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Fermi-liquid shell model for N electrons on a spherical surface: A test of the composite fermion hierarchy picture
Physica B 249—251 (1998) 32—35
Fermi-liquid shell model for N electrons on a spherical surface: A test of the composite fermion hierarchy picture P. Sitko!,",*, K.-S. Yi#, J.J. Quinn! ! University of Tennessee, Knoxville, Tennessee 37996, USA " Institute of Physics, Technical University of Wroc!aw, WybrzezR e Wyspian& skiego 27, 50-370 Wroc!aw, Poland # Pusan National University, Pusan 609-735, South Korea
Abstract The extension of the Jain theory of fractional quantum Hall states to all odd denominator filling states is proposed in analogy to the Haldane hierarchy. The composite fermion hierarchy is a natural consequence of the Fermi-liquid quasiparticle picture when applied to composite fermion excitations on the sphere. It is shown that the “finite size corrections” defined as 1/l!2S/(N!1), where l is the filling factor, 2S the magnetic monopole strength in flux quanta, and N the number of electrons, exhibit periodicity as a function of 2S with period 2(N!1). The results of numerical diagonalization for l" 4 and l" 4 (N"8) show no indication of a condensed state. It is also the case for the system of 11 13 4QE of the 2 Jain state (N"12) seen as the “half-filled state” of quasielectrons. ( 1998 Elsevier Science B.V. All rights 3 reserved. Keywords: Composite fermions
The idea of the hierarchy of fractional quantum Hall states was first proposed by Haldane [1] as the extension of the Laughlin theory of incompressible states of electrons within the lowest Landau level [2]. The Haldane theory predicted in principle that all odd denominator filling fractions can be seen as condensed states. That was certainly not true; only a subset of odd denominator fractions was observed in experiment. Good agreement with the experiment was achieved by the Jain theory of
* Correspondence address: University of Tennessee, Knoxville, Tennessee 37996, USA.
composite fermions [3]. The mean field composite fermion picture identifies immediately all observed condensed states. However, the possibility of the full hierarchy still exists. Jain and Goldman [4] proposed a hierarchy picture by making use of trial wave functions. We propose to define the composite fermion hierarchy in analogy to Haldane method, operating in terms of quasiparticles (composite fermion excitations) [5]. In a spherical system (originally introduced by Haldane for the purpose of the hierarchy) the lowest shell (Landau level) angular momentum is given by S and 2S
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 7 ) 0 0 0 6 1 - 1
(1)
P. Sitko et al. / Physica B 249—251 (1998) 32—35
is an integer number of flux quanta (strength of the magnetic monopole) piercing the sphere. When performing a CF transformation the effective field is found to be 2S*"2S!2p(N!1),
(2)
where 2p is an even parameter of the Chern— Simons transformation [6]. Thus, in the mean field approximation, one deals with the magnetic field (2) instead of real magnetic flux (1). This leads to different occupations of effective shells and gives support for defining the fractional quantum Hall states as the states with an integer number of CF shells filled in the field 2S* [3]. This prescription (mean field) works very well for electrons on the sphere and is confirmed in a number of numerical studies. It was also verified that the mean composite fermion theory can be used for values of the magnetic field different than for Jain states, where the Fermi-liquid shell model is proposed for many-quasiparticle systems [7]. The agreement of many-quasiparticle spectra obtained for phenomenological (Fermi-liquid) interactions with numerical results is good enough to suggest that quasiparticle description works around all Jain states. Since quasiparticles partially occupy an angular momentum shell (angular momentum is predicted by the mean field CF approach) we apply the CF transformation to quasiparticles in a partially filled CF shell in analogy with the treatment for electrons. The important difference here, however, is the form of quasiparticle interaction. The CF mean field does not explicitly involve interaction, so that its prediction of the condensed states of quasiparticles is analogous to that of electrons. We list such states for the case of 12 electrons in Table 1. The filling fractions are obtained in the following way. The Jain states are found for 1 1 "2p$ , l n
33
Table 1 N"12 2S
l
2S
l
2S
l
11 12 13 14 15 16 17 18
1 ! 12/13 6/7 4/5 ! 13/19 2/3
19 20 21 22 23 24 25 26
15/23 ! 3/5 ! 3/7 ! 15/37 2/5
27 28 29 30 31 32 33 34
13/33 ! 4/11 6/17 12/35 ! 1/3 !
states are obtained when 1 1 "2p$ , l n#l QE
(4)
where l is the filling fraction of quasiparticles and QE has a form of Jain fraction (3). The procedure (4) can be repeated on l to generate the full hierQE archy of odd denominator fractions. It is interesting to consider the “finite size corrections” to the filling fraction defined as 1 2S ! . l N!1
(5)
We plot the results for 12 particles for 1*l*1 in 5 Fig. 1. We find the periodicity of Eq. (5) as a function
(3)
where n is the number of filled effective shells, the $ sign reflects the sign of the effective field with respect to the real one. The first order hierarchy
Fig. 1. The finite size corrections: 1/l!2S/(N!1) for 12 particles and 1*l*1. Note the periodicity of this function with 5 2S.
34
P. Sitko et al. / Physica B 249—251 (1998) 32—35
Fig. 2. The results of exact diagonalization: 2(A) the system of N"8 electrons and 2S"18 ( 4 state); (B) the system of N"8 electrons 11 and 2S"24 ( 4 state); (C) the system of N"12 electrons and 2S"20. Energies in respective units of e2/l (l is the magnetic length). 13 0 0
of 2S when 2S@!2S"(N!1)](even number).
(6)
Relation (6) is obviously the composite fermion ansatz (2). Since the hierarchy fractions (Table 1) can be derived within the Haldane method [1] we can say that the composite fermion theory appears naturally within hierarchy. Let us discuss two cases of the filling (4) obtained for N"8 electrons (Fig. 2A and Fig. 2B). In Fig. 2A, we have 3QE of the 1 Laughlin state with 3 angular momentum l "3.0. Then the filling l is QE QE 1 as it is for three electrons for the same shell 3 (l" 4 ). Fig. 2B can be seen in the reversed effective 11 field (2S*(0, p"2) as a system of three quasiparticles with angular momentum l "3.0. Now, QE however, l" 4 . In Fig. 2C we present the case of 13 4QE of the 2 state (2S*(0, p"1 — we have two 3 filled shells in the reversed effective field and 4 particles are in the angular momentum shell l "3.0) QE or as conjugate system of 3QH of the 3 state 5 (l "3.0). In the spirit of Rezayi and Read [8] this QH
is the “half-filled state” of quasielectrons (for analogous system of four electrons the CF approach predict an ¸"0 ground state — two filled CF shells). In all figures we can see very similar spectra of three quasiparticles in the angular momentum shell l "3.0. None of them give an ¸"0 multiplet as QP the lowest energy state in contrast to the analogous system of three electrons with angular momentum l"3.0. The CF hierarchy predictions are not valid here (for Fig. 2A and Fig. 2B). Though, in all three figures quasiparticles have a different origin, the quasiparticle interaction is similar, as the spectra have a similar form. It is clear that this quasiparticle interaction belongs to a class of interactions for which mean composite fermion theory is not adequate for description of the lowest energy states. One of the authors (P.S.) would like to acknowledge support by the KBN grant No. PB 674/P03/96/10; one (J.J.Q.) would like to acknowledge the support of the Division of Material Sciences, Office of Basic Energy Sciences, of the US
P. Sitko et al. / Physica B 249—251 (1998) 32—35
Department of Energy. K.S.Y. acknowledges the support by the BSRI-97-2412 program of the Ministry of Education, Korea. References [1] F.D.M. Haldane, Phys. Rev. Lett. 51 (1983) 605.
[2] [3] [4] [5] [6] [7]
35
R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. J.K. Jain, Phys. Rev. Lett. 63 (1989) 199. J.K. Jain, V.J. Goldman, Phys. Rev. B 45 (1992) 1255. P. Sitko, K.S. Yi, J.J. Quinn, unpublished. A. Lopez, E. Fradkin, Phys. Rev. B 44 (1991) 5246. P. Sitko, S.N. Yi, K.S. Yi, J.J. Quinn, Phys. Rev. Lett. 76 (1996) 3396. [8] E. Rezayi, N. Read, Phys. Rev. Lett. 72 (1994) 900.
Fermi-liquid shell model for N electrons on a spherical surface: A test of the composite fermion hierarchy picture P. Sitko!,",*, K.-S. Yi#, J.J. Quinn! ! University of Tennessee, Knoxville, Tennessee 37996, USA " Institute of Physics, Technical University of Wroc!aw, WybrzezR e Wyspian& skiego 27, 50-370 Wroc!aw, Poland # Pusan National University, Pusan 609-735, South Korea
Abstract The extension of the Jain theory of fractional quantum Hall states to all odd denominator filling states is proposed in analogy to the Haldane hierarchy. The composite fermion hierarchy is a natural consequence of the Fermi-liquid quasiparticle picture when applied to composite fermion excitations on the sphere. It is shown that the “finite size corrections” defined as 1/l!2S/(N!1), where l is the filling factor, 2S the magnetic monopole strength in flux quanta, and N the number of electrons, exhibit periodicity as a function of 2S with period 2(N!1). The results of numerical diagonalization for l" 4 and l" 4 (N"8) show no indication of a condensed state. It is also the case for the system of 11 13 4QE of the 2 Jain state (N"12) seen as the “half-filled state” of quasielectrons. ( 1998 Elsevier Science B.V. All rights 3 reserved. Keywords: Composite fermions
The idea of the hierarchy of fractional quantum Hall states was first proposed by Haldane [1] as the extension of the Laughlin theory of incompressible states of electrons within the lowest Landau level [2]. The Haldane theory predicted in principle that all odd denominator filling fractions can be seen as condensed states. That was certainly not true; only a subset of odd denominator fractions was observed in experiment. Good agreement with the experiment was achieved by the Jain theory of
* Correspondence address: University of Tennessee, Knoxville, Tennessee 37996, USA.
composite fermions [3]. The mean field composite fermion picture identifies immediately all observed condensed states. However, the possibility of the full hierarchy still exists. Jain and Goldman [4] proposed a hierarchy picture by making use of trial wave functions. We propose to define the composite fermion hierarchy in analogy to Haldane method, operating in terms of quasiparticles (composite fermion excitations) [5]. In a spherical system (originally introduced by Haldane for the purpose of the hierarchy) the lowest shell (Landau level) angular momentum is given by S and 2S
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 7 ) 0 0 0 6 1 - 1
(1)
P. Sitko et al. / Physica B 249—251 (1998) 32—35
is an integer number of flux quanta (strength of the magnetic monopole) piercing the sphere. When performing a CF transformation the effective field is found to be 2S*"2S!2p(N!1),
(2)
where 2p is an even parameter of the Chern— Simons transformation [6]. Thus, in the mean field approximation, one deals with the magnetic field (2) instead of real magnetic flux (1). This leads to different occupations of effective shells and gives support for defining the fractional quantum Hall states as the states with an integer number of CF shells filled in the field 2S* [3]. This prescription (mean field) works very well for electrons on the sphere and is confirmed in a number of numerical studies. It was also verified that the mean composite fermion theory can be used for values of the magnetic field different than for Jain states, where the Fermi-liquid shell model is proposed for many-quasiparticle systems [7]. The agreement of many-quasiparticle spectra obtained for phenomenological (Fermi-liquid) interactions with numerical results is good enough to suggest that quasiparticle description works around all Jain states. Since quasiparticles partially occupy an angular momentum shell (angular momentum is predicted by the mean field CF approach) we apply the CF transformation to quasiparticles in a partially filled CF shell in analogy with the treatment for electrons. The important difference here, however, is the form of quasiparticle interaction. The CF mean field does not explicitly involve interaction, so that its prediction of the condensed states of quasiparticles is analogous to that of electrons. We list such states for the case of 12 electrons in Table 1. The filling fractions are obtained in the following way. The Jain states are found for 1 1 "2p$ , l n
33
Table 1 N"12 2S
l
2S
l
2S
l
11 12 13 14 15 16 17 18
1 ! 12/13 6/7 4/5 ! 13/19 2/3
19 20 21 22 23 24 25 26
15/23 ! 3/5 ! 3/7 ! 15/37 2/5
27 28 29 30 31 32 33 34
13/33 ! 4/11 6/17 12/35 ! 1/3 !
states are obtained when 1 1 "2p$ , l n#l QE
(4)
where l is the filling fraction of quasiparticles and QE has a form of Jain fraction (3). The procedure (4) can be repeated on l to generate the full hierQE archy of odd denominator fractions. It is interesting to consider the “finite size corrections” to the filling fraction defined as 1 2S ! . l N!1
(5)
We plot the results for 12 particles for 1*l*1 in 5 Fig. 1. We find the periodicity of Eq. (5) as a function
(3)
where n is the number of filled effective shells, the $ sign reflects the sign of the effective field with respect to the real one. The first order hierarchy
Fig. 1. The finite size corrections: 1/l!2S/(N!1) for 12 particles and 1*l*1. Note the periodicity of this function with 5 2S.
34
P. Sitko et al. / Physica B 249—251 (1998) 32—35
Fig. 2. The results of exact diagonalization: 2(A) the system of N"8 electrons and 2S"18 ( 4 state); (B) the system of N"8 electrons 11 and 2S"24 ( 4 state); (C) the system of N"12 electrons and 2S"20. Energies in respective units of e2/l (l is the magnetic length). 13 0 0
of 2S when 2S@!2S"(N!1)](even number).
(6)
Relation (6) is obviously the composite fermion ansatz (2). Since the hierarchy fractions (Table 1) can be derived within the Haldane method [1] we can say that the composite fermion theory appears naturally within hierarchy. Let us discuss two cases of the filling (4) obtained for N"8 electrons (Fig. 2A and Fig. 2B). In Fig. 2A, we have 3QE of the 1 Laughlin state with 3 angular momentum l "3.0. Then the filling l is QE QE 1 as it is for three electrons for the same shell 3 (l" 4 ). Fig. 2B can be seen in the reversed effective 11 field (2S*(0, p"2) as a system of three quasiparticles with angular momentum l "3.0. Now, QE however, l" 4 . In Fig. 2C we present the case of 13 4QE of the 2 state (2S*(0, p"1 — we have two 3 filled shells in the reversed effective field and 4 particles are in the angular momentum shell l "3.0) QE or as conjugate system of 3QH of the 3 state 5 (l "3.0). In the spirit of Rezayi and Read [8] this QH
is the “half-filled state” of quasielectrons (for analogous system of four electrons the CF approach predict an ¸"0 ground state — two filled CF shells). In all figures we can see very similar spectra of three quasiparticles in the angular momentum shell l "3.0. None of them give an ¸"0 multiplet as QP the lowest energy state in contrast to the analogous system of three electrons with angular momentum l"3.0. The CF hierarchy predictions are not valid here (for Fig. 2A and Fig. 2B). Though, in all three figures quasiparticles have a different origin, the quasiparticle interaction is similar, as the spectra have a similar form. It is clear that this quasiparticle interaction belongs to a class of interactions for which mean composite fermion theory is not adequate for description of the lowest energy states. One of the authors (P.S.) would like to acknowledge support by the KBN grant No. PB 674/P03/96/10; one (J.J.Q.) would like to acknowledge the support of the Division of Material Sciences, Office of Basic Energy Sciences, of the US
P. Sitko et al. / Physica B 249—251 (1998) 32—35
Department of Energy. K.S.Y. acknowledges the support by the BSRI-97-2412 program of the Ministry of Education, Korea. References [1] F.D.M. Haldane, Phys. Rev. Lett. 51 (1983) 605.
[2] [3] [4] [5] [6] [7]
35
R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. J.K. Jain, Phys. Rev. Lett. 63 (1989) 199. J.K. Jain, V.J. Goldman, Phys. Rev. B 45 (1992) 1255. P. Sitko, K.S. Yi, J.J. Quinn, unpublished. A. Lopez, E. Fradkin, Phys. Rev. B 44 (1991) 5246. P. Sitko, S.N. Yi, K.S. Yi, J.J. Quinn, Phys. Rev. Lett. 76 (1996) 3396. [8] E. Rezayi, N. Read, Phys. Rev. Lett. 72 (1994) 900.