Within the newly formulated composite fermion hierarchy the filling fraction
of a spherical quantum Hall system is obtained when it can be expressed as an
odd or even denominator fraction. A plot of $\nu\frac{2S}{N-1}$ as a function
of $2S$ for a constant number of particles (up to N=10001) exhibits structure
of the fractional quantum Hall effect. It is confirmed that $\nu_e +\nu_h=1$
for all particle-hole conjugate systems, except systems with $N_e =N_h$, and
$N_e=N_h \pm 1$.Comment: 3 pages, Revtex, 7 PostScript figures, submitted to Phys. Rev. B
  Rapid Communicatio

D. C. Marinescu

E. Rezayi

F. D. M. Haldane

J. J. Quinn

J. K. Jain

P. Sitko

R. B. Laughlin

English

arXiv

Crossref

Quantum Hall spherical systems: The filling fraction

Within the recently formulated composite fermion hierarchy the ﬁlling fraction of a spherical quantum Hall system is obtained when it can be expressed as an odd or even denominator fraction. A plot of ␯(2S/NϪ1) as a function of 2S for a constant number of particles ͑up to Nϭ10 001͒ exhibits a structure of the fractional quantum Hall effect. It is conﬁrmed that ␯eϩ␯hϭ1 for all particle-hole conjugate systems, except systems with NeϭNh and NeϭNhϮ1. ͓S0163-1829͑97͒03132-9

Sitko, P

Quinn, J J

Marinescu, D C

Clemson University: TigerPrints

Clemson UniversityTigerPrintsPublications Physics and Astronomy8-15-1997Quantum Hall Spherical Systems: The FillingFractionP SitkoUniversity of TennesseeJ J. QuinnUniversity of TennesseeD C. MarinescuClemson University, dcm@clemson.eduFollow this and additional works at: https://tigerprints.clemson.edu/physastro_pubsThis Article is brought to you for free and open access by the Physics and Astronomy at TigerPrints. It has been accepted for inclusion in Publicationsby an authorized administrator of TigerPrints. For more information, please contact kokeefe@clemson.edu.Recommended CitationPlease use publisher's recommended citation.Quantum Hall spherical systems: The filling fractionP. SitkoUniversity of Tennessee, Knoxville, Tennessee 37996and Institute of Physics, Technical University of Wrocław, Wybrzez˙e Wyspian´skiego 27, 50-370 Wrocław, PolandJ. J. QuinnUniversity of Tennessee, Knoxville, Tennessee 37996and Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831D. C. MarinescuOak Ridge National Laboratory, Oak Ridge, Tennessee 37831~Received 25 February 1997!Within the recently formulated composite fermion hierarchy the filling fraction of a spherical quantum Hallsystem is obtained when it can be expressed as an odd or even denominator fraction. A plot of n(2S/N21) asa function of 2S for a constant number of particles ~up to N510 001! exhibits a structure of the fractionalquantum Hall effect. It is confirmed that ne1nh51 for all particle-hole conjugate systems, except systemswith Ne5Nh and Ne5Nh61. @S0163-1829~97!03132-9#During the past 15 years, systems of the quantum Halleffect were intensively studied,1 both for experimentally re-lated situations ~Laughlin droplet2! and for purely theoreticalspherical systems.3 The main problem of any quantum Hallrelated studies is the definition of the filling fraction. Forinfinite systems the filling fraction is defined asn215(B/r)@(h/e)#21, i.e., the number of the flux quantaper electron. For finite systems, however, this definition canno longer be used. The only way of assigning the nonintegerfilling fraction to the system with a given number of particlesand strength of the magnetic field is either intuitive2,3 orbased on the use of the composite fermion ~CF!transformation.4 We extend the formulation of the Jain fillingfractions to all odd denominator hierarchy fractions. Wemake use of recently formulated5 composite fermion hierar-chy. The further extension is made by generalizing the com-posite fermion hierarchy to all even denominator fractions byintroducing the idea of ‘‘half-filled’’6 states of quasiparticles.In such a way we can obtain the filling fraction for almost allpairs of N ~number of particles! and 2S ~the strength of themagnetic monopole for the sphere!.We define the filling fraction in the following way. First,perform the CF transformation ~if n,1! changing the valueof 2p ~the strength of the Chern-Simons field! to get at leastone filled effective Landau shell. Hence,1n52p1an1nQE, ~1!where a is the sign of the effective field ~with respect to thereal magnetic field!, n is the number of filled effective Lan-dau shells, and nQE is the filling fraction for quasielectronspartially occupying the ‘‘n11’’ effective shell. The processcan be repeated on nQE until at the mth step nQEm 50 ~thehierarchy odd denominator fraction! or aQEm 50 which corre-sponds to the ‘‘half-filled’’ @nQEm 5(1/2pQEm )# quasielectroncase ~even denominator fractions!. Hence, we get all frac-tions, except the cases when one quasielectron is left ~nofraction can be assigned to one particle system!.In Fig. 1 we plot the values ofn2SN21 ~2!for eight electrons for values of the filling starting at n51and going to n51/5. The function ~2! has the value of unitywhen n5@(N21)/2S# , which occurs for the Laughlin states~1,1/3,1/5! and for the ‘‘half-filled’’ ~1/2,1/4! states. Forother fractions the function ~2! varies from unity, but anabrupt change is clearly visible at n close to 1/2 and 1/4.Such ‘‘discontinuity’’ can be explained already with intro-duction of Jain states. The integer filling ~for real 2S oreffective 2S* field! is obtained whenN5n21n~2S !, ~3!FIG. 1. The values of n@2S/(N21)# for N58 within a range of1>n>1/5.PHYSICAL REVIEW B 15 AUGUST 1997-IVOLUME 56, NUMBER 7560163-1829/97/56~7!/3606~3!/$10.00 3606 © 1997 The American Physical Societyand N5n21n(2S*) for Jain states. Since 2S*52S22p(N21), we get2SN21 52pn11n2n221n~N21 ! , for 2S*.0, ~4!and2SN21 52pn21n1n221n~N21 ! , for 2S*,0. ~5!Hence, approaching the 1/2p states from both sides we findthe corrections to the filling fractions to be of opposite sign,with a maximum correction for maximum n .We plot similar results for N5101 ~Fig. 2!. The discon-tinuity at 1/2p states is not only confirmed by the Jain statesbut also by other hierarchy states. Additionally, similar dis-continuities can be seen at each 1/2pQE state of quasielec-trons ~even denominator fractions!. In the range of300,2S,500 the curve clearly repeats the results for100,2S,300, because in the formulation of the fraction wechange only the value of 2p . We confirm this by plotting theresults for N510001 within the range 10 000,2S,30 000~Fig. 3!, and within the range 30 000,2S,50 000 ~Fig. 4!with an adjusted scale for the function ~2!. Such a repeatingstructure can be also seen when looking at quasielectron fill-ing, i.e., n54/5 (nQE51/3) down to n52/3 (nQE51). Weplot the results for N510 001 and the range12 500,2S,15 000 in Fig. 5. The curves are not exactlythe same, however, due to the fact that the number ofquasielectrons changes with 2S , in contrast to all other fig-ures where N5const.In order to see the structure for n.1 we plot the resultsfor N5101 for 0,2S,500 in Fig. 6. A better view is ob-tained when N510 001 and 2000,2S,10 000 in Fig. 7.The discontinuities again come from analogous ‘‘half-filled’’states at higher Landau levels. The integer fillings are seen aslittle jumps at the curve. Such jumps are also seen in Fig. 2for Laughlin and in Figs. 3, 4 for Jain states. In fact, all odddenominator hierarchy states lead to such jumps ~each ofthem can be seen if the number of particles is large enough!and all even denominator fractions give discontinuities whenresolution of the curves ~the scale and the number of par-ticles! increases. Thus, for an infinite number of particles thecurve exhibits fractal structure.In order to confirm our method of defining the filling frac-tion we calculate the sum of ne1nh for particle-hole conju-gate states (ne,1). The sum is always one, as expected,except for the three cases when Ne5Nh , and Ne5Nh61,FIG. 2. The same as Fig. 1 for N5101.FIG. 3. The values of n@2S/(N21)# for N510 001 within arange of 1>n>1/3.FIG. 4. The same as Fig. 3 for 1/3>n>1/5.FIG. 5. The spectrum of the filling fraction multiplied by@2S/(N21)# for N510 001 and 12 500>2S>15 000. This rep-resents the systems of quasielectrons within a range of 1>nQE>1/3~note that nQE increases with 2S!.56 3607BRIEF REPORTSwhich correspond to difficulty in determing the filling frac-tion of 1/2. The filling ne51/2 is obtained when Ne5Nh11,hence, the filling fraction nh for Nh is necessarily less than1/2. A similar problem is for Ne5Nh when the CF hierarchyfraction is less than 1/2. It is worth noting, however, that noproblems occur at other fillings 1/2p (1/4,1/6, . . .) whichrepresent exactly the same problem interms of composite fermions.This work was supported in part by Oak Ridge NationalLaboratory, managed by Lockheed Martin Energy ResearchCorp. for the U.S. Department of Energy under Contract No.DE-AC05-96OR22464. P.S. acknowledges support by theCommittee for Scientific Research, Poland, Grant No. PB674/P03/96/10.1 The Quantum Hall Effect, edited by R. E. Prange and S. M.Girvin ~Springer-Verlag, New York, 1987!.2 R. B. Laughlin, Phys. Rev. Lett. 50, 1395 ~1983!.3 F. D. M. Haldane, Phys. Rev. Lett. 51, 605 ~1983!.4 J. K. Jain, Phys. Rev. Lett. 63, 199 ~1989!.5 P. Sitko, K. S. Yi, and J. J. Quinn ~unpublished!.6 E. Rezayi and N. Read, Phys. Rev. Lett. 72, 900 ~1994!.FIG. 6. The whole spectrum 0,2S<500 for N5101. FIG. 7. The range of integer filling for N510 001,2000<2S<10 000.3608 56BRIEF REPORTS

Quantum Hall Spherical Systems: The Filling Fraction

https://tigerprints.clemson.edu/cgi/viewcontent.cgi?article=1175&amp;context=physastro_pubs

Quantum Hall Spherical Systems: the Filling Fraction

Quantum Hall Spherical Systems: the Filling Fraction

Abstract

Similar works

Full text

Available Versions

Crossref

Clemson University: TigerPrints