PERGAMON Solid State Communications 122 (2002) 401±406 www.elsevier.com/locate/ssc Coulomb energy of quasiparticle excitations in Chern±Simons composite states

Orion Ciftja*, Carlos Wexler

Department of Physics and Astronomy, University of Missouri±Columbia, Columbia, MO 65211, USA Received 5March 2002; accepted 23 March 2002 by A.H. MacDonald

Abstract The attachment of ¯ux tubes to electrons by a Chern±Simons (CS) singular gauge transformation of the wavefunction opened up the ®eld theoretical description of the fractional (FQHE). Nevertheless, in Jain's composite fermion (CF) theory, quasiparticles are believed to be vortices carrying a fractional charge in addition to the winding phase of the CS ¯ux tubes. The different structure of the wavefunction in these two cases directly affects the excitation energy gaps. By using a simple ansatz we were able to evaluate analytically the Coulomb excitation energies for the mean-®eld level CS wavefunction, thus allowing a direct comparison with corresponding numerical results obtained from Jain's CF picture. The considerable difference between the excitation energies found in these two cases demonstrates in quantitative terms the very different impact that the internal structure of the wavefunction has in these two approaches, often used interchangeably to describe the FQHE. q 2002 Elsevier Science Ltd. All rights reserved.

PACS: 73.43. 2 f; 73.21. 2 b Keywords: A. Surfaces and interfaces; D. Electron±electron interactions; D. Fractional quantum Hall effect

2 p 1. Introduction given by r n†ˆn=‰2pl0 B† Š; where l0 B†ˆ h= eB† is the magnetic length. At these ®lling factors the electrons Two-dimensional electronic systems (2DES) subject to a condense into a strongly correlated incompressible quantum strong perpendicular magnetic ®eld display remarkable liquid, giving rise to quantized Hall resistivity and thermally phenomena, re¯ecting the great importance of electronic activated longitudinal resistivity. correlations. The most important among them is the frac- Much of the theoretical work on FQHE is based on the tional quantum Hall effect (FQHE) [1,2] which results from study of the properties of a 2D fully spin-polarized (effec- a strongly correlated incompressible liquid state [3,4] tively spinless) system of N interacting electrons embedded formed at special densities of the 2DES. The basic physics in a uniform positive neutralizing background. The electrons of FQHE is well understood: the kinetic energy of the elec- with charge 2e e . 0† and mass me are assumed to be trons is quenched by the strong perpendicular magnetic ®eld con®ned in the xy-plane and subject to a strong perpendicu- and the Coulomb interaction dominates the physics of the lar magnetic ®eld B ˆ 0; 0; B†: Normally, the symmetric highly degenerate partially ®lled Landau level. gauge is adopted and the vector potential is A r†ˆ The dominant sequence of FQHE states occurs when the 2By=2; Bx=2; 0†: We will consider the thermodynamic ®lling factor of the lowest Landau level (LLL) is n ˆ limit N ! 1 and V ! 1; while N=V ˆ r n†; where N is p= 2mp 1 1†; where p ˆ 1; 2; ¼ and m ˆ 1; 2; ¼ are integers. the number of electrons and V is the area of the 2D sample. By de®nition n is the ratio of the number of electrons to the The many-electron system is described by the Hamiltonian degeneracy of each Landau level: the electron density is H^ ˆ K^ 1 V^; where K^ is the kinetic energy operator.

1 XN K^ ˆ ‰2ih7 1 eA r †Š2; 1† 2m j j * Corresponding author. Tel.: 11-573-884-2278; fax: 11-573- e jˆ1 882-4195. E-mail address: [email protected] (O. Ciftja). and V^ is the potential energy operator representing the 0038-1098/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S0038-1098(02)00139-4 402 O. Ciftja, C. Wexler / Solid State Communications 122 (2002) 401±406 electron±electron, electron±background and background± local particle density, f0 ˆ h=e is the magnetic ®eld ¯ux background interaction potentials quantum and ez is a unit vector perpendicular to the plane. In a translationally invariant system, the mean-®eld XN XN Z Hamiltonian may be obtained by replacing b r† by its V^ ˆ v ur 2 r u† 2 r n† d2rv ur 2 ru† j k j mean value kbl ˆ 2mf r n†e ; where r n† is the uniform j,k jˆ1 0 z electronic density. This reduces the magnetic ®eld to Bp ˆ 2 Z Z p r n† 2 2 B 1 2 2mn†; thus increasing the effective ®lling factor n for 1 d r1 d r2v ur1 2 r2u†; 2† p p 2 the CFs. When n ˆ r n†f0=B ˆ p with p an integer, the corresponding electron ®lling factor is n ˆ r n†f0=B ˆ 2 where v urj 2 rku†ˆ e =4pe0euzj 2 zku† is the Coulomb p= 2mp 1 1†; which is precisely the Jain's series of FQHE potential, zj ˆ xj 1 iyj is the location of the jth electron in states. In other words, at the mean-®eld level, the system can complex coordinates and e is medium's dielectric constant. be described as a system of subject to a reduced It has become clear in recent years that many features of magnetic ®eld Bp ˆ B= 2mp 1 1†: Therefore the CS ®eld the FQHE can be understood in terms of a new kind of theory solution (at the mean-®eld level) will be the follow- particle called a composite fermion (CF), which is a ing CS CF wavefunction. bound state of an electron and 2m vortices of the many- body quantum wavefunction [5,6]. The basic property of YN z 2 z †2m C ˆ j k F Bp†; 5† the CFs is that they experience a reduced effective magnetic n 2m p j,k uzj 2 zku ®eld Bp ˆ B 1 2 2mn†: Since the degeneracy of each Landau level N is proportional to the magnetic ®eld, the p s where Fp B † is the Slater determinant of p ®lled CF p p degeneracy Ns of each CF Landau level will be Ns ˆ Landau levels evaluated at the magnetic ®eld shown in the Ns 1 2 2mn†: As a result, at ®lling factors corresponding argument. to the main sequence of FQHE states, n ˆ p= 2mp 1 1† Such singular gauge transformation of the wavefunction (with p an integer), the effective ®lling factor of CFs will corresponds to attaching 2m magnetic ¯ux tubes to each be np ˆ p; that is, an integer number of CF LLs. In other electron. Flux tubes are neither charged, nor low-energy words, this transformation maps the FQHE of electrons onto excitations, so in reality it is not clear how electrons can an integer QHE of CFs. bind to them. In the CF theory, the quasiparticle is believed There are two calculational schemes based on the intui- to be an electron bound to 2m vortices, instead of 2m ¯ux tive physics presented earlier. The ®rst constructs explicit tubes. Differently from a ¯ux tube, a vortex brings a zero wavefunctions [5] while the second method employs a into the wavefunction, is therefore charged and, as a result, Chern±Simons (CS) ®eld theory [7] approach to investigate the attachment of vortices (instead of ¯ux tubes) brings a the CF state. Although the two schemes are based on the signi®cantly different physics. In fact, attachment of same physics, a precise quantitative relationship between vortices forms the basis of the Jain's CF wavefunction them has not been made clear. In the CS approach proposed theory [5]. Under general circumstances, however, Jain's by Halperin, Lee and Read [7] (HLR) to describe even- CF wavefunction also needs a complete LLL projection in denominator-®lled states, the electron system is subjected order to generate results directly comparable with experi- to a (mathematical) singular gauge transformation which ments. The LLL projection is quite far from being a trivial converts it into a new system of fermions interacting with operation and after projection it is not clear how to associate a CS magnetic-like ®eld b r† in addition to the original electrons and vortices in an unambiguous way. Neverthe- magnetic ®eld. The transformation can be thought of as less, both projected and unprojected Jain's CF wavefunc- the binding of 2m ®ctitious magnetic ¯ux quanta to an elec- tions describe binding of vortices to electrons and a ^ tron. If Cn is a solution of the SchroÈdinger equation HCn ˆ signi®cant amount of work concerning both groundstate ECn it will be related by and excited state properties of these wavefunctions has been reported in the literature [8±11]. YN z 2 z †2m C ˆ j k C0; 3† On the other hand, based on ®eld-theoretical grounds [12] n 2m j,k uzj 2 zku the attachment of ¯ux tubes, as in the CS CF wavefunction, precedes the attachment of vortices. Although it is expected to the wavefunction C0 which is a solution of the SchroÈdinger that the attachment of ¯ux tubes to the wavefunction does equation H^ 0C0 ˆ EC0; with Hamiltonian H^ 0 ˆ K^ 0 1 V^ and not capture the complete physics obtained by attachment of transformed kinetic energy operator, vortices, a detailed quantitative comparison of the two processes is highly desirable from a theoretical point of 1 XN K^ 0 ˆ { 2 ih7 1 e‰A r † 2 a r †Š}2; 4† view and has not yet been realized. Since the nature of 2m j j j e jˆ1 quasiparticles in the FQHE crucially depends on whether ¯ux tubes or vortices are attached to electrons, a study of where a r† is the CS vector potential, b r†ˆ7 £ a r†ˆ the low-energy excitations and excited state properties on

2mf0r r†ez is the ®ctitious CS magnetic ®eld, r r† is the both cases seems to be quite appealing on its own. O. Ciftja, C. Wexler / Solid State Communications 122 (2002) 401±406 403 Exact calculations using Jain's CF approach have only been possible in systems with a small number of electrons [8]. However, for ¯ux-bound electrons as in the CS CF picture, we point out that calculations can be readily performed by adopting a technique ®rstly introduced by Friedman and Pandharipande [13] in the context of nuclear matter. Suppose we are studying the FQHE state with ®lling n ˆ p= 2mp 1 1† having p CF Landau levels ®lled. The uppermost ®lled CF Landau level is the one with quantum index p 2 1†: The promotion of a CF from the CF Landau level with quantum index p 2 1† to the one with quantum ph index p produces a correlated wavefunction Cn that describes the quasiparticle±quasihole excitation. Instead of promoting a single CF from the uppermost ®lled CF Landau level to the next empty one, we choose to promote p Fig. 1. The characteristic integral C p† (Eq. (11)) plotted in a loga- a fraction DN ˆ xN p Ns † of such CFs. By consequence rithmic scale as a function of p. The ®t corresponds to C p†/ the quasiparticle±quasihole excitation energy will be given p20:38: by:

The intent of this work is to provide a comparison 2 Eg n†ˆ lim DU x; N†; 6† between the excitation energies of ¯ux-bound CFs described DN!1 2DN by the mean-®eld level CS CF wavefunction of Eq. (5) and those obtained for wavefunctions where vortices instead of where the change on the total correlation energy due to this ¯uxes are attached. We provide analytical expressions for process is DU x; N† and x is the small fraction of displaced the Coulomb excitation energy gaps corresponding to a CFs. Eq. (6) may be rewritten as speci®c CS CF wavefunction that incorporates the idea of  ¯ux binding to electrons and discuss the results in the 2 DU x; N† context of the HLR theory. We compare our analytic results Eg n†ˆlim ; x!0 2x N with corresponding numerical results obtained using Jain's 7† picture of CFs where vortices bind to electrons. We show in DU x; N† 1 kC ph x†uV^uC ph x†l 1 kC uV^uC l ˆ n n 2 n n ; quantitative terms how the excitation energy gaps are ph ph N N kCn x†uCn x†l N kCnuCnl affected by the internal structure of the wavefunction. ph where Cn x† denotes the quasiparticle±quasihole wave- function for xN quasiparticle±quasihole excitations (by 2. Coulomb excitation energies ph construction Cn x ˆ 0† ; Cn). Only the ®rst term in Eq. (7) is dependent on the fraction x; therefore one can write the The excitation energy gap is de®ned as the energy to add quasiparticle±quasihole excitation energy as one quasiparticle and one quasihole far away from each other (we will neglect effects) to the FQHE ground- 2 state at the given ®lling factor. In the CF language, the Eg n†ˆlim un x†; 8† x!0 2x excitation energy gap corresponds to the quasiparticle± quasihole excitation obtained by promoting a single CF where u x† is the correlation energy per particle corre- from the uppermost ®lled CF Landau level to the next higher n sponding to the quasiparticle±quasihole wavefunction, CF Landau level. The excited state wavefunction constructed using CFs 1 kC ph x†uV^uC ph x†l (whether we have ¯ux tubes or vortices bound to electrons) u x†ˆ n n : 9† n ph ph is not entirely in the LLL. Such intrinsic Landau level N kCn x†uCn x†l mixing in the wavefunction brings in some kinetic energy contribution into the excitation energy gap. In general, the The mathematical details of the calculation of un x† are energy gap can be written as D n†ˆEg n† 1 DKE n†; where given in Appendix A (see Eq. (A10)). The ®nal result is Eg n† is due to Coulomb correlation effects and DKE n†ˆ that the quasiparticle±quasihole Coulomb energy gap O "vc† is related to the kinetic energy associated with the corresponding to the CS CF wavefunction (Eq. (5)) is Landau level mixing. In the present work, we consider only given by: intra-LL excitations consistent with the (generally used) 2 assumption hvc ! 1: In this case only the Coulomb corre- e C p† Eg n†ˆ p ; 10† lation energies are signi®cant and D n† ù Eg n† [9]. 4pe0el0 B† 2mp 1 1 404 O. Ciftja, C. Wexler / Solid State Communications 122 (2002) 401±406 (is not projected) and has no special short-distance correla- tions when compared to Jain's CF wavefunction [5], obtained by throwing away the denominators in Eq. (5) and performing a projection onto the LLL [8,17]. In parti- cular, the mean-®eld CS CF formalism does not lead to screening of the quasiparticles' or quasiholes' charge, differ- ently from the fractionalization of charge which occurs beyond mean-®eld. Nevertheless both Jain's and the mean-®eld CS wavefunctions describe at some level CFs, and these have been generally assumed to be equivalent to a ®rst degree of approximation. The results presented in this paper show how these assumptions can be misleading in some circumstances.

3. Asymptotic behavior of the gap

Fig. 2. Exact excitation energy gaps, Eg n† corresponding to the As the ®lling factor of the main sequence of FQHE states ¯ux-bound CS CF wavefunction Cn obtained at the mean-®eld approaches the even denominator fractions 1/2 and 1/4 level of the CS ®eld theory for ®lling factors n ˆ p= 2mp 1 1† as (obtained as p ! 1 and m ˆ 1 and 2, respectively) it is function of p for m ˆ 1 and 2 and p $ 1; see Eq. (10). The results expected that the quasiparticle±quasihole excitation gap are compared to corresponding values for energy gaps [8] obtained should vanish. From the behavior of C p† shown in Fig. 1 for the LLL projected Jain's CF wavefunction, where vortices and Eq. (10), it is evident that this is, indeed, the case: our instead of ¯ux tubes are bound to electrons. All energies are in analysis of the exact CS excitation energies shows that E n† units of e2=‰4pe el B†Š: For large p: E n†/p20:88: g 0 0 g decays as p20:88 to a very good approximation for 50 & p & where the m-independent C p† is given by 2000: These results can be compared to the HLR theory at both !"# ! ! Z1 2 2 2 mean-®eld [7] and beyond [18]. At the mean-®eld level, the 2t2=2 1 t t t C p†ˆ dt e Lp21 Lp21 2 Lp ; HLR theory [7] predicts 0 2 2 2  11† p 1 1 e2 E n ˆ , ; 12† g 2mp 1 1 2mp 1 1† 4pe el B† a 0 0 and Ln x† are associated Laguerre polynomials [14]. In what follows, we calculated exactly the integrals as p ! 1: This behavior is consistent with the fact that p appearing in Eq. (11) for increasing values of p up to within this mean-®eld theory the effective mass m is 2000. The dependence of C p† on p is shown in Fig. 1. In unrenormalized by the ¯uctuations of the CS gauge ®eld the large-p limit C p†/p20:38: (see Eq. (4)) and should be p-independent. Since our approach neglects all ¯uctuation effects, the resulting In Fig. 2, we show the dependence of Eg n† on p for ®lling factors n ˆ p= 2mp 1 1† for m ˆ 1 and 2 with the excitation large p energy gaps should decay at most as fast as 1=p: 2 Our results are consistent with this argument. energy values expressed in units of e =‰4pe0el0 B†Š: A strik- ing feature of the results is the fact that the excitation energy When interactions beyond the mean-®eld and/or RPA gaps decrease very slowly as we approach the even- level are included and ¯uctuations in the transverse CS denominator-®lled states n ˆ 1=2 m ˆ 1† and n ˆ 1=4 m ˆ gauge ®eld are considered, a considerable renormalization 2† where, as expected, they vanish. To give a quantitative of the effective mass is to be expected. In fact, using a estimate of how the energy excitation gap is affected when fermionic CS approach, Stern and Halperin [18] considered we switch from ¯ux attachment to vortex attachment, we these ¯uctuations and found a divergent renormalization of p compare our analytic results for ¯ux-bound CFs to Monte m as p ! 1: Therefore in this limit, the low energy excita- Carlo results [8,15,16,] based on Jain's CF wavefunctions tions vanish faster as where vortex attachment is considered. One notes that the p 1 1 e2 energy gaps for ¯ux-bound CFs are considerably above the E < p : 13† g 2m m 2mp 1 1†ln 2mp 1 1† 4pe el B† energy gaps obtained in the vortex-bound case. Our analytic 0 0 results for ¯ux-bound CFs are based on a mean-®eld wave- At present time, a fully exact calculation of energy gaps, function, so we have to expect an overestimation of the starting from a microscopic wavefunction that incorporates energy gaps since we are neglecting ¯uctuations, however, effects beyond mean-®eld theory appears to be a very dif®- the large difference was not completely anticipated. cult task.

The wavefunction Cn obtained at the mean-®eld level of Quite recently, a sophisticated calculation for the FQHE the CS theory (Eq. (5)) has a high occupation of higher LLs energy gaps has been performed by Shankar [19] using the O. Ciftja, C. Wexler / Solid State Communications 122 (2002) 401±406 405 Hamiltonian theory of CFs that he and Murthy developed Acknowledgements [20]. In their Hamiltonian approach one ®rstly performs a 2m-fold ¯ux attachment by a CS transformation, and then This work was supported in part by the University of attaches the necessary number of zeros to transform the Missouri Research Board and Research Council (CW). ¯uxes into full vortices [20]. In this approach, the Hamil- tonian contains complicated expressions for the charge and other operators, while the wavefunction remains quite Appendix A simple (this treatment seems complimentary to Jain's approach, where the Hamiltonian is simple, but the wave- For convenience we list here the calculation of the function is very complicated because of the LLL projection Coulomb correlation energy per electron un x† (Eq. (9)) operator). The formalism was then used to compute activa- which is given by tion gaps for FQHE states and analytic expressions were derived for all fractions of the form n ˆ p= 2mp 1 1†: ph ^ ph 1 kCn x†uVuCn x†l Based on this approach various quantities of interest were un x†ˆ N kC ph x†uC ph x†l calculated to a reasonable accuracy: in particular it was n n Z shown that for the Zhang±Dasp Sarma [21] (ZDS) interaction r n† 2 2 ˆ d r ‰g x; r † 2 1Šv r †; A1† potential, v r†ˆe = e r2 1 l2†; the gap energies were 2 12 n 12 12 accurate to within 10±20% as compared to Monte Carlo work [22] for l . 1: Favorable electronic correlations in where the radial distribution function gn x; r12† is de®ned in ph the LLL are build up only through vortex attachment, as a terms of Cn x† by result the energy gaps derived from the Hamiltonian theory R are in better agreement and directly comparable to experi- N N 2 1† d2r ¼d2r uC ph x†u2 g x; r †ˆ R 3 N n : A2† mental and/or Monte Carlo results. n 12 2 2 ¼ 2 ph 2 r n† d r1 d rN uCn x†u On the other hand, we limited ourselves to a mean-®eld model with a speci®c choice of the wavefunction incorpor- In the thermodynamic limit, the above quantity depends on ating attachment of 2m-fold ¯ux tubes to electrons. We were the 2D interparticle distance r12 ˆ ur1 2 r2u and the quasi- thus able to provide analytical results strictly valid for the particle±quasihole fraction parameter x. In our case, one speci®c choice of the mean-®eld derived CS CF wavefunc- ph 2 p 2 observes that uCn x†u ˆ uFp x; B †u ; where the latter tion, without intending to directly compare the obtained term corresponds to a squared Slater determinant represent- energy gap values to realistic calculations where 2m-fold ing p 2 1† fully ®lled CF Landau levels, the pth CF Landau vortices are attached to the electrons and full projection level (with CF Landau level index p 2 1†) ®lled except for into the LLL is performed. The objective of this calculation p DN ˆ xN p Ns † CF holes, and the p 1 1†th CF Landau was to demonstrate the signi®cant effect that the lack of level (with CF Landau level index p) having the DN CFs vortex attachment has on the excitation properties of the that were removed from the underlying level. As a standard system. This is important due to the generalized impression result, it follows that for fully spin-polarized electrons (CFs) that most of the physics of the FQHE is contained in the simple CS mean-®eld model. 2 gn x; r12†ˆ1 2 ulp x; r12†u ; A3†

where the `statistical exchange' factor is computed from

4. Conclusions lp x; r12†ˆr^p x; r1; r2†=r n†: The (reduced) one-body density matrix r^p x; r1; r2† which corresponds to the dyna- p In summary, by using a simple ansatz we were able to mically uncorrelated state Fp x; B † is given by calculate analytically the excitation energy gaps for a mean- p ®eld CS CF wavefunction where 2m-fold ¯ux tubes are pX2 1 NXs 2 1 p attached to electrons. This provides a direct quantitative r^p x; z1; z2†ˆ wn;l z1†´wn;l z2† comparison with corresponding numerical results for energy nˆ0 lˆ0 gaps where vortices, and not ¯ux tubes, are bound to elec- " p DN NXs 2 1 trons as in Jain's CF wavefunction case. Based on these 1 wp z †´w z † Np p;l 1 p;l 2 analytical results, we can directly estimate in quantitative s lˆ0 terms the sizeable effect that the internal structure of CFs p # NXs 2 1 has on FQHE energy gaps, when switching from ¯ux attach- p 2 w p21†;l z1†´w p21†;l z2† ; A4† ment to vortex attachment. The large discrepancy in the lˆ0 excitation energies thus obtained clearly shows the limita- p tions of the commonly held belief that most of the important where DN=Ns ˆ xp is the average occupancy of a quantum physics of the FQHE is contained in the simple mean-®eld state in the partially ®lled CF Landau level with index p. The approximation of the CS CF picture. symmetric gauge single-particle eigenstates for magnetic 406 O. Ciftja, C. Wexler / Solid State Communications 122 (2002) 401±406

p ®eld B are The calculation of un x† follows from Eq. (A1). After writ- "# p 2 p ing r n†ˆp=‰2pl0 B † Š; we introduce the dimensionless 1 zz variable t ˆ r =l Bp†; and by rescaling the magnetic length wn;l z†ˆ p exp 12 0 2nn! 4l Bp†2 p 2 2 0 l0 B † ˆ l0 B† 2mp 1 1† we obtain ()"# n p !( ! 2 zz Z 2 2  2l Bp† w z†exp 2 ; A5† p 1 t 1 t 0 0;l p 2 u x†ˆ2 p dt exp 2 L1 2z 4l0 B † n p21 2 2mp 1 1 0 2 p 2 where " ! !#) 2 2 2 2 "# t t 1 e 1 z l zzp 1 x Lp 2 Lp21 : w z†ˆ p exp 2 ; A6† 2 2 4pe0 el0 B† 0;l l p 2 l Bp† 4l Bp†2 2 l!2pl0 B † 0 0 A10† where n ˆ 0; 1; ¼ denotes the index of various CF Landau p The results in Eqs. (10) and (11) follow immediately. levels and l ˆ 0; 1; ¼ Ns 2 1† is their angular momentum quantum number. Noting that

p !p NXs 2 1 r2 NXs 2 1 wp z †w z †ˆL 12 ´ wp z †w z †; n;l 1 n;l 2 n p 2 0;l 1 0;l 2 References lˆ0 2l0 B † lˆ0 A7† [1] R.E. Prange, S.M. Girvin (Eds.), The Quantum Hall Effect one ®nds that Springer, New York, 1990. [2] T. Chakraborty, P. PietilaÈinen, The Fractional Quantum Hall p ( ! NXs 2 1 2 Effect, Springer, New York, 1988. p 1 r12 r^ x; z ; z †ˆ w z †w z † L [3] D.C. Tsui, H.L. Stormer, A.C. Gossard, Phys. Rev. Lett. 48 p 1 2 0;l 1 0;l 2 p21 2l Bp†2 lˆ0 0 (1982) 1559. " ! !#) [4] R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. r2 r2 1 xp L 12 2 L 12 ; [5] J.K. Jain, Phys. Rev. Lett. 63 (1989) 199. p p 2 p21 p 2 2l0 B † 2l0 B † [6] J.K. Jain, Phys. Rev. B 41 (1990) 7653. A8† [7] B.I. Halperin, P.A. Lee, N. Read, Phys. Rev. B 47 (1993) 7312. x n n n 2x where Ln x†ˆ e =n!† d =dx † x e † are the Laguerre poly- [8] J.K. Jain, R.K. Kamilla, Phys. Rev. B 55 (1997) R4895. k k k k nomials of order n ˆ 0; 1; ¼ and Ln x†ˆ 21† d =dx †Â [9] N.E. Bonesteel, Phys. Rev. B 51 (1995) 9917. ‰Ln1k x†Š are the generalized Laguerre polynomials of [10] R.K. Kamilla, J.K. Jain, S.M. Girvin, Phys. Rev. B 56 (1997) order n ˆ 0; 1; ¼ and degree k ˆ 0; 1; ¼ 12411. In the thermodynamic limit [23], both density and ®lling [11] K. Park, J.K. Jain, Phys. Rev. Lett. 81 (1998) 4200. factor n ˆ N=N are kept constant as the number of electrons [12] A. Lopez, E. Fradkin, Phys. Rev. B 44 (1991) 5246. s [13] B. Friedman, V.R. Pandharipande, Phys. Lett. 100B (1981) 3. N and the Landau level degeneracy Ns go to in®nity. Since p [14] I.S. Gradshteyn, I.M. Ryzhik, in: A. Jeffrey (Ed.), Table of the degeneracy Ns of each CF Landau level is directly p Integrals, Series, and Products, ®fth Ed, Academic Press, New proportional to Ns then also Ns 2 1† goes to in®nity in York, 1980, pp. 1064±1065. this limit. At this point the summation over l in Eq. (A8) [15] J.K. Jain, R.K. Kamilla, in: O. Heinonen (Ed.), Composite is extended from 0 to 1 and one obtains the one-body Fermions, World Scienti®c, New Jersey, 1998. p density matrix r^p x; z1; z2† that no longer depends on Ns : [16] J.K. Jain, R.K. Kamilla, Int. J. Mod. Phys. B 11 (1997) 2621. The statistical exchange term lp x; z1; z2†ˆr^ x; z1; z2†=r n† [17] R.K. Kamilla, J.K. Jain, Phys. Rev. B 55 (1997) 9824. is a complex quantity, but since the radial distribution func- [18] A. Stern, B.I. Halperin, Phys. Rev. B 52 (1995) 5890. tion is found by squaring it (see Eq. (A3)), the phase factor [19] R. Shankar, Phys. Rev. B 63 (2001) 085322 and references therein. of lp x; z1; z2† vanishes and we obtain " #( ! [20] R. Shankar, G. Murthy, Phys. Rev. Lett. 79 (1997) 4437. r2 1 r2 [21] F.C. Zhang, S. Das Sarma, Phys. Rev. B 33 (1986) 2908. g x; r †ˆ1 2 exp 2 12 L1 12 n 12 p 2 p21 p 2 [22] K. Park, N. Meskini, J.K. Jain, J. Phys.: Condens. Matter 11 2l0 B † p 2l0 B † (1999) 7283. " ! !#)2 [23] O. Ciftja, Phys. Rev. B 59 (1999) 8132. r2 r2 1 x L 12 2 L 12 : p p 2 p21 p 2 2l0 B † 2l0 B † A9†