Physica E 9 (2001) 701-708
The Fermion-Boson transformation in fractional quantum Hall systems
John J. Q~inn~.~,*, Arkadiusz WOJS~~~,Jennifer J. Quinnd, Arthur T. Benjamine
'Department of Pl~y~ics,200 South College. University of Tennessee, Knoxuille, TN 37996-1501, USA hOak Ridge National Laboratorjl, Oak Ridge, TN 37831. USA Wroclaw University of Technology, Wroclatv 50-370, Poland Occidental College, Los Angeles, CA 90041, USA 'Haroej~Mudd College, Claremont, CA 91711, USA Received 15 August 2000; accepted 30 September 2000
Abstract A Fermion-to-Boson transformation is accomplished by attaching to each Fermion a single flux quantum oriented opposite to the applied magnetic field. When the mean field approximation is made in the Haldane spherical geometry, the Fermion angular momentum IF is replaced by le = IF - f (N - 1). The set of allowed total angular momentum multiplets is identical in the two different pictures. The Fermion and Boson energy spectra in the presence of many-body interactions are identical if and only if the pseudopotential is "harmonic" in form. However, similar low-energy bands of states with Laughlin correlations occur in the two spectra if the interaction has short range. The transformation is used to clarify the relation between Boson and Fermion descriptions of the hierarchy of condensed fractional quantum Hall states. O 2001 Elsevier Science B.V. All rights reserved. PACS: 71.10.Pm; 73.20.D~;73.40.Hm Kej~ic~ords:Boson-Fermion mapping; Low-dimensional structure; Fractional quantum Hall effect
I 1. Introduction quantum Hall effect (FQHE) [2]. Shortly after the in- C troduction of the CF picture, Xie et al. [3] introduced The transformation of electrons into composite a Fermion-+Boson (F+B) mapping connecting a 2D Fermions (CF) by attaching to each particle a flux Fermion system at filling v~ with a 2D Boson sys- tube carrying an even number of flux quanta has tem at filling vg, where vF' = v;' + 1. These authors led to a simple intuitive picture [I] of the fractional stated that the sizes of the many-body Hilbert spaces for the Boson and Fermion systems were identical, and that their numerical calculations verified that ' Corresponding author. Tel.: +1-865-974-4089; fax: +1-865- 974-6378. the mapping accurately transformed the ground state E-mail address: [email protected] (J.J. Quinn). of the Fermion system into the ground state of the
1386-9477/01/$-see front matter @ 2001 Elsevier Science B.V. All rights reserved. PII: S 1386-9477(00)00293-9 702 J.J. Quinn et al. l Pltysica E 9 (2001) 701-708
Boson system if and only if these ground states were F+B transformation (with p = O), IFis replaced by incompressible FQH states. In this paper we show 1~ = 11~- $(N - 111. that the F+B transformation leads to identical en- ergy spectra if and only if the pseudopotential V(L12) describing the interactions among the particles is of 3. Some useful theorems the "harmonic" form VH(L12)=A + BL12(L12+ I), where A and B are constants, and L12 is the to- When a shell of angular momentum 1 contains N tal angular momentum of the interacting pair [4]. identical particles (Fermions or Bosons), the result- Laughlin correlations [5] occur when the actual pseu- ing N particle states can be classified by eigenvectors dopotential V(LI2)rises more quickly with increas- IL, M, a), where L is the total angular momentum, M ing L12 than VH(LIZ).Anharmonic effects (due to its z-component, and a a label which distinguishes in- AV(L12)= V(L12)- VH(L12))cause the interacting dependent multiplets with the same total angular mo- Fermion and interacting Boson spectra to differ for mentum L. In the mean field CF (CB) transformation every value of the filling factor VF = vB(l + VB)-'. lF (lB)is transformed to 1; (I:). In trying to under- However, for appropriately chosen (short-range) stand why the mean field CF picture correctly pre- model pseudopotentials, the F-tB mapping accurately dicted the low-lying band of states in the interacting transforms the ground state of the Fermion system electron spectrum, the following theorem was impor- to that of the Boson system both for incompressible tant [9,10]. FQH states and for other low-lying states. The F-tB mapping is also very useful in understanding the re- Theorem 1. The set of allowed total angular mo- lation between the Haldane [6] hierarchy of Boson mentum multiplets ofN Fermions each with angular quasiparticle (QP) condensates and the CF hierarchy momentum 1; is a subset of the set of allowed [7,8] of Fermion QP condensed states. multiplets of N Fermions each with angular momen- tum lF = 1; + (N - 1).
2. Gauge transformations in two-dimensional Thus, if we define givr(L)as the number of indepen- systems dent multiplets of total angular momentum L formed by addition of the angular momenta of N Fermions, By attaching to each Fermion or Boson of charge each with angular momentum I, then ~NI*(L) < g,v/(L) -e, a fictitious flux tube carrying an even number 2p for every value of L. A few examples for small sys- of flux quanta oriented opposite to the applied field, tems suggest that the theorem is correct, but a gen- the eigenstates and particle statistics are unchanged. eral mathematical proof is non-trivial. A proof using The "gauge field" interactions between the charge on the methods of combinatorics and the KOH [11,12] one particle and the vector potential due to the flux theorem has been given recently [13]. The same quanta on every other particle make the Hamiltonian method allows the proof of a second theorem. more complicated. Only when the mean field approxi- mation is made does the problem simplify. In addition Theorem 2. The set of allowed totalangular momen- to these (CF and CB) transformations, a F+B trans- tum multiplets of N Bosons each with angtilar mo- formation can be made by attaching to each Fermion mentum lB is identical to the set of multiplets for an odd number 2p + 1 of flux quanta (one flux quan- N Fermions each with angular momentum IF = lB + tum changes the statistics; other 2p flux quanta de- $(N - 1) [13] (see Footnote 1). scribe an additional CF or CB transformation). If the particles are confined to the surface of a sphere con- From Theorem 2, it follows immediately that Theo- taining at its center a magnetic monopole of strength rem 1 also applies to Bosons. Theorem 2 is a stronger 2SF (for Fermions) or ~SB(for Bosons) flux quanta, then the lowest shell of mean field composite particles has angular momentum 1; = 1 lF- p(N - 1)I where ' The second theorem has been arrived at independently by IF =SF0r 1; = IIB - p(N - 1)1 where IB = SB.Inthe B. Wyboume, private communication. J.J. Quinn et al. l Physica E 9 (2001) 701-708 703
statement than a simple equality of the sizes of the V(9), VH(~),and AV(W), respectively. It is more many body Hilbert spaces [3]. reasonable to make simple models for AV(9) (e.g. I assume that it vanishes for all .% greater than some value) than for V(B) itself. From Eq. (2) and the 4. Interaction effects equation for the total energy,
In studying why the mean field CF picture correctly 1 EL, = N(N - 1) %LZ(~)V(a), (3 predicts the low-lying states of a 2D electron system in .4 a magnetic field [4,9], the "harmonic pseudopotential" where %L,(B) is the coefficient of fractional grand- parentage (CFGP) [4,9,14], it is readily ascertained that the interacting Boson and interacting Fermion sys- was introduced. Here A and B are constants and Ll2 tems cannot have identical spectra when AV(W) is is the total angular momentum operator of the pair of non-zero. particles. It was shown that for the harmonic pseu- Xie et al. [3] determined the Boson and Fermion dopotential the energy of any multiplet of angular mo- eigenhnctions by exact numerical diagonalization for mentum L was given by six particle systems connected through the F-+B trans- formation. They then transformed the Boson eigen- functions into Fermion eigenhnctions by multiplying them by ni
i6.51- ,0': C\! a, ...... W . .
I -8 0 I (a) Coulomb (a')Coulomb
Fig. I, The eilel.gy spectra (energy E 1-ersus ailgulal. momcntum L ) of the correspond~ngeight Fermion (left ) and eight Bt7ion (right) systems at the monopole strengths 2SF = 21 and 2Sn = I? (fillins fac(ors 1.1 = i and i.n = 4) for the Coulotnb pseudopotcnt~nlIn the
lowest Landau lehel ((I -- '1'). and for the nod el psrudopotentials HI (h- 11'). and H: (c - c '). i is the magnetic length.
tials. We have used the full Coulomb pseudopotential. same for thc state containing two Laughlin quasielec-
HI.Hi, H5,and a model pseudopotcntial L( in which trons (QE). The lowest states in (cr - tr') and (h - h') Y(l)= 1: Y(.X35)=0, and K(~)=.Y.LJH(3) is an are quite similar consisting of a Laughlin L = 0 ground arbitrary fraction r of the "harmonic" value. We per- state in Fig. I and two-QE states with Igr = ;(N - form thc same calculations for eight Boson systcnls I)=? givingL=N-2, S -4. ...= 0,2,4; and6 at 2SF3- 12-16 (here, 130) = 1, 1;(6 34)= 0, and in Fig. 2. The magnetoroton band (at 2 Fermion and Boson systems at = (yn = 4) for the dopotential used in (r - c') gives very different results Coulomb pseudopotential appropriate for the lowest both in Figs. 1 and 2. As mentioned before, this re- Landau level (u-a'). and for the tnodel pseudopoten- sults because L'(3 ) used in Fennion pseudopotential tials HI (0 - h') and H; (c - c'). In Fig. 2, we do the H: is too large to lead to Laughlin coi-relations. J.J. Quinn et al. IPhysica E 9 (2001) 701-708 8 Fermions, 2S=19 8 Bosons, 2S=12 *-___..__ _,_I__.._ -. -.. ...-.--+.-----'-.. (a) Coulomb '.- (a') 9.2 6.8 - Coulomb - Fig. 2. The energy spectra (energy E versus angular momentum L) of the corresponding eight Fermion (left) and eight Boson (right) systems at the monopole strengths 2SF = 19 and 2S0 = 12 (two Laughlin quasielectrons in the v~ = and ve = state) for the Coulomb pseudopotential in the lowest Landau level (a - a'), and the model pseudopotentials HI (b - b'), and H3 (c - c'). ,i is the magnetic length. To illustrate this point, we have calculated the en- From the eigenfunctions, we can determine CFGP's ergy spectra using pseudopotential g with different gLu(W)for each state IL,ct). In Fig. 4, we plot the values of x. In Fig. 3, we show the spectra at VF = x-dependence of the CFGP's YL~(W)from pair states j.I (VB= $) for x = $,I, and l. For x < 1, g(W) is at three smallest values of W calculated for the low- superharmonic at W = 3 (for Fermions; x = 43)) or est energy L = 0 state of eight Fermions at 2SF= 21 at W = 2 (for Bosons; x r x(2)), and Laughlin cor- (vF = i)and eight Bosons at 2SB = 14 (vB = i).In relations with an L = 0 ground state occur. For x 2 1 both systems, a Laughlin incompressible state with there is little resemblance between the numerical spec- vanishing 9(1) (for Fermions) or 9(O) (for Bosons) tra and that associated with the full Coulomb interac- occurs at small x, and a rather abrupt transition occurs tion. Furthermore, the Fermion and Boson spectra are at x= 1, implying a change of the nature of the corre- quite different from one another. lations when the pseudopotential K(W) changes from J.J. Quinn et a/. l Physica E 9 12001) 701-708 8 Fermions, 2S=21 8 Bosons, 2S=14 1. (a) x=0.5 1 (a') x=0.5 .* (b) ~=1.0/I- (b') ~=1.014.0 Fig. 3. The energy spectra (energy E versus angular momentum L) of the corresponding eight Fermion (left) and eight Boson (right) systems at the monopole strengths 2SF = 21 and 2Ss = 14 (filling factors v~ = f and ve = $) for model interaction pseudopotentials K(d) with x = f (a - a'), x = 1 (b- b'), and x = 2 (c - c'). super- to subharmonic. At x> 1, the correlations in the condensed states having VF = (2p+ 1 )- ' (where p two systems are quite different and, for example, an- is a positive integer) occur at 2SF = (2p+ 1 )(N - 1) other abrupt transition occurs in the Boson system at in the Haldane spherical geometry. The CF transfor- xx4 (not shown in the figure), which is absent in the mation [l] gives an effective angular momentum 1; = Fermion system. S,* = S - p(N - 1) = :(N - 1) when 2pflux quanta are attached to each electron and oriented opposite to the applied magnetic field. Thus the N CFs fill the 5. Quasiparticles 21" + 1 states of the lowest CF shell giving an L = 0 incompressible ground state. The F 4 B transformation allows us to better un- The F B transformation gives 2s~= 2SF - derstand the Boson [6] versus Fermion [7,8] descrip- (N - 1) = 2p(N - 1) and a Boson filling factor of tion of QPs in incompressible FQH states. Laughlin vg = (2P)-'. Making a CB transformation gives J. J. Quinn et a!. l Physica E 9 (2001) 701-708 8 Fermions, 2S=21 8 Bosons, 2S=14 Fig. 4. The coefficients of fractional grandparentage 9(1)from the pair states at three smallest values of J? calculated for the lowest energy L = 0 state of the corresponding eight Fermion (a) and eight Boson (a') systems at the monopole strengths 2SF = 21 (vF= 4) and 2SB = 14 (ve = i)for the model interaction I(, as a function of x. I: = S: = SB- p(N - 1) = 0. This also gives an lier work [7,8], we have emphasized that both the L = 0 incompressible ground state because each CB Haldane hierarchy and CF hierarchy schemes assume has 1: = 0. Thus the CF description of a Laughlin the validity of the mean field approximation, and we state has one filled CF shell of angular momentum have shown that this approximation is expected to fail 1; = ;(N - l), while the CB description has N CBs when the QP-QP interaction is subharmonic. Numer- each with angular momentum 1; = 0. ical results show when the mean field approximation For 2SB = 2n(N - 1) h n~p,where the + and - is valid and when it fails. occur for quasiholes (QH) and QE, respectively, we define 21; = 12S:I = nqp. This gives exactly the same set of angular momentum multiplets as obtained in the 6. Summary CF picture with 2SF = (2n + 1 )(N - 1) + nq~.How- ever, it gives a larger set of multiplets than are allowed We have shown that the F + B transformation re- by 2s~= (2n + 1)(N - 1 ) - ~QE.For example, for places the single Fermion angular momentum IF by HQE = 2,1; = 1 and the allowed values of the pair an- lB = IF - ;(N - 1), and that this transformation leads gular momentum of the two QPs are N, N - 2, N - to an identical set of total angular momentum multi- 4,. . . . For a Fermion system with I; = ;(N - 1 ) - 2, plets. The Fermion and Boson systems have identical the allowed values of the QP pair angular momentum spectra in the presence of many-body interactions are N - 2, N - 4,. . . . The two sets can be made iden- only when the pseudopotential is harmonic, i.e. lin- tical only if a hard core repulsion forbids the Boson ear in squared pair angular momentum, LI2(Ll2+ 1). QP pair from having the largest allowed pair angu- However, similar low-energy bands of states with i lar momentum LflAX = N [15]. This behavior is ob- Laughlin correlations occur in the two spectra if served in Fig. 2, where the Boson treatment of two the interaction pseudopotential is superharmonic, QEs (i.e., the CB transformation) would predict states i.e. has short range. We have studied numerically at L = 0,2,4,6, and 8, but the L = 8 state does not eight particle systems for different model interac- occur in the low-energy band. tions and shown the relation between the spec- Since the description of CBs (with hard core QE tra and coefficients of fractional grandparentage interaction) and CFs give identical sets of QP states, for the Fermion and Boson systems. Finally, we filled QP levels (implying daughter states) occur at have used the F + B transformation to clarify identical values of the applied magnetic field. In ear- the relation between the Haldane Boson picture 708 J. J. Quinn et al. 1 Physica E 9 (2001) 701-708 and the CF picture of the hierarchy of condensed [3] X.C. Xie, S. He, S. Das Sarma, Phys. Rev. Len. 66 (1991 ) states. 389. [4] A. Wojs, J.J. Quinn, Solid State Commun. 110 (1999) 45. [5] R. Laughlin, Phys. Rev. Len. 50 (1983) 1395. [6] F.D.M. Haldane, Phys. Rev. Lett. 51 (1983) 605. Acknowledgements [7] P. Sitko, K.-S. Yi, J.J. Quinn, Phys. Rev. B 56 (1997) 12417. [8] A. Wojs, J.J. Quinn, Phys. Rev. B 61 (2000) 2846. Research supported by the US Department of En- [9] J.J. Quinn, A. Wojs, Physica E 6 (2000) 1. ergy under contract DE-AC05-000R22725 with Oak [lo] J.J. Quinn, A. Wojs, J. Phys.: Condens. Mater. 12 (2000) Ridge National Laboratory managed by UT-Batelle, R265. [I 11 K. O'Hara, J. Combin. Theory Ser. A 53 (1990) 29. LLC. John J. Quinn acknowledges partial support from [12] D. Zeilberger, Amer. Math. Monthly 96 (1989) 590. the Materials Research Program of Basic Energy Sci- [13] J.J. Quinn, A. Benjamin, J.J. Quinn, A. Wi?js, to be published. ences, US Department of Energy. [14] A. Wojs, J.J. Quinn, Philos. Mag. B 80 (2000) 1405. [I51 S. He, X.-C. Xie, F.-C. Zhang, Phys. Rev. Len. 68 (1992) 3460. References [l] J. Jain, Phys. Rev. Lett. 63 (1989) 199. [2] R.E. Prange, S.M. Girvin (Eds.), The Quantum Hall Effect, Springer, New York, 1987.