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Coulomb Energy of Quasiparticle Excitations in Chern±Simons Composite Fermion States

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Coulomb Energy of Quasiparticle Excitations in Chern±Simons Composite Fermion States PERGAMON Solid State Communications 122 (2002) 401±406 www.elsevier.com/locate/ssc Coulomb energy of quasiparticle excitations in Chern±Simons composite fermion 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 quantum Hall effect (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 nn=2pl0 B ; where l0 B 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 0; 0; B: Normally, the symmetric highly degenerate partially ®lled Landau level. gauge is adopted and the vector potential is A r 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 limit N ! 1 and V ! 1; while N=V r n; where N is p= 2mp 1 1; where p 1; 2; ¼ and m 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^ K^ 1 V^; where K^ is the kinetic energy operator. 1 XN K^ 2ih7 1 eA r 2; 1 2m j j * Corresponding author. Tel.: 11-573-884-2278; fax: 11-573- e j1 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 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^ v ur 2 r u 2 r n d2rv ur 2 ru j k j mean value kbl 2mf r ne ; where r n is the uniform j,k j1 0 z electronic density. This reduces the magnetic ®eld to Bp 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 r nf0=B p with p an integer, the corresponding electron ®lling factor is n r nf0=B 2 where v urj 2 rku e =4pe0euzj 2 zku is the Coulomb p= 2mp 1 1; which is precisely the Jain's series of FQHE potential, zj 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 fermions subject to a reduced It has become clear in recent years that many features of magnetic ®eld Bp 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 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 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 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 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 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 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 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 EC0; with Hamiltonian H^ 0 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 { 2 ih7 1 eA r 2 a r }2; 4 view and has not yet been realized. Since the nature of 2m j j j e j1 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 r7 £ a r the low-energy excitations and excited state properties on 2mf0r rez is the ®ctitious CS magnetic ®eld, r r is the both cases seems to be quite appealing on its own.
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