VOLUME 63. NUMBER 2 PHYSICAL REVIEW LETTERS O JULY 1989 Composite-Fermion Approach for the fractional Quantum Hall Effect Section of Applied Physics, Yale University, P.O. Box 2157 Yale Station, New Haven, Connecticut 06520 (Received 24 January 198 m hal effect of the quasiparticle of the dargent state. in this appr esults from the fractional quan a new possible approach for ur derstanding the fractional quantum Hall effect is presented. It is proposed that the fractional quantum Hall effect of electrons can be physically understood as a manifestation of the integer quantum Hall effect of composite fermionic objects consisting of electrons bound to an even number of flux quanta. PACS numbers: 73. 50.Jt. 73.20 Dx Even though the experimental observations of the in- corresponds to magnetic field in the Fz direction) in teger' and the fractional quantum Hall effect(QHE) this situation is rather insensitive to the details of the in are essentially identical, except for the value of the quan- terelectron interactions and is determined mainly by vir tized Hall resistance there are, roughly speaking, three tue of the fermionicity of the electrons. Thus, the long different theoretical schemes for their explanation. range correlations due to the Fermi statistics provide ri While the integer QHE (IQHE)is thought of essentially gidity to the electron system at integer filling factors as a noninteracting electron phenomenon, 4 the fractional which results in the phenomenon of IQHE. It is useful QHE (FQHE) is believed to arise from a condensation to think in the path-integral language: The partition of the two-dimensional (2D) electrons into a"new col- function gets contributions from the closed paths in the lective state of matter"5 as a result of interelectron in- configuration space (for example, a path in which one teractions. Even within the Fqhe the "fundamental" electron moves in a loop while the others are held fixed, fractions play a special role and the other frac- or a cooperative ring exchange path ) The phase associ- tions are obtained in a hierarchical scheme in which a ated with each closed path has two contributions: the daughter state is obtained at each step from a condensa Aharonov-Bohm phase which depends on the flux en- tion of the quasiparticles of the parent state into a corre- closed in the loop and the statistical phase which de- lated low-energy state. pends on how many electrons participate in the path. An The purpose of this Letter is to present a theoretical incompressible state is obtained at integer filling factors framework which enables an understanding of both the presumably because of some special correlations (which IQHE and the FQHE in a unified scheme as two may not be easily identified) built in the phase factors different manifestations of the same underlying physics. corresponding to the various paths. Now attach to each It is argued that the possibility of QHE at fractional electron an infinitely thin magnetic solenoid carrying a filling factors p/(2mp+t1), where m and p are integers, flux apo(pointed in the -z direction). For lack of a arises because the correlations in the phase factors at better name, we term an electron bound to a flux tube a hese filling factors are very similar to the correlations composite particle. As is well known, the statistics present at integer filling factors p. This approach not of the composite particles is in general fractional, and is only gives all the observed fractions (except?3, which such that an exchange of two composite particles pro- therefore requires some additional physics ), and ex- duces a phase factor (-1)ta(Ref. 11). The relevant plains in doing so why only fractions with odd denomina- case here is when a is equal to an even integer (a=2m), tors are observed, but also provides the order of their sta- and the composite particles are fermions. It is easy bility, in agreement with experi it see that in this case the phase factor acquired along a suggests a generalization of the Laughlin wave functions given closed path is identical to the phase factor acquired to other fractions in the absence of the flux tubes, implying that the corre- I start by proposing a remarkably simple picture for lations in the phase factors for a=2m are the same as understanding the origin of the FQHE. The important those for a=0. Since these correlations are responsible parameter is the ratio of the total number of flux quanta for rigidity and QHE at integer filling factors, one can inverse of the filling factor v(in the thermodynamic lim- tained by adding to each electron in y+p a flux 2moo, to it)and specifies the average number of flux quanta avail- also be rigid and show QHE ble to each electron. Consider a 2D electron gas in the To determine the filling factor of ys we exploit an of a transverse magnetic field at an int ngenious observation due to Arovas et al. and Laugh filling factor v=p, so that there is an average flux po/p lin: A (uniform) liquid of electrons, each carrying per electron. The electronic wave function y+p(+ with it a flux ayo, is equivalent, in a mean field sense to @1989 The American Physical Society
VOLUME 63, NUMBER 2 PHYSICAL REVIEW LETTERS 10 JULY 1989 Composite-Fermion Approach for the Fractional Quantum Hall Effect J. K. Jain Section of Applied Physics, Yale University, P 0 Bo.x.2157 Yale Station, New Haven, Connecticut 06520 (Received 24 January 1989) In the standard hierarchical scheme the daughter state at each step results from the fractional quantum Hall effect of the quasiparticles of the parent state. In this paper a new possible approach for understanding the fractional quantum Hall effect is presented. It is proposed that the fractional quantum Hall effect of electrons can be physically understood as a manifestation of the integer quantum Hall effect of composite fermionic objects consisting of electrons bound to an even number of Aux quanta. PACS numbers: 73.50.Jt, 73.20.Dx Even though the experimental observations of the integer' and the fractional quantum Hall effect (QHE) are essentially identical, except for the value of the quantized Hall resistance, there are, roughly speaking, three different theoretical schemes for their explanation. While the integer QHE (IQHE) is thought of essentially as a noninteracting electron phenomenon, the fractional QHE (FQHE) is believed to arise from a condensation of the two-dimensional (2D) electrons into a "new collective state of matter" as a result of interelectron interactions. Even within the FQHE the "fundamental" fractions —, ' 5, . . . play a special role and the other fractions are obtained in a hierarchical scheme in which a daughter state is obtained at each step from a condensation of the quasiparticles of the parent state into a correlated low-energy state. The purpose of this Letter is to present a theoretical framework which enables an understanding of both the IQHE and the FQHE in a unified scheme as two diA'erent manifestations of the same underlying physics. It is argued that the possibility of QHE at fractional filling factors p/(2mp ~ 1), where m and p are integers, arises because the correlations in the phase factors at these filling factors are very similar to the correlations present at integer filling factors p. This approach not only gives all the observed fractions (except —', , which therefore requires some additional physics ), and explains in doing so why only fractions with odd denominators are observed, but also provides the order of their stability, in agreement with experiments. Furthermore, it suggests a generalization of the Laughlin wave functions to other fractions. I start by proposing a remarkably simple picture for understanding the origin of the FQHE. The important parameter is the ratio of the total number of flux quanta (po =bc/e) to the total number of electrons, which is the inverse of the filling factor v (in the thermodynamic limit) and specifies the average number of flux quanta available to each electron. Consider a 2D electron gas in the presence of a transverse magnetic field at an integer filling factor v=p, so that there is an average flux po/p per electron. The electronic wave function ++. ~ (+ corresponds to magnetic field in the + z direction) in this situation is rather insensitive to the details of the interelectron interactions and is determined mainly by virtue of the fermionicity of the electrons. Thus, the longrange correlations due to the Fermi statistics provide rigidity to the electron system at integer filling factors which results in the phenomenon of IQHE. It is useful to think in the path-integral language: The partition function gets contributions from the closed paths in the configuration space (for example, a path in which one electron moves in a loop while the others are held fixed, or a cooperative ring exchange path ). The phase associated with each closed path has two contributions: the Aharonov-Bohm phase which depends on the flux enclosed in the loop, and the statistical phase which depends on how many electrons participate in the path. An incompressible state is obtained at integer filling factors presumably because of some special correlations (which may not be easily identified) built in the phase factors corresponding to the various paths. Now attach to each electron an infinitely thin magnetic solenoid carrying a flux ago (pointed in the —z direction). For lack of a better name, we term an electron bound to a flux tube a "composite particle. " As is well known, ' the statistics of the composite particles is in general fractional, and is such that an exchange of two composite particles produces a phase factor (—1)'+' (Ref. 11). The relevant case here is when a is equal to an even integer (a =2m), and the composite particles are fermions. It is easy to see that in this case the phase factor acquired along a given closed path is identical to the phase factor acquired in the absence of the flux tubes, implying that the correlations in the phase factors for a=2m are the same as those for a=0. Since these correlations are responsible for rigidity and QHE at integer filling factors, one can expect the composite fermion state +~~, which is obtained by adding to each electron in @~~ a flux 2mpo, to also be rigid and show QHE. To determine the filling factor of +wz we exploit an ingenious observation due to Arovas et al. ' and Laughlin:'' A (uniform) liquid of electrons, each carrying with it a flux ago, is equivalent, in a mean fteld sense to 1989 The American Physical Society 199
Ⅴ OLUME63, NUMBER2 PHYSICAL REVIEW LETTERS 10JULY1989 a (uniform) liquid of electrons in a magnetic field of this implies possible stability at fractions n+p/ strength such that there is an average flux of aoo per (2mp±1)andn+1-p/(2mp±1) in the nth Landau electron. A uniform electron density is required to pro- level (LL). Notice that only fractions with odd denomi duce a uniform Alux density. Since in the state yp nators appear. In fact, in the present framework QHE at fractional values of v with odd denominators is as nat- we identify it with the mean-field state of electrons at ural as the Qhe at integer values of v. Besides explain mind that the true electron state is not as rigid as the dict the order of stability of the fractions, or the ordp. fractional filling v-p/(2mp+1). It must be borne in ing the"odd denominator rule, we are also able to composite fermion state, because in the true state the which new fractions should appear as the sample quality flux tubes are not strictly bound to the electrons, and the is improved. Since a collapse of the gap due to an"un phase factors simulate IQHE only on average. However, binding transition"is more likely for larger values of m, provided that the true electron state is also incompre s, served ctions p/(2m'pt 1)with m'<m must also be ob- te fermion state should correct description of the essential physics at the mean- served. One also expects weaker correlations for higher field level, as it contains the correlations giving rise to values of p. Thus in Fig. I a given fraction in the right the incompressibility. On the other hand, when the true (left)half is more stable than the one directly above electron state is not incompressible, identification of the and the one on its right (left). This is quite generally composite fermion state with the mean-field state of the borne out in experiments. 7.15-17 This also identifies the electrons is no longer valid, or meaningful in fractions to be observed next, if any, as the sample quali is approach there are two types of correla- ty is further improved. Read 8 has pointed out that the tions essential for FQHE. The first type of correlations, fractions obtained here are only the first level of a new hich have been widely appreciated in the field, 2, I3 in ierarchy, and all other fractions with odd denominators volve binding of electrons and zeros of the wave function, can be obtained within the present formalism.However or, equivalently, of electrons and flux tubes, which is a it is interesting to note that all the observed fractions, ex very useful way of incorporating the effect of repulsive cept i and $(Ref. 19), are obtained in this scheme at interactions. Thus the role of repulsive interactions in the very first level, which is to be contrasted with the the present framework is assumed to be to generate com- standard hierarchy in which one needs to go down many impart rigidity to the composite fermion system and thus lead to FQHe, are the correlations due to their Fermi statistics. These are included in the present scheme by mimicking the statistical correlations present in the noninteracting electron system at integer filling factors. This is the central idea of this work it is best summa rized by saying that the FQHE of electrons is a manifes tation of the IoHE of composite fermions. The Hall plateaus at fractional filling factors appear in his model precisely as at integer filling factors except for the trivial modification that now each electron carries with it 2m flux quanta. Following the argument of Laughlin and Halperin- consider geometry. The Hall resistance is related to the charge transported from one edge to the other as one flux quan- tum is adiabatically pierced through the center. At in- teger filling factor P, p electrons are transported across he sample in this process. For y2mp, as each electron carries 2m Aux quanta, one must supply 2mp additional flux quanta (in all 2mp t 1 Alux quanta)to transport p electrons across the sample. This gives RH=h/ve2 with v-p/(2mp+1). Just as in the IQHE, sample impuri- ties and inhomogeneities create localized states, which produce a quantized Hall plateau so long as the Fermi level lies in a mobility gap P/(2mp+1)and p/(2mp-1)are The stable fractional filling factors obtained in this filled circles and the left half, respectively. The ctions that have been observed in the manner are p/(2mp+ 1), and due to electron-hole sym etry, 1-p/(2mp+ 1). [As indicated by Haldane 8. I4 lowest LL. The predicted values of the next most stable frac tions at this level are shown near empty circles
VOLUME 63, NUMBER 2 PHYSICAL REVIEW LETTERS 10 JULY 1989 a (uniform) liquid of electrons in a magnetic field of strength such that there is an average flux of ago per electron. A uniform electron density is required to produce a uniform flux density. Since in the state P+'p there are a total of 2m ~p ' flux quanta per electron, we identify it with the mean-field state of electrons at fractional filling v=p/(2mp ~ 1). It must be borne in mind that the true electron state is not as rigid as the composite fermion state, because in the true state the flux tubes are not strictly bound to the electrons, and the phase factors simulate IQHE only on average. However, provided that the true electron state is also incompressible, the composite fermion state should provide the correct description of the essential physics at the meanfield level, as it contains the correlations giving rise to the incompressibility. On the other hand, when the true electron state is not incompressible, identification of the composite fermion state with the mean-field state of the electrons is no longer valid, or meaningful. Thus in this approach there are two types of correlations essential for FQHE. The first type of correlations, which have been widely appreciated in the field, ' involve binding of electrons and zeros of the wave function, or, equivalently, of electrons and flux tubes, which is a very useful way of incorporating the effect of repulsive interactions. Thus the role of repulsive interactions in the present framework is assumed to be to generate cornposite fermions. The second type of correlations, that impart rigidity to the composite fermion system and thus lead to FQHE, are the correlations due to their Fermi statistics. These are included in the present scheme by mimicking the statistical correlations present in the noninteracting electron system at integer filling factors. This is the central idea of this work; it is best summarized by saying that the FQHE of electrons is a manifes ration ofthe IQHE of composite fermions The Hall plateaus at fractional filling factors appear in this model precisely as at integer filling factors except for the trivial modification that now each electron carries with it 2m flux quanta. Following the argument of Laughlin and Halperin consider a corbino disk geometry. The Hall resistance is related to the charge transported from one edge to the other as one flux quantum is adiabatically pierced through the center. At integer filling factor p, p electrons are transported across the sample in this process. For ++p, as each electron carries 2m Aux quanta, one must supply 2mp additional flux quanta (in all 2mp ~1 flux quanta) to transport p electrons across the sainple. This gives RH=h/ve with v=p/(2mp+1). Just as in the IQHE, sample impurities and inhomogeneities create localized states, which produce a quantized Hall plateau so long as the Fermi level lies in a mobility gap. The stable fractional filling factors obtained in this manner are p/(2mp+ 1), and due to electron-hole symmetry, 1 —p/(2mp+ 1). [As indicated by Haldane ' 8 15 7 7 13 6 11 5 9 4 4 15 7 3 3 11 5 7 15 6 '13 5 11 9 3 7 3 13 2 2 11 7 2 '3 2 5 2 9 2 13 1 7 1 5 1 '3 1 '3 1 5 1 7 1 9 FIG. 1. The fractions p/(2mp+ I ) and p/(2mp —I) are shown in the right half and the left half, respectively. The filled circles show the fractions that have been observed in the lowest LL. The predicted values of the next most stable fractions at this level are shown near empty circles. this implies possible stability at fractions n +p/ (2mp+ 1) and n+1 —p/(2mp ~ 1) in the nth Landau level (LL).l Notice that only fractions with odd denominators appear. In fact, in the present framework QHE at fractional values of v with odd denominators is as natural as the QHE at integer values of v. Besides explaining the "odd denominator rule, " we are also able to predict the order of stability of the fractions, or the order in which new fractions should appear as the sample quality is improved. Since a collapse of the gap due to an "unbinding transition" is more likely for larger values of m, if a fraction p/(2mp ~ 1) is observed for a given p then all fractions p/(2m'p ~ 1) with m' (m must also be observed. One also expects ~eaker correlations for higher values of p. Thus in Fig. 1 a given fraction in the right (left) half is more stable than the one directly above it and the one on its right (left). This is quite generally borne out in experiments. ' ' This also identifies the fractions to be observed next, if any, as the sample quality is further improved. Read' has pointed out that the fractions obtained here are only the first level of a new hierarchy, and all other fractions with odd denominators can be obtained within the present formalism. However, it is interesting to note that all the observed fractions, except —, ', and —,', (Ref. 19), are obtained in this scheme at the very first level, which is to be contrasted with the standard hierarchy in which one needs to go down many 200
VOLUME 63, NUMBER 2 PHYSICAL REVIEW LETTERS 10JULY1989 gels in order to obtain some of the observed fractions. occupied single-particle states in each LL is N/p.Since It is also worth mentioning that the present scheme natu- the largest power of a z; in Zm is 2m(N-1), y has rally produces the experimentally observed sequences 2m(N-1)+Np single-particle states occupied in of fractions converging to (for m-1), to (for each LL, which immediately yields a filling factor 2), to 4(hole analog of 4) P/(2mp+1)in the thermodynamic limit. Thus the state In the following I will construct explicit trial wave yi(unlike ptp) satisfies the fundamental requirement functions, analogous to the Laughlin wave functions, that the filling factor obtained by counting the total which have the correlations discussed above. The Ham- number of states agrees with that obtained from the iltonian for N noninteracting electrons (N-oo)at Alux-counting argument (i. e the number of flux quanta +eA, /c), where A; is chosen so as to produce a un? piercing the sample is equal to the number of single- form magnetic field in the +z direction of strength such partially projects the single-particle states of the higher that there is an average flux of p po per electron. The LLs into the lowest LL. Write corresponding ground-state wave functions are y+p with y-p=Y+p. We first consider the fractions p/(2mp+1) Az2m∏∏5-1(z1+) hich are obtained by starting from y+p. Gauge flux tubes 2moo are attached to each electron by adding to the vector potential A a singular gauge potential 0, II (2n2+11)-12-112.-n2, 2 VOjk, where A is the anitsymmetrization operator, Sl,s are the single-particle states, 7=0, 1 is the LL ind where Ojk is defined by (zj -z)-1z-Ik lexp(ie/k), and s-O,., N/p-1 is the angular momentum quan and zj=;+iy denotes the position (xj, yj) of the jth tum number. Zm is a sum of terms of type Ilf-1z7 particle as a complex number. The new ground-state with 2i t -mN(N-1), where t, is typically a large wave function is! power (in the thermodynamic limit) of order mN. Thus, in each term of y 4p the coordinate zi of a particle ap- 0-12-x1my+ pears as the product z, S,s(z,). For ty of order mN, this product lies almost entirely in the lowest LL. Expanding Clearly this is not an appropriate wave function for it as a sum of single-particle states, describing the FQHE of electrons. Following the analo- gy of the Laughlin wave functions, and for the reasor z/5, ak5+1+k-1, related (unnormalized trial wave function one can show that the ratio ak +ilak is of order 1/vN e,, the amplitude of z, 5i, s (zy is smaller by a factor of order I/vN in each successively higher LL. Thus in here Z=ll <k(z -zk)a This wave function has the yi p the amplitude is expected to be in general much same topological structure as dm, and also describes larger for the terms which have a greater number of the electrons carrying gauge flux tubes of strength 2moo. I lowest LL states occupied. Furthermore, there are mani this state addition of flux tubes is accompanied by a festly terms, with extremely large amplitudes, which change in the size of the system in such a way as to keep have only the lowest LL occupied. This implies that, un he total flux per unit area (i. e, the magnetic field)con- less there are some very strange cancellations, the state w2 lies predominantly in the lowest LL in the thermo This state has the following properties: (i)For p=1, dynamic limit. (vi)Last, y i p is expected to be a good w2? is identical with the corresponding Laughlin state. variational state in the presence of repulsive interactions, (ii)Since V+p is determined almost completely by the because both the factors Zm and Y+p are very efficient Pauli principle, and has little dependence on interelec- in keeping the electrons apart. This is a direct sense in tron interactions, y*p is also largely insensitive to the in- which the correlations of the higher LL are utilized to teractions. This is explicitly the case for the Laughlin obtain a low-energy state. Thus we believe that the states which have been found to be very accurate for a states y p possess all the necessary properties of a variety of interelectron interactions. (i) It describes an reasonable trial state. At present we are working to- electron gas of uniform density. This follows straightfo wards a quantitative test, which is complicated due to wardly from the fact that y+p describes an electron gas the complex structure and the inherent thermodynamic of uniform density. o an eigenstate of the angu- nature of these states. The form of the incompressible ar momentum. (iv) One can read off the filling factor state at v-p/(2mp-1), which is obtained starting from from the wave function. Take y +p with p LL's com- y-p, is not as obvious. pletely occupied in a disk-shaped region; the number of Normally the ground state is expected to be complete
VOLUME 63, NUMBER 2 PHYSICAL REVIEW LETTERS 10 JULY 1989 levels in order to obtain some of the observed fractions. It is also worth mentioning that the present scheme naturally produces the experimentally observed sequences ' of fractions converging to —, ' (for m=1), to —, ' (for m =2), to —, ' (hole analog of —, ' ), etc. In the following I will construct explicit trial wave functions, analogous to the Laughlin wave functions, which have the correlations discussed above. The Hamiltonian for N noninteracting electrons (N ~ ~) at filling factor p is given by Ho =Pj.=i (2m, ) ' (pj +eAj/c), where Aj is chosen so as to produce a uniform magnetic field in the T- z direction of strength such that there is an average flux of p '&0 per electron. The corresponding ground-state wave functions are W ~~ with We first consider the fractions p/(2mp+1) which are obtained by starting from ++~. Gauge flux tubes 2m&0 are attached to each electron by adding to the vector potential Ai a singular gauge potential' " i gi, =(2ir2'+'lis!) ' e!' 2 s —iz i'/2 occupied single-particle states in each LL is N/p. Since the largest power of a zj in Z is 2m(N —1), +~~ has 2m (N —1)+Np ' single-particle states occupied in each LL, which immediately yields a filling factor p/(2mp+1) in the thermodynamic limit. Thus the state ++p (unlike @+„)satisfies the fundamental requirement that the filling factor obtained by counting the total number of states agrees with that obtained from the flux-counting argument (i.e., the number of flux quanta piercing the sample is equal to the number of singleparticle states in each LL). (v) The factor Z in ++~ partially projects the single-particle states of the higher LL's into the lowest LL. Write A, = —2m 2z p(I ~, ) where Oik is defined by (zj —zk) = izj —zk i exp(i8jk), and zj =xj+iyj denotes the position (xj,yj) of the jth particle as a complex number. The new ground-state wave function is ' ' (zj —zk)' Clearly this is not an appropriate wave function for describing the FQHE of electrons. Following the analogy of the Laughlin wave functions, and for the reasons mentioned below, we write instead the following closely related (unnormalized) trial wave function @2m+p Z2m@ +p where Z' =Qj & k (zi —zk ) '. This wave function has the same topological structure as ++~, and also describes electrons carrying gauge flux tubes of strength 2m&0. In this state addition of flux tubes is accompanied by a change in the size of the system in such a way as to keep the total flux per unit area (i.e., the magnetic field) constant. This state has the following properties: (i) For p =1, ++i is identical with the corresponding Laughlin state. (ii) Since ++~ is determined almost completely by the Pauli principle, and has little dependence on interelectron interactions, ++~ is also largely insensitive to the interactions. This is explicitly the case for the Laughlin states which have been found to be very accurate for a variety of interelectron interactions. (iii) It describes an electron gas of uniform density. This follows straightforwardly from the fact that ++~ describes an electron gas of uniform density. It is also an eigenstate of the angular momentum. (iv) One can read off the filling factor from the wave function. Take ++~ with p LL's completely occupied in a disk-shaped region; the number of where A is the anitsymmetrization operator, gi, are the single-particle states, l =0, . . . ,p —1 is the LL index, and s=O, . . . ,N/p —1 is the angular momentum quantum number. Z2 is a sum of terms of type Qj ——izji' with Pitj =mN(N 1), where—tj is typically a large power (in the thermodynamic limit) of order mN. Thus, in each term of I+p the coordinate zj of a particle appears as the product zPgi, (zj). For tj of order rnN, this product lies almost entirely in the lowest LL. Expanding it as a sum of single-particle states, fj Zj gi, s X ak4k, + is+kS—i i k=0 one can show that the ratio akil/ak is of order I/JN; i.e., the amplitude of zji'gi, (zj) is smaller by a factor of order I/JN in each successively higher LL. Thus in +~~ the amplitude is expected to be in general much larger for the terms which have a greater number of the lowest LL states occupied. Furthermore, there are manifestly terms, with extremely large amplitudes, which have only the lowest LL occupied. This implies that, unless there are some very strange cancellations, the state ++~ lies predominantly in the lowest LL in the thermodynamic limit. (vi) Last, ++~ is expected to be a good variational state in the presence of repulsive interactions, because both the factors Z and ++~ are very efficient in keeping the electrons apart. This is a direct sense in which the correlations of the higher LL are utilized to obtain a low-energy state. Thus we believe that the states ++~ possess all the necessary properties of a reasonable trial state. At present we are working towards a quantitative test, which is complicated due to the complex structure and the inherent thermodynamic nature of these states. The form of the incompressible state at v p/(2mp —1), which is obtained starting from +—~, is not as obvious. Normally the ground state is expected to be complete- 201
VolUMe 63. NUMBER 2 PHYSICAL REVIEW LETTERS 10 JULY 198 ly spin polarized, and is obtained by choosing in y +p the IK. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett p lowest LL's with the same spin orientation. However, 45,494(1980) g Is D. C. Tsui. H. L. St to consider situations in which y+e has LL's with both Lett. 48, 1559(1982) spin orientations occupied, so that in y p the two lowest SFor a review, see The Quantum Hall Efect, edited by R.E spin-split Landau bands are occupied. Thus, for small Prange and s. M. Girvin(Springer-Verlag, New York, 1987) spin splitting, there are in general many candidates for 4R. B. Laughlin, Phys. Rev. B 23, 5632(1981);B IHalpe the incompressible state rin, Phys. Rev. B 25, 2185(1982) completely spin polarized states, spin unpolarized SR. B Laughlin, Phys. Rev. Lett. 50, 1395(1983) 6F. D. M. Haldane, Phys. Rev. Lett. 51, 605(1983)B states, 4 and partially spin polarized states There seems to be a close analogy between the com- Surf. Sci. 141, 11(1984). and in Ref. 3. There have also been posite fermion states proposed in this paper and the stan nonhierarchical states proposed for certain higher-level frac- dard hierarchical states. To illustrate this we consider tions: see.A.H.MacDonald C. Aers, and M.w. C. the example of the state which is obtained by multi- Dharma-wardana, Phys. Rev. B 31, 5529(1985);Y. Yoshioka, plying y+I by Z. It can be shown that the state Y+ A. H. MacDonald, and S. M. Girvin, Phys. Rev. B 38, 3636 with one hole similarly produces the state with a (1988): Phys. Rev. B 39, 1932(1989);R. Morf, N. D'Ambru quasielectron would be obtained by me state with a menil, and B. I.Halperin, Phys. Rev. B 34, 3037(1986) he state with the lowest (I-0)LL fully occupied and 8F. D M. Haldane and E. H. Rezay Phys. Rev. Lett. 60 956(1988) one electron in the I-1 LL. The state with fully occu 9S. Kivelson, C. Kallin, D. P. Arovas, and J.R.Schrieffer pied lowest LL and 8 electrons in the /l LL then cor- Phys. Rev. Lett. 56, 873(1986) responds to the state with 8 quasielectrons. Thus the 10F. wilczek, Phys. Rev. Lett.49, 957(1982);F,Wilczek s state can be viewed in the present approach as the and A. Zee, Phys. Rev. Lett. 51, 2250(1983);D.P.Arovas,J state with N/2 quasielectrons, and similarly the, state R. SchriefFer, F. wilczek, and A. Zee, Nucl. Phys. B251,117 can be viewed as the 3 state with N/3 quasielectrons This assignment is in exact agreement with that of the IR. B. Laughlin, Phys. Rev. Lett. 60, 2677(1988) standard hierarchy theory. One can also show that the 12B. 1. Halperin, Helv. Phys. Acta. 56, 75(1983) quasiparticles described above have the sam 13S. M. Girvin and A. H. MacDonald, Phys. Rev.Lett.5 e char those in the standard scheme. The analogy is, howev- 1252(1987); E H. Rezayi and F. D M. Haldane, Phys. Rev er,not complete. In the standard picture stability is ob- Kivelson, Phys. Rev. Lett. 62, 82(1989): N Read, Phys. Rev tained when the quasielectrons form a Laughlin-type Lett 62, 86(1989) state, whereas in the composite-fermion scheme the 14F D. M. Haldane in ref. 3 derive their arrangement from the higher LL. Also, tak ISR. willett et al., Surf. Sci. 196, 257(1988); A.M.Chang ing the above example, in the standard picture one aL, Phys. Rev, Lett, 53, 997(1984) obtain both the, and the 3 states from the state 6R.G. Clark et al,Sur.Sci.196,219(1988);196,257 whereas the composite-fermion approach does not yield (1988) the experimentally unobserved i3 state at this level r7V. J. Goldman, M. Shayegan, and D. C. Tsui, Phys.Rev In conclusion, this paper proposes that the FQhe car Lett.61,881(1988). 18N Read(private communication be accessed from the IQHE by adding an even number 19v. J. Goldman, D. C.Ts ui, and M. Shayegan, in Proceed of flux quanta to each electron. This analogy between ings of the Nineteenth International Conference on the Physics FQHE and IQHE suggests a natural generalization of of Semiconductors Warsaw, Poland. 1988(World Scientific the Laughlin states. Singapore, to be published) I thank N. Read, D. Stone, S. Kivelson, A. Mac- 20R. G. Clark et al., Phys. Rev. Lett. 62, 1536(1989);J.P Donald, A. Chang, V. Goldman, and H. Stormer for Eisenstein et al., ibid. 62, 1540(1989) helpful conversations 2 J. K. Jain(to be published)
VOLUME 63, NUMBER 2 PHYSICAL REVIEW LETTERS IO JULY 1989 ly spin polarized, and is obtained by choosing in + w~ the p lowest LL's with the same spin orientation. However, when the spin splitting is insignificant, it may be useful to consider situations in which y~~ has LL's with both spin orientations occupied, so that in ++~ the two lowest spin-split Landau bands are occupied. Thus, for small spin splitting, there are in general many candidates for the incompressible state' ' for a given fraction: the completely spin polarized states, spin unpolarized states, ' ' and partially spin polarized states. There seems to be a close analogy between the composite fermion states proposed in this paper and the standard hierarchical states. To illustrate this, we consider the example of the 3 state which is obtained by multiplying +~~ by Z . It can be shown ' that the state @+1 with one hole similarly produces the 3 state with a Laughlin quasihole. By analogy, the 3 state with a quasielectron would be obtained by multiplying by Z the state with the lowest (l=0) LL fully occupied and one electron in the I=1 LL. The state with fully occupied lowest LL and 8 electrons in the I=I LL then corresponds to the 3 state with 6 quasielectrons. Thus the state can be viewed in the present approach as the state with N/2 quasielectrons, and similarly the 7 state can be viewed as the —' , state with N/3 quasielectrons. This assignment is in exact agreement with that of the standard hierarchy theory. One can also show that the quasiparticles described above have the same charge as those in the standard scheme. ' The analogy is, however, not complete. In the standard picture stability is obtained when the quasielectrons form a Laughlin-type state, whereas in the composite-fermion scheme they derive their arrangement from the higher LL. Also, taking the above example, in the standard picture one can obtain both the 7 and the, '& states from the 5 state, whereas the composite-fermion approach does not yield the experimentally unobserved —,', state at this level. In conclusion, this paper proposes that the FQHE can be accessed from the IQHE by adding an even number of flux quanta to each electron. This analogy between FQHE and IQHE suggests a natural generalization of the Laughlin states. I thank N. Read, D. Stone, S. Kivelson, A. MacDonald, A. Chang, V. Goldman, and H. Stormer for helpful conversations. 'K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980). 2D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982). 3For a review, see The Quantum Hal! Effect, edited by R. E. Prange and S. M. Girvin (Springer-Verlag, New York, 1987). 4R. B. Laughlin, Phys. Rev. B 23, 5632 (1981);B. I. Halperin, Phys. Rev. B 25, 2185 (1982). sR. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). 6F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983); B. I. Halperin, Phys. Rev. Lett. 52, 1583 (1984); R. B. Laughlin, Surf. Sci. 141, 11 (1984), and in Ref. 3. There have also been nonhierarchical states proposed for certain higher-level fractions; see, A. H. MacDonald, G. C. Aers, and M. W. C. Dharma-wardana, Phys. Rev. B 31, 5529 (1985); Y. Yoshioka, A. H. MacDonald, and S. M. Girvin, Phys. Rev. B 38, 3636 (1988); Phys. Rev. B 39, 1932 (1989); R. Morf, N. D'Ambrumenil, and B. I. Halperin, Phys. Rev. B 34, 3037 (1986). 7R. Willett et al. , Phys. Rev. Lett. 59, 1776 (1987). 8F. D. M. Haldane and E. H. Rezayi, Phys. Rev. Lett. 60, 956 (1988). S. Kivelson, C. Kallin, D. P. Arovas, and J. R. SchrieAer, Phys. Rev. Lett. 56, 873 (1986). 'OF. Wilczek, Phys. Rev. Lett. 49, 957 (1982); F. Wilczek, and A. Zee, Phys. Rev. Lett. 51, 2250 (1983); D. P. Arovas, J. R. Schrieff'er, F. Wilczek, and A. Zee, Nucl. Phys. B251, 117 (1985). ''R. B. Laughlin, Phys. Rev. Lett. 60, 2677 (1988). '2B. I. Halperin, Helv. Phys. Acta. 56, 75 (1983). ' S. M. Girvin and A. H. MacDonald, Phys. Rev. Lett. 58, 1252 (1987); E. H. Rezayi and F. D. M. Haldane, Phys. Rev. Lett. 61, 1985 (1988); S. C. Zhang, T. H. Hansson, and S. Kivelson, Phys. Rev. Lett. 62, 82 (1989); N. Read, Phys. Rev. Lett. 62, 86 (1989). '4F. D. M. Haldane, in Ref. 3. '5R. Willett er al. , Surf. Sci. 196, 257 (1988); A. M. Chang et al. , Phys. Rev. Lett. 53, 997 (1984). '6R. G. Clark et al. , Surf. Sci. 196, 219 (1988); 196, 257 (1988). ' V. J. Goldman, M. Shayegan, and D. C. Tsui, Phys. Rev. Lett. 61, 88 1 (1988). 'sN. Read (private communication). ' V. J. Goldman, D. C. Tsui, and M. Shayegan, in Proceedings of the Nineteenth International Conference on the Physics of Semiconductors Warsaw, Poland, 1988 (World Scientific, Singapore, to be published). 2oR. G. Clark et al. , Phys. Rev. Lett. 62, 1536 (1989);J. P. Eisenstein er al. , ibid 62, 1540 (198.9). 2'J. K. Jain (to be published). 202
