Physica E 9 (2001) 226–230 www.elsevier.nl/locate/physe Theoretical estimates for the correlation energy of the unprojected composite fermion wave function Orion Ciftjaa; b; ∗ a Ames b Department Laboratory, Iowa State University, Ames, IA 50011, USA of Physics, Texas A&M University, College Station, TX 77843, USA Received 18 December 1999; accepted 14 June 2000 Abstract The most prominent lling factors of the fractional quantum Hall e ect are very well described by the Jain’s microscopic composite fermion wave function. Through these wave functions, the composite fermion theory recovers the Laughlin’s wave function as a special case and exposes itself to rigorous tests. Considering the system as being in the thermodynamic limit and using simple arguments, we give theoretical estimates for the correlation energy corresponding to all unprojected composite fermion wave functions in terms of the accurately known correlation energies of the Laughlin’s wave function. The provided theoretical estimates are in very good agreement with available Monte Carlo data extrapolated to the thermodynamic limit. These results can be quite instructive to test the reliability and accuracy of di erent computational methods employed on the study of these phenomena. ? 2001 Elsevier Science B.V. All rights reserved. PACS: 73.40.Hm; 73.20.Dx Keywords: Quantum Hall e ect; Composite fermions; Laughlin states 1. Introduction The fractional quantum Hall e ect [1,2] (FQHE) results from a strongly correlated incompressible liquid state [3,4] formed at special uniform densities () of a two-dimensional (2D) electronic system which is subjected to a very strong perpendicular magnetic Correspondence address: Department of Physics, Texas A&M University, College Station, TX 77843, USA. Tel.: +1-409-845-7785; fax: +1-409-845-2590. E-mail address: ociftja@rainbow.physics.tamu.edu (O. Ciftja). ∗ eld B. The dominant sequence of fractional Hall states occurs when the lling of the lowest Landau level (LLL) is  = p=(2mp + 1), where p = 1; 2; : : : and m = 1; 2; : : : are integers. Much of the theoretical work on the FQHE is based on the study of the properties of a 2D fully spin-polarized (spinless) system of N interacting electrons embedded in a uniform positive background. The electrons with charge −e (e ¿ 0) and mass me are considered con ned in the x–y plane of area and subjected to a magnetic eld, B = (0; 0; B) which is generated from the symmetric gauge vector potential A(r) = (−By=2; Bx=2; 0). We 1386-9477/01/$ - see front matter ? 2001 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 0 ) 0 0 1 9 8 - 3 227 O. Ciftja / Physica E 9 (2001) 226–230 will consider the thermodynamic limit of an in nite system de ned as the limit of N electrons in a sample of area , where N and go to in nity with the density of the sample, () = N= kept constant. By varying the strength B of the magnetic eld the lling factor of the LLL is changed and, in a more convenient form, the electronic density of the system may p be written as () = =[2l0 (B)2 ] where l0 (B) = ˜=(eB) is the electronic magnetic length. The many-electron system is described by the Hamiltonian Ĥ = K̂ + V̂ , where K̂ is the kinetic energy operator N 1 P [ − i˜Bj + eA(rj )]2 K̂ = 2me j=1 (1) and V̂ = N P j¡k v(|rj − rk |) − () N P j=1 Z d 2 r v(|rj − r|) Z Z ()2 d 2 r1 d 2 r2 v(|r1 − r2 |) (2) + 2 is the total electron–electron, electron–background and background–background interaction potential, where v(|rj − rk |) = e2 =(40 |zj − zk |) is the interaction potential, zj = xj + iyj is the location of the jth electron in complex coordinates and  is the dielectric constant of the background. It has become clear in recent years that many essential features of the FQHE can be understood straightforwardly in terms of a new kind of particle, called a composite fermion (CF), which is a bound state of an electron and an even number of vortices of the many-body quantum wave function [5,6] formed at the electronic densities (). The fundamental property of the CFs is that they experience a reduced e ective eld, B∗ = B(1 − 2m) so that the quantum liquid of strongly correlated electrons at B is equivalent to a quantum liquid of weakly interacting CFs at B∗ . Since the degeneracy Ns of each electronic Landau level is proportional to the magnetic eld, the degeneracy Ns∗ of each CF Landau level will be smaller than the corresponding degeneracy for the electrons and will be given by Ns∗ = Ns (1 − 2m). As a result the e ective lling factor of CFs will be an integer number ∗ = p and will correspond to stable electronic lling factors  = p=(2mp + 1). There are two calculational schemes based on the intuitive physics above. One constructs explicit wave functions [5] while the second scheme employs a Chern–Simons (CS) eld theory [7] approach to investigate the CF state. Although the two schemes are based on the same physics, a precise quantitative relationship between them is not clear. The microscopic wave function N Q CF (zj − zk )2m =p=(2mp+1) = P̂ LLL j¡k   |zj |2 exp −2m × p (B∗ ); 4l0 (B)2 j=1 N Q (3) where p (B∗ ) is the Slater determinant wave function of p lled CF Landau levels, evaluated at the magnetic eld shown in the argument and P̂ LLL is the LLL projection operator is due to Jain [5] and we shall refer to it as the Jain’s CF wave function. For the special case of the ground state at p = 1; the above CF wave function recovers the Laughlin wave function [4], which is already known to be a very accurate representation of the exact ground state at  = 1=(2m + 1). Note that the Gaussian factors in Q N 2 2 ∗ j=1 exp[ − 2m|zj | =4l0 (B) ] and p (B ) combine to produce a Gaussian factor corresponding to the external magnetic eld, since 1 1 2m + = : l0 (B)2 l0 (B∗ )2 l0 (B)2 (4) In the p → ∞ limit the e ective magnetic eld B∗ vanishes and we end up with lim p→∞ CF =p=(2mp+1) = Fermi =1=(2m) ; (5) where the latter is the Rezayi–Read (RR) Fermi-sealike wave function [8] that has the form   N N Q |zj |2 2m Q Fermi exp − (zj − zk ) =1=(2m) = P̂ LLL 4l0 (B)2 j=1 j¡k ×Det{’k (r)}; (6) where ’k (r) − s are 2D normalized plane waves which ll a Fermi disk up to |k|6kF [ = 1=(2m)]. In this paper, we neglect the LLL projection operator and we consider the unprojected CF wave function that again corresponds exactly to the Laughlin’s wave function for p = 1 and becomes the unprojected RR Fermi-sea-like wave function in the p → ∞ limit. By using simple arguments we prove that we can obtain very accurate theoretical estimates for the correlation energy per particle corresponding to the 228 O. Ciftja / Physica E 9 (2001) 226–230 unprojected CF wave function for any arbitrary lling factor  = p=(2mp + 1) in terms of the correlation energy per particle for the Laughlin’s wave function. The correlation energy per particle for the Laughlin states has been computed with di erent methods and very accurate estimates are available, but the study of the other fractional states, especially in the vicinity of the even-denominator- lled state is computationally very challenging to be performed with high accuracy in the thermodynamic limit. The provided theoretical estimates of the correlation energy per particle for such arbitrary lling factors may prove quite useful to test the accuracy of di erent nite number computational schemes and the extrapolation of their results to the thermodynamic limit. 2. The correlation energy for the unprojected composite fermion wave function Considering the 2D system of electrons as being in the thermodynamic limit, the correlation energy per particle corresponding to the unprojected CF wave function describing the state with lling factor  = p=(2mp + 1) may be calculated from the expression 1 h CF |V̂ | CF i N h CF | CF i Z () d 2 r12 [g (r12 ) − 1]v(r12 ); = 2 u() = (7) where g (r12 ) is the radial distribution that is given by R N (N − 1) d 2 r3 · · · d 2 rN | CF |2 R ; (8) g (r12 ) = ()2 d 2 r1 · · · d 2 rN | CF |2 and for the system under consideration will depend only on the interparticle spatial distance. For all the unprojected CF wave functions describing states with lling factor  = p=(2mp + 1) the short-range behavior of the radial distribution function is determined by (2m) zeros of the same Jastrow factor in addition to the usual antisymmetry zero coming from the Pauli principle, so as a result g=p=(2mp+1) (r12 ) ∼ (r12 )2m+2 : (9) With the only plausible assumption that the interaction between electrons is governed by the short-range behavior of the radial distribution function, one would expect to have the correlation energy per particle given by   p e2 u = = C(m) 2mp + 1 40 R0 () r e2  ; (10) = C(m) 2 40 l0 (B) where the xed density () was expressed in terms of the ion disk radius R0 ()2 = 1=() and     Z  r12 r12 g=p=(2mp+1) −1 : C(m)= d R0 () R0 () (11) With this assumption, C(m), that represents the above one-dimensional integral should not depend on the value of p, but only on the value of m and, as a result, can be determined by considering the state with lling factor  = 1=(2m + 1), (p = 1) that is described by the Laughlin wave function. So all correlation energies per particle corresponding to the Jain’s unprojected CF wave function for states with arbitrary lling factor  = p=(2mp + 1) can be expressed in terms of the correlation energy per particle of the Laughlin’s wave function by the simple formula  s  (2m + 1)p p = u = 2mp + 1 2mp + 1   1 ; ×u  = 2m + 1 (12) where u( = 1=(2m + 1)) is the correlation energy per particle of the Laughlin states. The corresponding correlation energies per particle for the unprojected RR Fermi-sea-like wave function are easily obtained by performing the p → ∞ limit and are given by  r    2m + 1 1 1 = : (13) u = u = 2m 2m 2m + 1 As value for the correlation energy per particle for the Laughlin’s wave function we have taken the essentially exact results reported by Levesque et al. [9], u( = 1=3) = −0:4100(1=40 )e2 =l0 (B) and obtained u( = 1=5) = −0:3277(1=40 )e2 =l0 (B), after extensive Variational Monte Carlo (VMC) simulations with up to 256 particles and generating as many as 5 million con gurations. O. Ciftja / Physica E 9 (2001) 226–230 229 Table 1 Theoretical estimates for the correlation energy per particle u() corresponding to the Jain’s unprojected CF wave function, CF at lling factor  = p=(2p + 1). The correlation energy per particle is expressed in units of (1=40 )e2 =l0 (B), where l0 (B) is the magnetic length of the electrons. The correlation energy per particle corresponding to the Laughlin’s wave function was taken u( = 1=3) = −0:4100(1=40 )e2 =l0 (B) as reported from the extensive VMC simulations of Levesque et al. [9]. In the fth column we report the available results of Kamilla and Jain [10] obtained from a VMC simulation in the spherical geometry m p  = p=(2mp + 1) Theoretical estimate Ref. [10] 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 ∞ 1=3 2=5 3=7 4=9 5=11 6=13 7=15 8=17 9=19 10=21 1=2 −0.410000 −0.449132 −0.464896 −0.473427 −0.478776 −0.482445 −0.485118 −0.487152 −0.488752 −0.490043 −0.502145 −0.4092(70) −0.4489(10) −0.4644(20) −0.4734(15) ··· ··· ··· ··· ··· ··· ··· Table 2 Theoretical estimates for the correlation energy per particle u() corresponding to the Jain’s unprojected CF wave function, CF at lling factor  = p=(4p + 1). The correlation energy per particle is expressed in units of (1=40 )e2 =l0 (B), where l0 (B) is the magnetic length of the electrons. The correlation energy per particle corresponding to the Laughlin’s wave function was taken u( = 1=5) = −0:3277(1=40 )e2 =l0 (B) as reported from the extensive VMC simulations of Levesque et al. [9] m p  = p=(2mp + 1) Theoretical estimate 2 2 2 2 2 2 2 2 2 2 2 1 2 3 4 5 6 7 8 9 10 ∞ 1=5 2=9 3=13 4=17 5=21 6=25 7=29 8=33 9=37 10=41 1=4 −0.327700 −0.345426 −0.352006 −0.355440 −0.357550 −0.358977 −0.360007 −0.360785 −0.361394 −0.361884 −0.366379 In Tables 1 and 2 we show the theoretical estimates for the correlation energy per particle obtained from Eq. (12) for the rst 10 states with lling factors of the form  = p=(2p + 1) and  = p=(4p + 1), respec- Fig. 1. Theoretical estimates for the correlation energy per particle corresponding to the unprojected Jain’s CF wave function CF for states with lling factors  = p=(2mp + 1) for m = 1 (square) and m = 2 (circle) as function of p. In the p → ∞ limit we obtain the theoretical estimates for the correlation energy per particle corresponding to the even-denominator- lled states described by the unprojected RR Fermi-sea-like wave function, respectively, for  = 21 (dotted line) and  = 14 (dashed line). The values of the correlation energy per particle corresponding to the Laughlin’s wave function were taken from Levesque et al. [9]. The energies are expressed in the standard units of (1=40 )e2 =l0 (B). tively. As seen from Table 1 such theoretical estimates for the correlation energy per particle of the Jain’s unprojected CF wave function are in excellent agreement with the corresponding available data from Kamilla and Jain [10] obtained using VMC techniques in the spherical geometry. The correlation energy per particle u( = p=(2mp + 1)) corresponding to the unprojected Jain’s CF wave function is shown in Fig. 1 for the series of states  = p=(2p + 1) (square) and  = p=(4p + 1) (circle). In the p → ∞ limit the results converge to the values given by Eq. (13), that correspond to the unprojected RR Fermi-sea-like wave function for lling factors  = 21 (dotted line) and  = 41 (dashed line), respectively. These theoretical estimates can be quite useful to gauge the accuracy of di erent numerical methods. Suppose that, with a given a numerical method, one can compute the correlation energy per particle for the Laughlin’s wave function with an absolute error
u( = 1=(2m + 1)). A theoretical estimate for the absolute error on computing the correlation energy per particle for the Jain’s unprojected CF wave function describing a state with arbitrary lling factor 230 O. Ciftja / Physica E 9 (2001) 226–230 believe that such theoretical estimates are essentially exact and can be used to test the reliability and to discriminate between di erent numerical methods employed on the study of FQHE.  = p=(2mp + 1) can be readily obtained as  s  (2m + 1)p p =
u  = 2mp + 1 2mp + 1  ×
u  = 1 2m + 1  : (14) One easily notes that if the same numerical method applied to the unprojected CF wave function with arbitrary lling factor  = p=(2mp + 1) has an absolute error larger than that given from Eq. (14) then the reliability of the results obtained with this method, especially for lling factors with increasing values of p, is certainly doubtful. 3. Conclusions With the only assumption that the correlation energy between electrons is governed by the short-range behavior of the radial distribution function we are able to obtain very accurate theoretical estimates for the correlation energy per particle for FQHE states studied in the thermodynamic limit at arbitrary lling factors described by the unprojected CF wave function. Taking as best values for the correlation energy for particle for the Laughlin’s states those reported by Levesque et al. [9] we provide tables of theoretical estimates of the correlation energy per particle for all these states described by the unprojected CF wave function. We Acknowledgements Most of this work was carried out at the Ames Laboratory, which is operated for the US Department of Energy by Iowa State University under Contract No. W-7405-Eng-82 and was supported by the Director for Energy Research, Oce of Basic Energy Sciences of the US Department of Energy. References [1] R.E. Prange, S.M. Girvin (Eds.), The Quantum Hall E ect, Springer, New York, 1990. [2] T. Chakraborty, P. Pietilainen (Eds.), The Fractional Quantum Hall E ect, Springer, New York, 1988. [3] D.C. Tsui, H.L. Stormer, A.C. Gossard, Phys. Rev. Lett. 48 (1982) 1559. [4] R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. [5] J.K. Jain, Phys. Rev. Lett. 63 (1989) 199. [6] J.K. Jain, Phys. Rev. B. 41 (1990) 7653. [7] B.I. Halperin, P.A. Lee, N. Read, Phys. Rev. B 47 (1993) 7312. [8] E. Rezayi, N. Read, Phys. Rev. Lett. 72 (1994) 900. [9] D. Levesque, J.J. Weis, A.H. MacDonald, Phys. Rev. B 30 (1984) 1056. [10] R.K. Kamilla, J.K. Jain, Phys. Rev. B 55 (1997) 9824.