A Fermion to Boson transformation is accomplished by attaching to each
Fermion a tube carrying a single quantum of flux oriented opposite to the
applied magnetic field. When the mean field approximation is made in Haldane's
spherical geometry, the Fermion angular momentum l_F is replaced by
l_B=l_F-(N-1)/2. 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 only if the
pseudopotential V (interaction energy as a function of pair angular momentum
L_12) increases as L_12(L_12+1). Similar bands of low energy states occur in
the two spectra if V increases more quickly than this.Comment: 4 pages, 1 figure, poster at ARW in Queenstown, New Zealand (2001

Arkadiusz Wójs

Arthur T. Benjamin

Chen

Haldane

Jain

Jennifer J. Quinn

John J. Quinn

Laughlin

Sitko

Wilczek

Wójs

English

arXiv

Crossref

A Fermion to Boson transformation is accomplished by attaching to each Fermion a tube carrying a single quantum of flux oriented opposite to the applied magnetic field. When the mean field approximation is made in Haldane’s spherical geometry, the Fermion angular momentum lF is replaced by lB =lF − 1/2 (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 only if the pseudopotential V (interaction energy as a function of pair angular momentum L12) increases as L12(L12 + 1). Similar bands of low energy states occur in the two spectra if V increases more quickly than this

Quinn, John J

Wojs, Arkadiusz

Quinn, Jennifer J

Benjamin, Arthur T.

Scholarship@Claremont

Claremont CollegesScholarship @ ClaremontAll HMC Faculty Publications and Research HMC Faculty Scholarship10-1-2001Transformation of Statistics in Fractional QuantumHall SystemsJohn J. QuinnUniversity of Tennessee - KnoxvilleArkadiusz WojsWroclaw Technical UniversityJennifer J. QuinnOccidental CollegeArthur T. BenjaminHarvey Mudd CollegeThis Article is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusionin All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contactscholarship@cuc.claremont.edu.Recommended CitationQuinn, J.J., Wojs, A., Quinn, J.J., & Benjamin, A.T. (2001). Transformation of statistics in fractional quantum Hall systems. Physica E,11(2-3): 182-185. DOI: 10.1016/S1386-9477(01)00199-0Physica E 1 1  (2001) 182-1 85 Transformation of statistics in fractional quantum Hall systems John J. Q u i n n a ,  *, Arkadiusz W O J S " ~ ~ ,  Jennifer J. Quinnc, Arthur T. Benjamind 'Department ofphysics, University of Tennessee. 200 South College, Knoxville, TN 37996-1501. USA Wroclaw University of Technology, Wroclaw 50-370, Poland COccidental College, Los Angeles, CA 90041, USA ~ a r v e y  Mudd College, Claremon f, CA 91 711, USA Abstract A Fermion to Boson transformation is accomplished by attaching to each Fermion a tube carrying a single quantum of flux oriented opposite to the applied magnetic field. When the mean field approximation is made in Haldane's spherical geometry, the Fermion angular momentum IF is replaced by IB = IF - $(N - 1). The set of allowed total angular momentum multiplets is identical in the two different pictures. The Fermion and Boson energy spectra in the presence of many body interactions are identical only if the pseudopotential V (interaction energy as a function of pair angular momentum Ll2) increases as L I 2 ( L l 2  + 1). Similar bands of low energy states occur in the two spectra if V increases more quickly than this. @ 2001 Published by Elsevier Science B.V. PACS: 71.10.Pm; 73.20.D~; 73.40.Hm Keywords: Fermion-Boson mapping; Fractional quantum Hall effect; Composite Fermion 1. Introduction In two dimensional systems particle statistics can : be changed by making a Chem-Simons (CS) transfor- . mation (see for example [I] and references therein). This transformation can be described as attaching to each particle an infinitesimal flux tube carrying a fic- titious flux 4 and a fictitious charge q, which couples to the vector potential produced by the flux tubes on every other particle in the standard way. If q is equal to the electron charge and 4 is an even number times 4 0  = hcle, the quantum of flux, no change in statistics occurs. If 4 is and odd number times 4 0 ,  Fermions are transformed into Bosons. The "gauge field" inter- actions associated with the fictitious charge and vec- tor potential produced by the fictitious flux make the Hamiltonian of the system considerably more com- plicated. Only when the mean field approximation is made does the problem simplify. Jain [2] introduced 'Corresponding author. Tel.: +1-865-974-4089; fax: the mean field CS transformation to give a simple in- +I-865-974-6378. tuitive picture of the hierarchy of fractional quantum E-mail address: jjquinn@utk.edu (J.J. Quinn). Hall (FQH) states in terms of the resulting compos- 1386-9477/01/$-see front matter @ 2001 Published by Elsevier Science B.V. PII: S 1386-9477(01)00199-0 J. J Q~iinn et ui. I Plij,sicu E 11 (2001 ) 182-18.5 183 ite Fermions (CF). Shortly after the introduction of the CF picture, Xie et al. [3] introduced the Fermion- Boson mapping connecting a 2D Fermion system at filling factor 1 . b  to a 2D Boson system with filling fac- tor \ Q  = 1 3 ~ (  1 - I'F ) I .  These authors noted that the size of the Hilbert space for the Fermion and Boson systems was identical. and they found that the F+B mapping accurately transformed the ground state of the Fennion system into that of the Boson system only if these ground states were incompressible FQH states. In this paper we show that the F-B transformation leads to identical energy spectra if and only if the pseu- dopotential I'(LI1) describing the interaction energy as a function of the pair angular momentum Li. is of the "harmonic" form VH(L12) = A  + BLI2(Ll7  + 1 ) where A and B are constants. Laughlin [4] correlations occur when the actual pseudopotential V(L12) rises more quickly with increasing L I 2  than ITH(Ll2). Anhar- monic effects (due to AV(LII)  = V(LI?)  - VH(LI? ) )  cause the interacting Fermion and interacting Boson spectra to differ for every value of the filling factor. It is wcll known that in the Haldane spherical ge- ometry [5] the mean field CF transformation changes an electron of angular momentum I t  to a CF of angu- lar moincntum I ;  = I F  - y(2\7 - 1 )  [6]. Here I F  = S t ,  onc-half the strength (measured in units of 40) of the magnetic monopole u-hich produces thc radial magnetic field .8 = 2 ~ ~ $ ( , ( 4 n ~ ' ) ~ '  at the spherical surface of radius R on which the N electrons are con- fined, and p is an intcgcr. For an N Boson system, the composite Boson transformation replaces I n  by 1; - p(.V - 111. In the FIB mapping I t  is replaced by I n  = I t  - ; ( N  - 1 ) .  2. Angular momentum addition: useful theorems When a shell of angular momentum I contains N identical particles (Fermions or Bosons), the result- ing .V particle states can be classified by eigenvectors L .  .V. 2 ) .  where L is the total angular momentum, M its :-component, and x a label which distinguishes in- dependent multiplets with the same L. In the mean field CF (CB) transformation I F  (In) is transformed to 1; ( I ; ) .  In trying to understand why the mean field CF picture correctly predicted the low lying band of states in the electron spcctrum. scvcral simple conjec- tures were proposed on the basis of numerical stud- ies of a finite number of particles [7-101. These con- jectures have been elevated to the status of theorems by rigorous mathematical proof [ l l ]  using partition theory. Theorem 1. Tlze set qf' u l l o ~ ~ ~ e d  rortrl rrnglrlur mo- mentun1 ~ii~lltiplets qf'N Fer.ri~ions each 11,ith angtllar 17io1iiel~t~1111 1; is a sub.ser oj'rhe .set qf al1011,ed rntll- tiplets o f N  Fermions etrch 1 1 3 h  unglrlu~. rrionlent~un I F  = 1; + ('v - 1 ). Theorem 2. The ser c!f ollolt~cri totlrl urlj/~llur. 1110rr1en- t l ln~ n~~ll t iplets  of N Bosons each rt.itli N I I C / L / ~ ( I I .  1110- ri7cr7t1rrii I D  i.r iderztical to the set of rn~lltiplets j i ~ r  IV Fc1r.rt7ions each 11,ith u n g ~ ~ l u r  ~1on1ent~1nz It = I H  + 4(Av - 1 ). From Theorem 2 it follows immediately that Theo- rem I also applies to Bosons. Theorem 2 is a stronger statcmcnt than a simplc cquality of sizes of the many body Hilbert spaces [3]. 3. Interaction effects It has been shown that for the harmonic pseudopo- tential VH(LIZ) the energy of any multiplet of angular momentumL is given by E,(L)=A x IN(N-  ~ ) + B x  { N ( N 2 ) 1 ( 1 +  1 )+L(L+ 1 )). The energy is indepen- dent of x, so that every multiplet with the same value of L has the same energy. If BF = BB = B. then the spec- trum of N Bosons each with angular momcnturn I B  is identical (up to a constant) to that o f N  Fermions each with angular momentum IF = I B  + + ( N  - 1 ). This is a rather surprising result because Fermions and Bosons sample different sets of values of the pair angular mo- mcntum. For cxample, for N = 9 and I F  = 12 (corre- sponding to 1 % ~  = ) the allowed values of the Fermion pair angular mominturn consist of all odd integers be- tween 1 and 23: for the corresponding Boson system with I n  = 8 ( lqn  = f ). the allowed values of L 1 2  are all even integers between 0 and 16. Xie ct al. [3] detcrmined the Boson and Fermion eigenfunctions by exact numerical diagonalization for six particle systems connected through the F-B trans- formation. They then transformed the Boson eigen- functions into Fermion eigenfunctions by multiply- ing them by n,,,(-., - - ,)  as required by the B-F 184 J. J. Quinn et al. 1 Physica E 11 (2001) 182-185 8 Fermions, 2S=21 N W ... . . 8 Bosons, 2S=14 Q O ,  . 8 Fermions, 2S=l9  ...... .... 8 Bosons, 2S=12 9.9 . :.I. 0 .. . . :  O 5 2 9.2 .. t I Fig. I. The energy spectra (energy E vs. angular momentum L)  of eight Fermions at the monopole strength 2SF = 21 (filling factor v~ = ;; circles) and 19 (two Laughlin quasielectrons; lines), and of eight Bosons at 2SB = 14 (vB = i; circles) and 12 (two quasielectrons; lines) for the Coulomb pseudopotential in the lowest Landau level (a)-(d), and for the model pseudopotentials HI (a1)-(dl), and H3 (a1')- (d"). I is the magnetic length. transformation. The overlap of these transformed Bo- son eigenfunctions with the exact Fermion eigenfunc- tions was then evaluated. The overlap was close to unity for incompressible quantum fluid states when the full Coulomb interaction was used, but it was consid- erably smaller when certain model pseudopotentials were used. We have evaluated numerically the eigenstates of an eight electron system at 2SF = 19 to 23 (these states correspond to Laughlin VF = $ states with 0, 1, or 2 QPs) for a number of different pseudopotentials. In frames (a)-(a'') and (b)-(b") of Fig. 1 we contrast the energy spectra for the Fermion and Boson sys- tems at VF = 5 ( v B  = f ) for the Coulomb pseudopo- tential appropriate for the lowest Landau level and for the model pseudopotentials H I  and H3. H I  is de- fined to have only the largest pseudopotential coeffi- cient [V(L12 = 21F - 1) for Fermions and V ( L I 2  = 21B) for Bosons] equal to its Coulomb value and all other pseudopotential coefficients equal to zero. H3 has the two largest pseudopotential coefficients equal to their Coulomb values and all other coefficients equal to zero. In frames (c)-(cl') and (d)-(dl') we do the same for the state containing two Laughlin quasielectrons (QE). The lowest states in (a), (a'), (b), (b') are quite similar consisting of a Laughlin L = 0 ground state. The magnetoroton band (at 2 < L < 8) is apparent in all four spectra, although the gaps and band widths are different for different pseudopotentials. The lowest states in (c), (c'), (d), (dl) are also similar containing twoQE'swith lQE= ; ( N -  1)= f givingL=N-2,  N - 4, . . .  = 0, 2, 4, and 6. The pseudopotential H3 J. J. Quinn et al. 1 Physica E 11 (2001) 182-185 185 used in (a1')-(d") gives very different results. We be- lieve that this behavior results because the pseudopo- tential used in H3 is not "short-range" in the sense that it does not increase faster than VH(L12)) in the region of non-vanishing V(LI2). This behavior of the pseu- dopotential results in correlations that are very differ- ent from Laughlin correlations, and it accounts for the poor overlap found by Xie et al. [3] for certain model pseudopotentials. 4. Quasiparticles The F+B transformation allows us to better under- stand the Boson [5] vs. Fermion [12,13] description of QP's in incompressible FQH states. Laughlin con- densed states having v~ = ( 2 p  + I)-' (where p is a positive integer) occur at 2 s ~  = (2p+ 1 )(N - 1 ) in the Haldane spherical geometry. The CF transformation gives an effective angular momentum I ;  = S; = S - p(N - 1) = f ( ~  - 1) when 2 p  flux quanta are attached to each electron. Thus the N CFs fill the 21* + 1 states of the lowest CF shell giving an L = 0 incompressible ground state. The F-+B transformation gives 2SB =2SF - (N - 1)=2p(N - 1) and a Boson filling factor of V B  = (2p)-I. Making a CB transformation gives 1; = S: = SB - p(N - 1 ) = 0. This also gives an L = 0 incompressible ground state because each CB has 1; = 0. Thus the CF description of a Laughlin state has one filled CF shell of angular momentum 1; = f ( N  - 1 ), while the CB description has N CBs each with angular momentum 1; = 0. For 286 = 2p(N - 1 ) f n ~ p ,  where the + and - occur for quasiholes (QH) and quasielectrons (QE), respectively, we define 21; = 12SiI =nQp. This gives exactly the same set of angular momentum multiplets as obtained in the CF picture with 2SF = ( 2 p  + 1 )(N - 1) f nQp only if a hard core repulsion forbids the Boson QE pair from having the largest allowed pair angular momentum L E ~ ~  = N [14]. This behavior is observed in Fig. l(c)-(c'), (d)-(dl) where the Boson treatment of two QE's predicts states at L = 0,2,4 ,  6 ,  and 8, but the L = 8 state does not occur in the low energy band. Since the description of CBs (with hard core QE interaction) and CFs give identical sets of QP states, filled QP levels (implying daughter states) occur at identical values of the applied magnetic field. Acknowledgements John J. Quinn acknowledges partial support from the Materials Research Program of Basic Energy Sci- ences, US Department of Energy. References [I]  F. Wilczek, Fractional Statistics and Anyon Super- conductivity, World Scientific, Singapore, 1990. [2] J. Jain, Phys. Rev. Lett. 63 (1989) 199. [3] X.C. Xie, S. He, S. Das Sarma, Phys. Rev. Lett. 66 (1991) 389. [4] R. Laughlin, Phys. Rev. Len. 50 (1983) 1395. [5] F.D.M. Haldane, Phys. Rev. Len. 51 (1983) 605. [6]  X.M. Chen, J.J. Quinn, Solid State Commun. 92 (1994) 865. [7] A. WOJS, J.J. Quinn, Solid State Commun. 108 (1998) 493. [8] A. WOJS, J.J. Quinn, Solid State Commun. 110 (1999) 45. [9] A. WOJS, J.J. Quinn, Acta Phys. Pol. A 96 (1999) 593. [lo] A. WOjs, J.J. Quim, Phil. Mag. B 80 (2000) 1405. [11]J.J. Quinn, A. WOjs, J.J. Quinn, A.T. Benjamin, cond-mat/000748 1, unpublished. [12] P. Sitko, K.-S. Yi, J.J. Quinn, Phys. Rev. B 56 (1997) 12417. [13] A. WOjs, J.J. Quinn, Phys. Rev. B 61 (2000) 2846. [14] S. He, X.-C. Xie, F.-C. Zhang, Phys. Rev. Len. 68 (1992) 3460. 

Transformation of Statistics in Fractional Quantum Hall Systems

Keck Graduate Institute

Claremont Colleges
Scholarship @ Claremont
All HMC Faculty Publications and Research HMC Faculty Scholarship
10-1-2001
Transformation of Statistics in Fractional Quantum
Hall Systems
John J. Quinn
University of Tennessee - Knoxville
Arkadiusz Wojs
Wroclaw Technical University
Jennifer J. Quinn
Occidental College
Arthur T. Benjamin
Harvey Mudd College
This Article is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion
in All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact
scholarship@cuc.claremont.edu.
Recommended Citation
Quinn, J.J., Wojs, A., Quinn, J.J., & Benjamin, A.T. (2001). Transformation of statistics in fractional quantum Hall systems. Physica E,
11(2-3): 182-185. DOI: 10.1016/S1386-9477(01)00199-0
Physica E 1 1  (2001) 182-1 85 
Transformation of statistics in fractional quantum Hall systems 
John J. Q u i n n a ,  *, Arkadiusz W O J S " ~ ~ ,  Jennifer J. Quinnc, Arthur T. Benjamind 
'Department ofphysics, University of Tennessee. 200 South College, Knoxville, TN 37996-1501. USA 
Wroclaw University of Technology, Wroclaw 50-370, Poland 
COccidental College, Los Angeles, CA 90041, USA 
~ a r v e y  Mudd College, Claremon f, CA 91 711, USA 
Abstract 
A Fermion to Boson transformation is accomplished by attaching to each Fermion a tube carrying a single quantum of flux 
oriented opposite to the applied magnetic field. When the mean field approximation is made in Haldane's spherical geometry, 
the Fermion angular momentum IF is replaced by IB = IF - $(N - 1). The set of allowed total angular momentum multiplets 
is identical in the two different pictures. The Fermion and Boson energy spectra in the presence of many body interactions 
are identical only if the pseudopotential V (interaction energy as a function of pair angular momentum Ll2) increases as 
L I 2 ( L l 2  + 1). Similar bands of low energy states occur in the two spectra if V increases more quickly than this. @ 2001 
Published by Elsevier Science B.V. 
PACS: 71.10.Pm; 73.20.D~; 73.40.Hm 
Keywords: Fermion-Boson mapping; Fractional quantum Hall effect; Composite Fermion 
1. Introduction 
In two dimensional systems particle statistics can 
: be changed by making a Chem-Simons (CS) transfor- 
. mation (see for example [I] and references therein). 
This transformation can be described as attaching to 
each particle an infinitesimal flux tube carrying a fic- 
titious flux 4 and a fictitious charge q, which couples 
to the vector potential produced by the flux tubes on 
every other particle in the standard way. If q is equal 
to the electron charge and 4 is an even number times 
4 0  = hcle, the quantum of flux, no change in statistics 
occurs. If 4 is and odd number times 4 0 ,  Fermions 
are transformed into Bosons. The "gauge field" inter- 
actions associated with the fictitious charge and vec- 
tor potential produced by the fictitious flux make the 
Hamiltonian of the system considerably more com- 
plicated. Only when the mean field approximation is 
made does the problem simplify. Jain [2] introduced 
'Corresponding author. Tel.: +1-865-974-4089; fax: the mean field CS transformation to give a simple in- 
+I-865-974-6378. tuitive picture of the hierarchy of fractional quantum 
E-mail address: jjquinn@utk.edu (J.J. Quinn). Hall (FQH) states in terms of the resulting compos- 
1386-9477/01/$-see front matter @ 2001 Published by Elsevier Science B.V. 
PII: S 1386-9477(01)00199-0 
J. J Q~iinn et ui. I Plij,sicu E 11 (2001 ) 182-18.5 183 
ite Fermions (CF). Shortly after the introduction of 
the CF picture, Xie et al. [3] introduced the Fermion- 
Boson mapping connecting a 2D Fermion system at 
filling factor 1 . b  to a 2D Boson system with filling fac- 
tor \ Q  = 1 3 ~ (  1 - I'F ) I .  These authors noted that the 
size of the Hilbert space for the Fermion and Boson 
systems was identical. and they found that the F+B 
mapping accurately transformed the ground state of 
the Fennion system into that of the Boson system only 
if these ground states were incompressible FQH states. 
In this paper we show that the F-B transformation 
leads to identical energy spectra if and only if the pseu- 
dopotential I'(LI1) describing the interaction energy 
as a function of the pair angular momentum Li. is 
of the "harmonic" form VH(L12) = A  + BLI2(Ll7  + 1 ) 
where A and B are constants. Laughlin [4] correlations 
occur when the actual pseudopotential V(L12) rises 
more quickly with increasing L I 2  than ITH(Ll2). Anhar- 
monic effects (due to AV(LII)  = V(LI?)  - VH(LI? ) )  
cause the interacting Fermion and interacting Boson 
spectra to differ for every value of the filling factor. 
It is wcll known that in the Haldane spherical ge- 
ometry [5] the mean field CF transformation changes 
an electron of angular momentum I t  to a CF of angu- 
lar moincntum I ;  = I F  - y(2\7 - 1 )  [6]. Here I F  = S t ,  
onc-half the strength (measured in units of 40) of 
the magnetic monopole u-hich produces thc radial 
magnetic field .8 = 2 ~ ~ $ ( , ( 4 n ~ ' ) ~ '  at the spherical 
surface of radius R on which the N electrons are con- 
fined, and p is an intcgcr. For an N Boson system, 
the composite Boson transformation replaces I n  by 
1; - p(.V - 111. In the FIB mapping I t  is 
replaced by I n  = I t  - ; ( N  - 1 ) .  
2. Angular momentum addition: useful theorems 
When a shell of angular momentum I contains N 
identical particles (Fermions or Bosons), the result- 
ing .V particle states can be classified by eigenvectors 
L .  .V. 2 ) .  where L is the total angular momentum, M 
its :-component, and x a label which distinguishes in- 
dependent multiplets with the same L. In the mean 
field CF (CB) transformation I F  (In) is transformed 
to 1; ( I ; ) .  In trying to understand why the mean field 
CF picture correctly predicted the low lying band of 
states in the electron spcctrum. scvcral simple conjec- 
tures were proposed on the basis of numerical stud- 
ies of a finite number of particles [7-101. These con- 
jectures have been elevated to the status of theorems 
by rigorous mathematical proof [ l l ]  using partition 
theory. 
Theorem 1. Tlze set qf' u l l o ~ ~ ~ e d  rortrl rrnglrlur mo- 
mentun1 ~ii~lltiplets qf'N Fer.ri~ions each 11,ith angtllar 
17io1iiel~t~1111 1; is a sub.ser oj'rhe .set qf al1011,ed rntll- 
tiplets o f N  Fermions etrch 1 1 3 h  unglrlu~. rrionlent~un 
I F  = 1; + ('v - 1 ). 
Theorem 2. The ser c!f ollolt~cri totlrl urlj/~llur. 1110rr1en- 
t l ln~ n~~ll t iplets  of N Bosons each rt.itli N I I C / L / ~ ( I I .  1110- 
ri7cr7t1rrii I D  i.r iderztical to the set of rn~lltiplets j i ~ r  
IV Fc1r.rt7ions each 11,ith u n g ~ ~ l u r  ~1on1ent~1nz It = I H  + 
4(Av - 1 ). 
From Theorem 2 it follows immediately that Theo- 
rem I also applies to Bosons. Theorem 2 is a stronger 
statcmcnt than a simplc cquality of sizes of the many 
body Hilbert spaces [3]. 
3. Interaction effects 
It has been shown that for the harmonic pseudopo- 
tential VH(LIZ) the energy of any multiplet of angular 
momentumL is given by E,(L)=A x IN(N-  ~ ) + B x  
{ N ( N 2 ) 1 ( 1 +  1 )+L(L+ 1 )). The energy is indepen- 
dent of x, so that every multiplet with the same value of 
L has the same energy. If BF = BB = B. then the spec- 
trum of N Bosons each with angular momcnturn I B  is 
identical (up to a constant) to that o f N  Fermions each 
with angular momentum IF = I B  + + ( N  - 1 ). This is a 
rather surprising result because Fermions and Bosons 
sample different sets of values of the pair angular mo- 
mcntum. For cxample, for N = 9 and I F  = 12 (corre- 
sponding to 1 % ~  = ) the allowed values of the Fermion 
pair angular mominturn consist of all odd integers be- 
tween 1 and 23: for the corresponding Boson system 
with I n  = 8 ( lqn  = f ). the allowed values of L 1 2  are all 
even integers between 0 and 16. 
Xie ct al. [3] detcrmined the Boson and Fermion 
eigenfunctions by exact numerical diagonalization for 
six particle systems connected through the F-B trans- 
formation. They then transformed the Boson eigen- 
functions into Fermion eigenfunctions by multiply- 
ing them by n,,,(-., - - ,)  as required by the B-F 
184 J. J. Quinn et al. 1 Physica E 11 (2001) 182-185 
8 Fermions, 2S=21 
N 
W ... 
. . 
8 Bosons, 2S=14 Q O ,  . 8 Fermions, 2S=l9  
...... 
.... 
8 Bosons, 2S=12 
9.9 . 
:.I. 
0 
.. 
. . :  O 5 
2 
9.2 
.. t 
I 
Fig. I. The energy spectra (energy E vs. angular momentum L)  of eight Fermions at the monopole strength 2SF = 21 (filling factor v~ = ;; 
circles) and 19 (two Laughlin quasielectrons; lines), and of eight Bosons at 2SB = 14 (vB = i; circles) and 12 (two quasielectrons; lines) 
for the Coulomb pseudopotential in the lowest Landau level (a)-(d), and for the model pseudopotentials HI (a1)-(dl), and H3 (a1')- 
(d"). I is the magnetic length. 
transformation. The overlap of these transformed Bo- 
son eigenfunctions with the exact Fermion eigenfunc- 
tions was then evaluated. The overlap was close to 
unity for incompressible quantum fluid states when the 
full Coulomb interaction was used, but it was consid- 
erably smaller when certain model pseudopotentials 
were used. 
We have evaluated numerically the eigenstates of 
an eight electron system at 2SF = 19 to 23 (these states 
correspond to Laughlin VF = $ states with 0, 1, or 2 
QPs) for a number of different pseudopotentials. In 
frames (a)-(a'') and (b)-(b") of Fig. 1 we contrast 
the energy spectra for the Fermion and Boson sys- 
tems at VF = 5 ( v B  = f ) for the Coulomb pseudopo- 
tential appropriate for the lowest Landau level and 
for the model pseudopotentials H I  and H3. H I  is de- 
fined to have only the largest pseudopotential coeffi- 
cient [V(L12 = 21F - 1) for Fermions and V ( L I 2  = 21B) 
for Bosons] equal to its Coulomb value and all other 
pseudopotential coefficients equal to zero. H3 has the 
two largest pseudopotential coefficients equal to their 
Coulomb values and all other coefficients equal to 
zero. In frames (c)-(cl') and (d)-(dl') we do the same 
for the state containing two Laughlin quasielectrons 
(QE). The lowest states in (a), (a'), (b), (b') are quite 
similar consisting of a Laughlin L = 0 ground state. 
The magnetoroton band (at 2 < L < 8) is apparent in 
all four spectra, although the gaps and band widths 
are different for different pseudopotentials. The lowest 
states in (c), (c'), (d), (dl) are also similar containing 
twoQE'swith lQE= ; ( N -  1)= f givingL=N-2,  
N - 4, . . .  = 0, 2, 4, and 6. The pseudopotential H3 
J. J. Quinn et al. 1 Physica E 11 (2001) 182-185 185 
used in (a1')-(d") gives very different results. We be- 
lieve that this behavior results because the pseudopo- 
tential used in H3 is not "short-range" in the sense that 
it does not increase faster than VH(L12)) in the region 
of non-vanishing V(LI2). This behavior of the pseu- 
dopotential results in correlations that are very differ- 
ent from Laughlin correlations, and it accounts for the 
poor overlap found by Xie et al. [3] for certain model 
pseudopotentials. 
4. Quasiparticles 
The F+B transformation allows us to better under- 
stand the Boson [5] vs. Fermion [12,13] description 
of QP's in incompressible FQH states. Laughlin con- 
densed states having v~ = ( 2 p  + I)-' (where p is a 
positive integer) occur at 2 s ~  = (2p+ 1 )(N - 1 ) in the 
Haldane spherical geometry. The CF transformation 
gives an effective angular momentum I ;  = S; = S - 
p(N - 1) = f ( ~  - 1) when 2 p  flux quanta are attached 
to each electron. Thus the N CFs fill the 21* + 1 states 
of the lowest CF shell giving an L = 0 incompressible 
ground state. 
The F-+B transformation gives 2SB =2SF - 
(N - 1)=2p(N - 1) and a Boson filling factor 
of V B  = (2p)-I. Making a CB transformation gives 
1; = S: = SB - p(N - 1 ) = 0. This also gives an 
L = 0 incompressible ground state because each CB 
has 1; = 0. Thus the CF description of a Laughlin 
state has one filled CF shell of angular momentum 
1; = f ( N  - 1 ), while the CB description has N CBs 
each with angular momentum 1; = 0. 
For 286 = 2p(N - 1 ) f n ~ p ,  where the + and - 
occur for quasiholes (QH) and quasielectrons (QE), 
respectively, we define 21; = 12SiI =nQp. This gives 
exactly the same set of angular momentum multiplets 
as obtained in the CF picture with 2SF = ( 2 p  + 1 )(N - 
1) f nQp only if a hard core repulsion forbids the 
Boson QE pair from having the largest allowed pair 
angular momentum L E ~ ~  = N [14]. This behavior is 
observed in Fig. l(c)-(c'), (d)-(dl) where the Boson 
treatment of two QE's predicts states at L = 0,2,4 ,  6 ,  
and 8, but the L = 8 state does not occur in the low 
energy band. Since the description of CBs (with hard 
core QE interaction) and CFs give identical sets of 
QP states, filled QP levels (implying daughter states) 
occur at identical values of the applied magnetic field. 
Acknowledgements 
John J. Quinn acknowledges partial support from 
the Materials Research Program of Basic Energy Sci- 
ences, US Department of Energy. 
References 
[I]  F. Wilczek, Fractional Statistics and Anyon Super- 
conductivity, World Scientific, Singapore, 1990. 
[2] J. Jain, Phys. Rev. Lett. 63 (1989) 199. 
[3] X.C. Xie, S. He, S. Das Sarma, Phys. Rev. Lett. 66 (1991) 
389. 
[4] R. Laughlin, Phys. Rev. Len. 50 (1983) 1395. 
[5] F.D.M. Haldane, Phys. Rev. Len. 51 (1983) 605. 
[6]  X.M. Chen, J.J. Quinn, Solid State Commun. 92 (1994) 865. 
[7] A. WOJS, J.J. Quinn, Solid State Commun. 108 (1998) 493. 
[8] A. WOJS, J.J. Quinn, Solid State Commun. 110 (1999) 45. 
[9] A. WOJS, J.J. Quinn, Acta Phys. Pol. A 96 (1999) 593. 
[lo] A. WOjs, J.J. Quim, Phil. Mag. B 80 (2000) 1405. 
[11]J.J. Quinn, A. WOjs, J.J. Quinn, A.T. Benjamin, 
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[12] P. Sitko, K.-S. Yi, J.J. Quinn, Phys. Rev. B 56 (1997) 12417. 
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Transformation of Statistics in Fractional Quantum Hall Systems

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