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 SocietyVOLUME 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 quan￾tum Hall effect of the quasiparticles of the parent state. In this paper a new possible approach for un￾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 Aux quanta. PACS numbers: 73.50.Jt, 73.20.Dx Even though the experimental observations of the in￾teger' and the fractional quantum Hall effect (QHE) are essentially identical, except for the value of the quan￾tized 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 col￾lective state of matter" as a result of interelectron in￾teractions. Even within the FQHE the "fundamental" fractions —, ' 5, . . . play a special role and the other frac￾tions are obtained in a hierarchical scheme in which a daughter state is obtained at each step from a condensa￾tion of the quasiparticles of the parent state into a corre￾lated 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 ex￾plains in doing so why only fractions with odd denomina￾tors are observed, but also provides the order of their sta￾bility, 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 lim￾it) and specifies the average number of flux quanta avail￾able 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 in￾terelectron interactions and is determined mainly by vir￾tue of the fermionicity of the electrons. Thus, the long￾range correlations due to the Fermi statistics provide ri￾gidity 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 associ￾ated with each closed path has two contributions: the Aharonov-Bohm phase which depends on the flux en￾closed in the loop, and the statistical phase which de￾pends 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 pro￾duces 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 corre￾lations 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 ob￾tained 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 Laugh￾lin:'' 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