Abstract
Two microscopic theories have been proposed for the explanation of the fractional quantum Hall effect, namely the Haldane–Halperin hierarchy theory and the composite fermion theory. Contradictory statements have been made regarding the relation between them, ranging from their being distinct to their being completely equivalent. This article attempts to provide a clarification of the issue. It is shown that the two theories postulate distinct microscopic mechanisms for the origin of the fractional quantum Hall effect, and make substantially different predictions that have allowed experiments to distinguish between them.
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Notes
For example, it predicts universal behavior for the FQHE edge, which has been a subject of much experimental investigation [30].
The fractions \(1/(2p+1)\), which lie at the boundary separating regions spanned by composite fermions with different vorticity, have a dual interpretation: either as filling factor \(1\) of composite fermions carrying \(2p\) vortices, or as filling factor “\(-1\)” of composite fermions carrying \(2(p+1)\) vortices. Both produce similar wave functions, but the values of the gaps suggest the former interpretation as being more natural. That is the reason why the sequences \(n/(2pn -1)\) begin with \(n=2\) in Fig. 2.
To see that a Laughlin-type FQHE state is far from automatic, it may be noted that even the interaction between electrons fails to establish \(1/3\) FQHE in the third and higher Landau levels, and is barely able to do so in the second Landau level, because the wave function of a localized electron in these Landau levels is slightly more extended than that in the lowest Landau level. As another reference, the interaction between composite fermions in the second, third or higher CF Landau levels has been evaluated and found to be weak and attractive at short distances [41, 42], and does not seem to stabilize a Laughlin-like state [43, 44].
For example, the pair correlation function of the \(n/(2pn+ 1)\) FQHE states exhibits \(2k_F\) oscillations; see [75]
An earlier approach for the neutral excitation, based on a single-mode approximation, successfully predicted the “magnetoroton” minimum at 1/3. See, [83]
The agreement with experiments is less accurate than that with exact diagonalization results because the experimental numbers are also affected by finite width corrections, Landau level mixing and disorder, for which our understanding is not as precise.
As a reference, the energy of two electrons in the lowest Landau level in the state with unit relative angular momentum is \(\sim 0.44\,e^2/\epsilon l\).
See footnote 7.
Ref. [38] has compared the HH hierarchy wave functions with the 2/5 Coulomb ground state for 6 and 8 particles, and with the 3/7 Coulomb ground state for 6 particles. These systems are too small for a convincing test of the wave functions, as they have only 3, 8 and 2 uniform \(L=0\) eigenstates, respectively. (In contrast, the 16 particle 2/5 system and the 18 particle 3/7 system shown in Fig. 5 have, respectively, 70180 and 159018 eigenstates in the \(L=0\) sector.) Furthermore, the actual Coulomb state of 6 particles at total flux 9, which was assigned the filling factor 3/7 in this work, is more appropriately viewed as the 6 particle 2/3 state, and is accurately represented by the hole counterpart of the 4 particle 1/3 state. This illustrates how studies of small systems can fail to discriminate between wave functions based on very different physics, necessitating studies of larger systems and comparisons with experiments for a more substantive test of the theory.
The fact that the CF and the HH hierarchy theories produce the same values for \(e^*\) and \(\theta ^*\), despite their different physics for the origin of the FQHE, is less surprising than it might at first seem. The mere assumption of incompressibility at a given fractional filling \(\nu \) imposes stringent constraints on \(e^*\) and \(\theta ^*\) without regard to the mechanism responsible for incompressibility, and both the CF and the HH hierarchy theories obtain the simplest allowed values for them. The logic is as follows. Laughlin’s method of adiabatic flux insertion at some location in an incompressible state at filling factor \(\nu \) produces a localized object with fractional charge excess of \(\nu \) (in units of the electron charge), and one can give strong arguments that the bound state of a fractional charge \(\nu \) and a unit flux quantum obeys fractional braid statistics [4]. (Note that this derivation of fractional charge by adiabatic flux insertion does not depend on the origin of incompressibility.) In general the charge \(\nu \) object is not an elementary excitation, however. Requiring that an integer number of elementary excitations can be combined to produce a charge \(\nu \) excitation and another integer number of them produce an electron imposes further constraints on the allowed values of \(e^*\) and \(\theta ^*\) of the elementary excitations. For the fractions \(\nu =n/(2pn\pm 1)\) this shows that the allowed values of local charge are \(e^*=1/[r(2pn\pm 1)]\), where \(r\) is an integer; integer multiples of this can produce both \(\nu \) and 1. This closely follows Ref. [90], which also contains a discussion of general fractions. The CF and HH hierarchy theories obtain the simplest allowed values: \(e^*=1/(2pn\pm 1)\) and the corresponding \(\theta ^*=2p/(2pn\pm 1)\).
See footnote 2
The vortices are topological because a closed loop around a vortex produces a phase of \(2\pi \) independent of the size or the shape of the loop.
The assertion (e.g. see the articles in Refs. [24–26]) that the CF theory is a special case of, or can be derived from, the HH hierarchy theory is not true. Briefly, these articles proceed by assuming composite fermions and their CF Landau levels, and fill successive CF Landau levels to obtain the CF-IQHE states with wave functions given precisely by Eq. 7; to claim that this physics was contained in the HH hierarchy theory is factually incorrect.
See footnote 11
See footnote 11
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Acknowledgments
I am grateful to H. L. Stormer for permission to use Fig. 3 and for the list of observed fractions given in Fig. 4. I am indebted to B. I. Halperin for detailed and insightful comments on an earlier version of this manuscript, which have been incorporated into the article, and for his kind encouragement. I also thank J. R. Banavar, S. Das Sarma, G. Murthy, D. S. Weiss, A. Wójs, and Y.-H. Wu for useful discussions in this context. Financial support from DOE under Grant No. DE-SC0005042 is gratefully acknowledged.
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Appendices
Appendix 1: Composite fermions \(\ne \) Laughlin quasiparticles
The HH hierarchy and the CF theories have one common point, namely the Laughlin ground state at \(\nu =1/m\). This Appendix shows that the two begin to diverge as soon as the filling factor is increased, and that the divergence reflects a structural difference between the two theories. In particular, creating Laughlin quasiparticles (LQPs) results in states with nonzero amplitude in up to very high CF Landau levels.
The trial wave function for a single LQP at the origin for \(\nu =1/m\,(m=2p+1)\) is given by
The trial wave function for a single composite fermion in the second CF Landau level at the origin, labeled “CF quasiparticle” (CF-QP), is given by [106]
where the determinant is the wave function with one particle in the second Landau level, and the lowest Landau level projection is accomplished by the replacement \(\bar{z}_j\rightarrow 2\partial /\partial z_j\). The wave functions \(\Psi ^{\rm LQP}\) and \(\Psi ^{\rm CF-QP}\) are not identical; explicit calculation for Coulomb interaction in the lowest Landau level has shown [107] that the CF quasiparticle in Eq. 10 has \(\sim 15\, \%\) lower energy than the LQP of Eq. 9.
Is this quantitative difference an indication of a qualitatively different underlying structure? To gain an insight into this question let us analyze the LQP from the perspective of the CF theory. In the CF theory, the wave function of a composite fermion in the \((n+1)\)th CF Landau level contains derivatives with powers up to \([\partial /\partial z_j]^n\). The CFQP in Eq. 10 has precisely one composite fermion in the second CF Landau level, and none in higher ones. The LQP in Eq. 9 has no composite fermions in the third and higher CF Landau levels, but it has a non-zero probability of containing many composite fermions in the second CF Landau level. That the difference between the LQPs and composite fermions is qualitative and structural becomes indisputable as more LQPs are created. The state with two LQPs at \({\eta }_1\) and \({\eta }_2\)
has composite fermions occupying the third CF Landau level as well. (For nearby LQPs, the energy of this state is substantially higher than that of the state with two nearby composite fermions occupying the second CF Landau level [107].) Analogously, for \(N_{\rm LQP}\) LQPs, the wave function \(\Psi ^{\rm LQP}_{1/m}(\mathbf{r}_j ; \varvec{\eta }_{\mu } )\) in Eq. 3 has a nonzero occupation of the lowest \(N_{\rm LQP}+1\) CF Landau levels. Adding LQPs is thus very different from filling the second CF Landau level. To reach the 2/5 daughter, \(N_{\rm LQP}=N/2\) LQPs must be created, which produces a state that has amplitude extending up to \(\sim N/2\) excited CF Landau levels. This is to be contrasted with the CF description of the 2/5 state as lowest two filled CF Landau levels.
Appendix 2: Composite-fermion vorticity
A distinctive feature of the CF theory is to identify a new topological quantum number, namely the CF vorticity, which is intrinsic to the definition of the composite fermions themselves and manifests through the effective field \(B^*=B-2p\rho \phi _0\). The following facts demonstrate that the composite fermions and their vorticity are more general than the local charge and braid statistics (\(e^*\) and \(\theta ^*\)) of the excitations.
-
1.
We first note that we can derive \(e^*\) and \(\theta ^*\) from composite fermions [11]. One may ask: “The existence of objects with fractional \(e^*\) and \(\theta ^*\) can be deduced from general principles,Footnote 15 but what are these objects?” The CF theory tells us what they really are: they are isolated composite fermions in an otherwise empty CF Landau level or missing composite fermions in an otherwise filled CF Landau level. These are sometimes referred to as “CF quasiparticles” and “CF quasiholes” when viewed relative to the uniform \(\nu ^*=n\) “vacuum.” This description has been demonstrated to give a precise account of the excitations of all \(n/(2pn\pm 1)\) FQHE states – see, e.g., Refs. [11, 77]. (Laughlin’s “quasihole” of the \(1/m\) state is identical to the CF quasihole for this state. The “quasielectrons” for \(\nu =1/m\) in Refs. [108–110], or for the other CF states in Ref. [24, 25], also precisely match those of the CF theory.) The local charge (i.e. charge excess relative to the uniform “vacuum” FQHE state) of a CF quasiparticle at \(\nu =n/(2pn\pm 1)\) is the sum of the charges of an electron and \(2p\) vortices; recognizing that the charge of a vortex is simply \(\nu \), the local charge is given by \(e^*=-1+2p\nu =-1/(2pn\pm 1)\). A Berry phase calculation [111, 112] shows that the braid statistics of the CF quasiparticles is well defined provided their spatial overlap is negligible, and is given by the product of the vorticity and the local charge, \(\theta ^*=2p\times {1\over 2pn\pm 1}\) (see Ref. [11] for further details). Both \(e^*\) and \(\theta ^*\) thus inherit their quantized values from the quantized CF vorticity \(2p\).
The existence of composite fermions and their vorticity, on the other hand, does not follow from the knowledge of \(e^*\) and \(\theta ^*\). In particular, the general arguments outlined in Footnote Footnote 16 that give us \(e^*\) and \(\theta ^*\) for a given \(\nu \) do not give any indication of the existence of composite fermions. The CF theory thus contains whatever follows from \(e^*\) and \(\theta ^*\), but also much that does not.
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2.
Viewed solely through their \(e^*\) and \(\theta ^*\) quantum numbers, as would be the case if we did not know about composite fermions, it would seem that the excitations of different FQHE states are fundamentally distinct, producing \(\sim 70\) distinct particles. The CF theory reveals that they are all the same. Furthermore, they are also identical to the particles forming the ground states. The same composite fermions are used to build the ground states, the charged excitations, the neutral excitations, and multiple excitations for all states of the form \(n/(2pn\pm 1)\) with a given \(2p\). Instead of \(\sim 70\) fractionally charged anyons [35], it is thus sufficient to work with only a few flavors of composite fermions with different vorticity. Different values of \(e^*\) and \(\theta ^*\) occur simply because the charge of a vortex (\(\nu \)) depends on the filling factor of the background FQHE state.
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3
Composite fermions and their vorticity are well defined over a broader region than their fractional \(e^*\) and \(\theta ^*\). We illustrate with some examples:
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Composite fermions in a filled CF Landau level do not have any \(e^*\) or \(\theta ^*\), as they are part of the “vacuum”. Their vorticity is well defined, however – the resulting \(B^*\) is what gave us the filled CF Landau level state in the first place.
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Imagine only a few composite fermions in a CF Landau level, i.e. a few CF quasiparticles. They have well defined local charge \(e^*\) and braid angle \(\theta ^*\), but only provided they have negligible spatial overlap with one another. Explicit Berry phase calculations [111, 112] for the CF quasiparticles of the 1/3 and 2/5 FQHE states show that they must be farther than \(\sim 10\) magnetic lengths in order for \(\theta ^*\) to have a well defined value. For other FQHE states the CF quasiparticles have even larger sizes, requiring larger separations to ensure a well defined \(\theta ^*\). Furthermore, the Landau level mixing, always present, introduces corrections to \(\theta ^*\) that decay only as a power law in the distance between the quasiparticles [113]. Detailed calculations have also demonstrated that the interaction between the CF quasiparticles is weak and often attractive at short distances [42], implying that there exists no energy barrier keeping them far apart from one another.
In contrast, the description in terms of CF quasiparticles remains well defined and accurate even when they are nearby and overlapping. This is demonstrated by the accuracy of the CF theory in describing even small systems containing multiple CF quasiparticles and / or CF quasiholes [11, 77].
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As we start populating a CF Landau level with more composite fermions, at some point, it is not possible, even in principle, to keep all CF quasiparticles away from one another, and \(e^*\) and \(\theta ^*\) cease to be meaningful quantum numbers. However, composite fermions and their \(B^*\) remain sharply defined all the way to the filled CF Landau level state, and beyond. It is thus the vorticity (or \(B^*\)) and the exchange statistics of composite fermions that are responsible for incompressibility and FQHE at \(n/(2pn\pm 1)\).
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Last, the vorticity of composite fermions manifests itself, through an effective \(B^*\), also in compressible regions (e.g. the 1/2 CF Fermi sea), which cannot support, even as a matter of principle, excitations with well defined local charge and braid statistics.
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4
One can ask what relevance these quantum numbers have to experiments. Many of the experimental facts discussed in Sect. 4 are direct consequences of \(B^*\) and hence of the CF vorticity. The vorticity of composite fermions has been determined directly also in experiments that measure the cyclotron orbits of the objects responsible for transport [60–69] which are seen to correspond to the effective field \(B^*\). Shot noise and interference experiments have been designed for detecting the \(e^*\) and \(\theta ^*\) quantum numbers of the excitations (e.g., Refs. [98, 114]).
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Jain, J.K. A note contrasting two microscopic theories of the fractional quantum Hall effect. Indian J Phys 88, 915–929 (2014). https://doi.org/10.1007/s12648-014-0491-9
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DOI: https://doi.org/10.1007/s12648-014-0491-9