗ = 2 N − n n (7)where Q ∗ is the effective magnetic flux seen by the CFs.This in the thermodynamic limit implies: E un n n − = (cid:18) n + 12 n − (cid:19) / E un n n +1 (8) Appendix C RESULTS ON FULLY POLARIZEDSTATES We take this opportunity to also report certain resultsfor fully spin polarized states. We will present betterestimates for the Fermi wave vector of fully spin polar-ized composite fermions than those in Ref. , obtainedfrom more extensive calculations. We will also presentresults for the static structure factor for fully spin polar-ized FQH sates and its comparison with predictions fromfield theoretical approaches . A Updating results of Ref. Since the publication of Ref. , we have obtained re-sults for 8/17; we have studied larger systems for states k ∗ un F ℓ ν = / k ∗ un F ℓ ν = / k ∗ un F ℓ ν = / k ∗ un F ℓ ν = / FIG. 4. Same as Fig. 3 but for the thermodynamic extrapolation of the Fermi wave vector k ∗ unF (cid:96) for the unprojected Jain wavefunction defined in Eq. 6 at various filling factors along the sequence n/ (2 n + 1). projected states1 / N E n e r g y ( e / ǫ ℓ ) projected reverse fl ux attached states1 / N E n e r g y ( e / ǫ ℓ ) unprojected states1 / N E n e r g y ( e / ǫ ℓ ) FIG. 5. (Color online) Thermodynamic extrapolation of the Coulomb ground state energies in the spherical geometry forprojected (left and center panels) and unprojected (right panel) spin-singlet states in the sequence n/ (2 n ± 1) with even n . Theextrapolated energies are listed in Table I. along the sequence n/ (2 n +1) with n = 3 − 7; and we havecalculated the pair correlation function for the ν = 1 / N = 100. The updated results are given inFigs. 6 and 7. The upper panel of Fig. 6 shows thepair correlation function for the largest N and its fittingto g ( r ) = 1 + A ( r √ πρ e ) − α sin(2 k ∗ F r + θ ) to obtain k ∗ F ,and Fig. 7 shows thermodynamic extrapolations of k ∗ F forseveral fillings. The thermodynamic values of the Fermiwave vector obtained from these extrapolations is shownin the lower panel of Fig. 6 where the range shown isobtained from linear and quadratic fitting in 1 /N of theFermi wave vector obtained from the chord and arc dis-tance results. These calculations further corroborate theresults of Ref. in that the Fermi wave vector for Jainstates at ν = n/ (2 n + 1) is close to the value ascertainedfrom the minority carrier rule. Furthermore, we find that at ν = 1 / B static structure factor We have also considered the static structure factorfor the fully spin polarized state and compared it tothe predictions by Gromov et al. using a topolog-ical approach and those based on the Dirac CFdescription . We find that our results agree with thepredictions in the long wave length limit. filling factor ν per particle Coulomb energy ( e /(cid:15)(cid:96) )projected state unprojected state6/13 -0.45798(1) -0.47080(1)8/17 -0.46074(1) -0.47539(0)10/21 -0.46245(2) -0.47821(0)12/25 -0.46358(2) -0.48011(1)14/29 -0.46449(5) -0.48151(2)1/2 -0.46961(1) -0.49003(0)6/11 -0.48304(4) -0.51181(1)8/15 -0.47940(8) 0.50609(0)10/19 -0.47735(7) 0.50275(0)TABLE I. Coulomb interaction energies (in units of e /(cid:15)(cid:96) )obtained from a thermodynamic extrapolation of results onthe spherical geometry for spin-singlet ground states states at ν = n/ (2 n ± ν per particle Coulomb energy ( e /(cid:15)(cid:96) )3/7 -0.44226(1)4/9 -0.44751(1)5/11 -0.45081(1)6/13 -0.45309(2)7/15 -0.45475(1)8/17 -0.45604(2)1/2 -0.46566(10)TABLE II. Coulomb interaction energies (in units of e /(cid:15)(cid:96) )obtained from a thermodynamic extrapolation of results onthe spherical geometry for the fully polarized ground statesat ν = n/ (2 n + 1) using the Jain wave function. The structure factor S ( q ) is defined by therelation : S ( q ) = (cid:104) ρ q ρ − q (cid:105) N − N δ q , , ρ q = (cid:88) j e i q . r j (9)where (cid:104)· · · (cid:105) denotes the expectation value in the groundstate. It is related to the pair-correlation function g ( r )by the Fourier transform : S ( q ) − (cid:90) d r e i q . r ρ ( r )[ g ( r ) − 1] (10)Considering uniform incompressible states on the planewe get: S ( q ) − πρ (cid:90) dr rJ ( kr )[ g ( r ) − 1] (11)where J ( x ) is the zeroth order Bessel function of the firstkind. For the Fermi sea of non-interacting fully polarizedelectrons at zero magnetic field the static structure factor r /ℓ , r is the arc distance g ( r ) N = , ν = / = , ν = / = , ν = / = , ν = / = , ν = / = , ν = / = , ν = / FIG. 6. (a) The pair correlation function obtained from theprojected Jain wave functions of fully spin polarized compos-ite fermions, as a function the arc distance on the sphere. Thethick lines are fits to g ( r ) = 1 + A ( r √ πρ e ) − α sin(2 k ∗ F r + θ )in an intermediate range of r where oscillations are seen. Thecurves (except for 6 / 13) have been shifted up or down by mul-tiples of 0 . 02 for ease of viewing. (b) Thermodynamic valueof k ∗ F (cid:96) as a function of ν obtained from linear and quadraticfits to the arc and chord data. The mean-field values √ ν (blue) and (cid:112) − ν ) (green) are shown for reference. is given by : S ( q ) = , q ≥ k F π (cid:34) q k F (cid:114) − (cid:16) q k F (cid:17) + arcsin (cid:16) q k F (cid:17)(cid:35) , q < k F (12)which starts at zero for q = 0 and increases monotonicallytill it attains its maximum value of unity at 2 k F and thenstays there for q > k F . At q = 2 k F , S ( q ) as well as itsfirst derivative are both continuous.Using the pair-correlation functions calculated in thespherical geometry we can get the structure factor bynumerically evaluating the integral given in Eq. 11. Thisis a valid approach since the systems considered in thiswork are large and curvature effects are negiligible forthem, thereby allowing us to use the Fourier transformon the plane. One can also directly use the Fourier trans-form on the sphere and we have checked that these givesimilar results.We can also evaluate the structure factor directly fromits definition given in Eq. 9. The magnitude of the pla-nar wave vector q is related to the total orbital angular k ∗ F ℓ ν = / k ∗ F ℓ ν = / k ∗ F ℓ ν = / k ∗ F ℓ ν = / k ∗ F ℓ ν = / k ∗ F ℓ ν = / k ∗ F ℓ ν = / FIG. 7. Thermodynamic extrapolation of the Fermi wave vector k ∗ F (cid:96) for the projected fully polarized Jain wave function atvarious filling factors along the sequence n/ (2 n + 1) and the ν = 1 / /N . projected states1 / N E n e r g y ( e / ǫ ℓ ) FIG. 8. (Color online) Thermodynamic extrapolation of theCoulomb ground state energies in the spherical geometry forthe fully polarized Jain states in the sequence n/ (2 n + 1) (seeTable II for the extrapolated energies). momentum L on the sphere by the relation: L = qR where R = √ Q(cid:96) is the radius of the sphere and the static structure factor S L ≡ S qR is given by : S L = , L = 0 πN (cid:104)| (cid:80) j Y L, (Ω j ) | (cid:105) = (cid:80) i,j (cid:104) P k ( cos ( rijR )) (cid:105) N , L > Y l,m (Ω ≡ ( θ, φ )) are spherical monopole harmon-ics with θ and φ the polar and azimuthal angles on thesphere, P k ( x ) is the k th ordered Legendre polynomial,and r ij is the arc distance between electrons i and j onthe sphere. In the above equation we have chosen L z = 0without loss of generality since we are only interestedin uniform homogeneous states. In Fig. 9 we show thestatic structure factor calculated using Eq. 13 for a largesystem along the sequence n/ (2 n + 1) and at ν = 1 / et al. found, under certain assumptions, thatthe static structure factor in the q (cid:28) (cid:96) = 1) limit forthe Jain states is given by : S top n n +1 ( q ) = 12 q + n q + (cid:18) n + 2 n − n − (cid:19) q + · · · (14)where in the terms corresponding to q and q can berelated to various topological properties of the system.0Using the Dirac composite fermion theory S n/ (2 n +1) ( q ) can be derived exactly in the large n limit,where q (2 n + 1) ∼ 1. The static structure factor S ( q ) inthis limit is given by: S Dirac n n +1 ( q ) = [ q (2 n + 1)] [(4 n + 2) − [ q (2 n + 1)] ] J ([ q (2 n + 1)])32 n (2 n + 1) J ([ q (2 n + 1)]) + 1 − e − q (15)where J α ( z ) is the Bessel function of the first kind.In the n → ∞ limit (and consequently small q limit) S Dirac n/ (2 n +1) ( q ) is identical to S top n/ (2 n +1) ( q ). We find thatthe calculated structure factor agrees well with both S top n/ (2 n +1) ( q ) and S Dirac n/ (2 n +1) ( q ) in the regime where q(cid:96) (cid:46) . N = , ν = / / F S ( q / k F ) q ℓ S ( q ) S ( q ) : topological termsS ( q ) : Dirac CF theoryS ( q ) : CF theory N = , ν = / / F S ( q / k F ) q ℓ S ( q ) S ( q ) : topological termsS ( q ) : Dirac CF theoryS ( q ) : CF theory N = , ν = / 11 q / F S ( q / k F ) q ℓ S ( q ) S ( q ) : topological termsS ( q ) : Dirac CF theoryS ( q ) : CF theory N = , ν = / 13 q / F S ( q / k F ) q ℓ S ( q ) S ( q ) : topological termsS ( q ) : Dirac CF theoryS ( q ) : CF theory N = , ν = / 15 q / F S ( q / k F ) q ℓ S ( q ) S ( q ) : topological termsS ( q ) : Dirac CF theoryS ( q ) : CF theory N = , ν = / 17 q / F S ( q / k F ) q ℓ S ( q ) S ( q ) : topological termsS ( q ) : Dirac CF theoryS ( q ) : CF theory q / F S ( q / k F ) N = , ν = / FIG. 9. 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