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Fermi liquid shell model descriptions of composite Fermions
Pergmon
Solid State Communications, Vol. 102, No. 11, pp. 775-777, 1997 @ 1997 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/97 9;17.00+.00
FERMI LIQUID SHELL MODEL DESCRIPTIONS
OF COMPOSITE
FERMIONS
IS. S. Yi a*band J. J. Quinn a,c a University of Tennessee, Knoxville, Tennessee 37996, U.S.A. b Pusan National University, Pusan 609-735, Korea c Oak Ridge National Laboratory, Oak Ridge, Tennessee 3783 1, U.S.A. (Received
and accepted
13 March 1997 by A. Pinczuk)
Systems containing a small number of quasiparticles of a Laughlin v = l/m state can be described in terms of composite Fermions carrying either 2p_ or 2p+ flux quanta, where 2~~ = m t- 1, The Fermi liquid shell model is applied to these two descriptions for a system of the six electrons on a spherical magnetic surface, and predictions for the low energy spectra are compared with one another and with exact numerical results. @ 1997 Published by Elsevier Science Ltd Keywords: A. fractional quantum Hall effect.
beyond mean field theory phenomenologically. In this In earlier works [I, 21 it was noted that the Laughpaper the results of the CF, Fermi liquid descriplin [3] incompressible state of filling factor v = l/m could be equally well described (within Jain’s [4] com- tions are compared for states containing three or more posite Fermion picture) in terms of two different types quasiparticles. Table II of Ref. [l] presents the CF? descriptions of the lowest energy bands for a system of of composite Fermions (CFS). The two types of W’s, referred to here as CF+, differ in that the flux tubes at- the six electrons on a Haldane sphere with 11 5 2s _< tached to the electrons to create CF’s carry 2~~ = rn? 1 19. In both descriptions the system with 2s = 15 has a flux quanta. For a system of N electrons on a Haldane Laughlin L = 0 ground state corresponding to filling sphere [5] the strength, in flux quanta, of the magnetic monopole at the center is given by 2S = m(N- 1) for a factor v = l/3, while a system containing rz quasielecLaughlin v = l/m incompressible state. For the CF& trons (or quasiholes) of the v = l/3 state has 2s = 15+N). descriptions, the efictive monopole strength seen by a 15--n(or2S= In the Fermi liquid model a quasielectron is a CF, is given by 2S_T= 2S-2p,(N1) = T(N- 1). Since 15’: I plays the role of angular momentum [6] composite Fermion in the first excited CF shell, and a is an empty CF state in the lowest CF for the lowest CF energy shell, and since this shell can quasihole shell. As the table demonstrates, the transformation accomodate 2/S,* I + 1 = N composite Fermions, it has a filling factor vr = T 1. The negative sign on ST from CF._ to CF+ interchanges the roles of QE and and vf indicates that the effective magnetic field BT QH. For systems containing two or more QE’s or QH’s, seen by CF+ is oriented opposite to the applied magnetic field B. When the CF filling factor v* is an in- the,lowest energy band always contains a larger numteger [4] the electron filling factor v is given by v = ber of angular momentum multiplets for the QH dev* (1 + 2pv*)-1 so that both v2 = ~1, 2p+ = m + 1 scription than for the QE description. As noted in ReE [I,21 the allowed QE multiplets form a low energy correspond to v = 1/m. The Chem-Simons(CS) [7] interactions of the CF+ subset of the allowed QH multiplets. The phenomenological Fermi liquid parameters [9] and CF- are different because the strength of the CS gauge field is proportional to 2~~. A Fermi liquid EQE, EQH, VQE-QE(J), VQH-QHCJ), and VQE-QH(J) model [S] which uses exact numerical results as input can be determined by comparing the CF Fermi liquid model with exact numerical results [lo]. Here J data accounts for the Coulomb and CS interactions 775
DESCRIPTIONS
776
OF COMPOSITE
FERMIONS
Vol. 102, No. 11
o’2 1 2S=lQ
1
(4 0
5 L
10
0
5
10
L 0.2
1
25=18
-0.2t,.. 0 0.2 0.1 0.0
(b)
.". 5
."I
10
L
L
;ib’;iflf”::j
-0.1
0.2 t
'
5
4
0 2S=16
-0.2
0
I
5
10
L
*. 0
“.
5 L
'.
(9
' 10
Fig. 1. Low energy spectra of a system of the six electrons as a function of total angular momentum L. The crosses are exact numerical results; the open circles and solid squares are the results of the CF._ and CF+ picture, respectively. Lowest energy band for 2S = 12 (a) and 2S = 18 (b) corresponds to the states having three quasielectrons or three quasiholes. The same results are given for 2s = 11 (c) and 2s = 19 (d), corresponding to four quasiparticle states. For four quasielectrons, the mean field ground state is an L = 0 Laughlin incompressible state with filling factor v = 2/5 (for 2s = 11) and v = 2/7 (for 2s = 19). The ground state and first excited band for 2s = 14 and 2s = 16 are given in (e) and (f), respectively.
Fig. 2. Low energy spectra of a system of the six electrons as a function of total angular momentum L. The crosses are exact numerical results; the open circles and solid squares are the results of the CF_ and CF+ picture, respectively. Lowest energy band for 2s = 12 (a) and 2s = 18 (b) corresponds to the states having three quasielectrons or three quasiholes. The same results are given for 2s = 11 (c) and 2s = 19 (d), corresponding to four quasiparticle states. For four quasielectrons, the mean field ground state is an L = 0 Laughlin incompressible state with filling factor v = 2/5 (for 2s = 11) and v = 2/7 (for 2s = 19). The ground state and first excited band for 2s = 14 and 2s = 16 are given in (e) and (f), respectively.
is the angular momentum of a pair of quasiparticles. It is evident that E& = E&, and v&_eH(J) = V&r_pH (J). A little thought leads to the conclusion QE that V&-QE (Jf = V&_eH(J) for J = 0,2, .... JMAX for a system of even number of electrons(or J = 1, 3> .._f Jg/” for a system of odd number of electrons). However, because a pair of QH’s has Jf$kx = Jf&f” + 2, there is one additional interaction paMAX) which is rameter for the QH pair, V&_QH(JQH determined from the numerical data. Given the Fermi liquid parameters, the energies of all the states given in Table II of Ref. [l] (or Table IV of Ref. [2]) can be evaluated within the Fermi liquid shell model. In Fig. 1 we display the results, as a function of the total angular momentum L, for the lowest band of the six electron system with 2s = 12 (a), 25 = 18 (b), 2s = 11 (c) and
2s = 19 (d), and the results for the lowest state and the first excited band for 2s = 14 (e) and 2s = 16 (f). Figures (a) and (b) contain three QE’s or three QH’s depending on which description is used. Figures (c) and (d) contain four QE’s or four QH’s. The descriptions in terms of four QE’s yield a Laughlin incompressible ground state at L = 0 as the only state in the band. These Laughlin states have lee = 312 so that the four QE’s fill the 2& + 1 states of the first excited CF shell [4]. Figures (e) and (f) correspond to systems with a ground state containing a single QP of angular momentum &-JP= 3. The first excited band contains an additional QE-QH pair. There are a larger number of -t signs indicating exact numerical results than open circles (CF._ results) or solid squares (CF+ results) for some values of L. We believe that these energy values are associated with states that have been
Vol. 102, No. 11
DESCRIPTIONS
ignored in our simple Fermi liquid picture which includes only the lowest and first excited CF shells. For example, one of the + states at L = 4 must be associated with a single QE in the second excited shell; other states are probably associated with bound states that contain more than one additional QE-QH pair. The agreement of the two different Fermi liquid descriptions with the exact numerical calculations and with one another is reasonably good. Because we are not explicitly including three body interactions arising from the CS gauge field, we do not expect perfect agreement. However, the model calculations are remarkably good for the lowest energy sector, and they could certainly be used to identify a well defined Laughlin incompressible ground state. Because the CF description in terms of quasielectrons always contains fewer multiplets than that in terms of quasiholes both for the lowest energy band and for excited bands, it is probably more useful for systems containing a number of quasiparticles. This work suggests that by determining the Fermi liquid parameters for nine and ten electron systems from exact numerical data and plotting all of the Fermi liquid parameters as a function of l/N, one can obtain Fermi liquid shell model parameters that could be used in understanding the low energy part of the spectrum for larger systems. AcknowledgementsThis work was supported in part by Oak Ridge National Laboratory, managed by Lockheed Martin Energy Research Corp. for the US Department of Energy under contract No. DEAC05-960R22464. K.S.Y. acknowledges supports by the BSRI-96-2412 program of the Ministry of Education, Korea.
OF COMPOSITE
FERMIONS
777
REFERENCES 1.
K. S. Yi, P Sitko, A. Khurana, and J. J. Quinn, Phys. Rev. B 54, 16432 (1996). 2. K. S. Yi and J. J. Quinn, Conference Book of the 8th Seoul International Symposium on the Physics of Semiconductor and ApplicationsfSeoul, 1996) p.69. 3. R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). 4. J. K. Jain, Phys. Rev. Lett. 63, 199 (1989). 5. F. D. M. Haldane, Phys. Rev. Lett. 51,605 (1983). 6. X. M. Chen and J. J. Quinn, Solid State Commun. 92, 865 (1994). 7. A. Lopez and E. Fradkin, Phys. Rev. B 44, 5246 (1991). 8. P Sitko, S. N. Yi, K. S. Yi, and J. J. Quinn, Phys. Rev. Lett. 76, 3396 (1996). 9. S. N. Yi, X. M. Chen, and J. J. Quinn, Phys. Rev. B 53, 9599 (1996). 10. S. He, X. C. Xie, and F. C. Zhang, Phys. Rev. Lett. 68, 3460 (1992); G. Fano, F. Ortolani, and E. Colombo, Phys. Rev. B 34, 2670 (1986); F. D. M. Haldane and E. H. Rezayi, Phys. Rev. Lett. 54, 237 (1985) and Phys. Rev. 31,2529 (1985).
Solid State Communications, Vol. 102, No. 11, pp. 775-777, 1997 @ 1997 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/97 9;17.00+.00
FERMI LIQUID SHELL MODEL DESCRIPTIONS
OF COMPOSITE
FERMIONS
IS. S. Yi a*band J. J. Quinn a,c a University of Tennessee, Knoxville, Tennessee 37996, U.S.A. b Pusan National University, Pusan 609-735, Korea c Oak Ridge National Laboratory, Oak Ridge, Tennessee 3783 1, U.S.A. (Received
and accepted
13 March 1997 by A. Pinczuk)
Systems containing a small number of quasiparticles of a Laughlin v = l/m state can be described in terms of composite Fermions carrying either 2p_ or 2p+ flux quanta, where 2~~ = m t- 1, The Fermi liquid shell model is applied to these two descriptions for a system of the six electrons on a spherical magnetic surface, and predictions for the low energy spectra are compared with one another and with exact numerical results. @ 1997 Published by Elsevier Science Ltd Keywords: A. fractional quantum Hall effect.
beyond mean field theory phenomenologically. In this In earlier works [I, 21 it was noted that the Laughpaper the results of the CF, Fermi liquid descriplin [3] incompressible state of filling factor v = l/m could be equally well described (within Jain’s [4] com- tions are compared for states containing three or more posite Fermion picture) in terms of two different types quasiparticles. Table II of Ref. [l] presents the CF? descriptions of the lowest energy bands for a system of of composite Fermions (CFS). The two types of W’s, referred to here as CF+, differ in that the flux tubes at- the six electrons on a Haldane sphere with 11 5 2s _< tached to the electrons to create CF’s carry 2~~ = rn? 1 19. In both descriptions the system with 2s = 15 has a flux quanta. For a system of N electrons on a Haldane Laughlin L = 0 ground state corresponding to filling sphere [5] the strength, in flux quanta, of the magnetic monopole at the center is given by 2S = m(N- 1) for a factor v = l/3, while a system containing rz quasielecLaughlin v = l/m incompressible state. For the CF& trons (or quasiholes) of the v = l/3 state has 2s = 15+N). descriptions, the efictive monopole strength seen by a 15--n(or2S= In the Fermi liquid model a quasielectron is a CF, is given by 2S_T= 2S-2p,(N1) = T(N- 1). Since 15’: I plays the role of angular momentum [6] composite Fermion in the first excited CF shell, and a is an empty CF state in the lowest CF for the lowest CF energy shell, and since this shell can quasihole shell. As the table demonstrates, the transformation accomodate 2/S,* I + 1 = N composite Fermions, it has a filling factor vr = T 1. The negative sign on ST from CF._ to CF+ interchanges the roles of QE and and vf indicates that the effective magnetic field BT QH. For systems containing two or more QE’s or QH’s, seen by CF+ is oriented opposite to the applied magnetic field B. When the CF filling factor v* is an in- the,lowest energy band always contains a larger numteger [4] the electron filling factor v is given by v = ber of angular momentum multiplets for the QH dev* (1 + 2pv*)-1 so that both v2 = ~1, 2p+ = m + 1 scription than for the QE description. As noted in ReE [I,21 the allowed QE multiplets form a low energy correspond to v = 1/m. The Chem-Simons(CS) [7] interactions of the CF+ subset of the allowed QH multiplets. The phenomenological Fermi liquid parameters [9] and CF- are different because the strength of the CS gauge field is proportional to 2~~. A Fermi liquid EQE, EQH, VQE-QE(J), VQH-QHCJ), and VQE-QH(J) model [S] which uses exact numerical results as input can be determined by comparing the CF Fermi liquid model with exact numerical results [lo]. Here J data accounts for the Coulomb and CS interactions 775
DESCRIPTIONS
776
OF COMPOSITE
FERMIONS
Vol. 102, No. 11
o’2 1 2S=lQ
1
(4 0
5 L
10
0
5
10
L 0.2
1
25=18
-0.2t,.. 0 0.2 0.1 0.0
(b)
.". 5
."I
10
L
L
;ib’;iflf”::j
-0.1
0.2 t
'
5
4
0 2S=16
-0.2
0
I
5
10
L
*. 0
“.
5 L
'.
(9
' 10
Fig. 1. Low energy spectra of a system of the six electrons as a function of total angular momentum L. The crosses are exact numerical results; the open circles and solid squares are the results of the CF._ and CF+ picture, respectively. Lowest energy band for 2S = 12 (a) and 2S = 18 (b) corresponds to the states having three quasielectrons or three quasiholes. The same results are given for 2s = 11 (c) and 2s = 19 (d), corresponding to four quasiparticle states. For four quasielectrons, the mean field ground state is an L = 0 Laughlin incompressible state with filling factor v = 2/5 (for 2s = 11) and v = 2/7 (for 2s = 19). The ground state and first excited band for 2s = 14 and 2s = 16 are given in (e) and (f), respectively.
Fig. 2. Low energy spectra of a system of the six electrons as a function of total angular momentum L. The crosses are exact numerical results; the open circles and solid squares are the results of the CF_ and CF+ picture, respectively. Lowest energy band for 2s = 12 (a) and 2s = 18 (b) corresponds to the states having three quasielectrons or three quasiholes. The same results are given for 2s = 11 (c) and 2s = 19 (d), corresponding to four quasiparticle states. For four quasielectrons, the mean field ground state is an L = 0 Laughlin incompressible state with filling factor v = 2/5 (for 2s = 11) and v = 2/7 (for 2s = 19). The ground state and first excited band for 2s = 14 and 2s = 16 are given in (e) and (f), respectively.
is the angular momentum of a pair of quasiparticles. It is evident that E& = E&, and v&_eH(J) = V&r_pH (J). A little thought leads to the conclusion QE that V&-QE (Jf = V&_eH(J) for J = 0,2, .... JMAX for a system of even number of electrons(or J = 1, 3> .._f Jg/” for a system of odd number of electrons). However, because a pair of QH’s has Jf$kx = Jf&f” + 2, there is one additional interaction paMAX) which is rameter for the QH pair, V&_QH(JQH determined from the numerical data. Given the Fermi liquid parameters, the energies of all the states given in Table II of Ref. [l] (or Table IV of Ref. [2]) can be evaluated within the Fermi liquid shell model. In Fig. 1 we display the results, as a function of the total angular momentum L, for the lowest band of the six electron system with 2s = 12 (a), 25 = 18 (b), 2s = 11 (c) and
2s = 19 (d), and the results for the lowest state and the first excited band for 2s = 14 (e) and 2s = 16 (f). Figures (a) and (b) contain three QE’s or three QH’s depending on which description is used. Figures (c) and (d) contain four QE’s or four QH’s. The descriptions in terms of four QE’s yield a Laughlin incompressible ground state at L = 0 as the only state in the band. These Laughlin states have lee = 312 so that the four QE’s fill the 2& + 1 states of the first excited CF shell [4]. Figures (e) and (f) correspond to systems with a ground state containing a single QP of angular momentum &-JP= 3. The first excited band contains an additional QE-QH pair. There are a larger number of -t signs indicating exact numerical results than open circles (CF._ results) or solid squares (CF+ results) for some values of L. We believe that these energy values are associated with states that have been
Vol. 102, No. 11
DESCRIPTIONS
ignored in our simple Fermi liquid picture which includes only the lowest and first excited CF shells. For example, one of the + states at L = 4 must be associated with a single QE in the second excited shell; other states are probably associated with bound states that contain more than one additional QE-QH pair. The agreement of the two different Fermi liquid descriptions with the exact numerical calculations and with one another is reasonably good. Because we are not explicitly including three body interactions arising from the CS gauge field, we do not expect perfect agreement. However, the model calculations are remarkably good for the lowest energy sector, and they could certainly be used to identify a well defined Laughlin incompressible ground state. Because the CF description in terms of quasielectrons always contains fewer multiplets than that in terms of quasiholes both for the lowest energy band and for excited bands, it is probably more useful for systems containing a number of quasiparticles. This work suggests that by determining the Fermi liquid parameters for nine and ten electron systems from exact numerical data and plotting all of the Fermi liquid parameters as a function of l/N, one can obtain Fermi liquid shell model parameters that could be used in understanding the low energy part of the spectrum for larger systems. AcknowledgementsThis work was supported in part by Oak Ridge National Laboratory, managed by Lockheed Martin Energy Research Corp. for the US Department of Energy under contract No. DEAC05-960R22464. K.S.Y. acknowledges supports by the BSRI-96-2412 program of the Ministry of Education, Korea.
OF COMPOSITE
FERMIONS
777
REFERENCES 1.
K. S. Yi, P Sitko, A. Khurana, and J. J. Quinn, Phys. Rev. B 54, 16432 (1996). 2. K. S. Yi and J. J. Quinn, Conference Book of the 8th Seoul International Symposium on the Physics of Semiconductor and ApplicationsfSeoul, 1996) p.69. 3. R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). 4. J. K. Jain, Phys. Rev. Lett. 63, 199 (1989). 5. F. D. M. Haldane, Phys. Rev. Lett. 51,605 (1983). 6. X. M. Chen and J. J. Quinn, Solid State Commun. 92, 865 (1994). 7. A. Lopez and E. Fradkin, Phys. Rev. B 44, 5246 (1991). 8. P Sitko, S. N. Yi, K. S. Yi, and J. J. Quinn, Phys. Rev. Lett. 76, 3396 (1996). 9. S. N. Yi, X. M. Chen, and J. J. Quinn, Phys. Rev. B 53, 9599 (1996). 10. S. He, X. C. Xie, and F. C. Zhang, Phys. Rev. Lett. 68, 3460 (1992); G. Fano, F. Ortolani, and E. Colombo, Phys. Rev. B 34, 2670 (1986); F. D. M. Haldane and E. H. Rezayi, Phys. Rev. Lett. 54, 237 (1985) and Phys. Rev. 31,2529 (1985).