A note contrasting two microscopic theories of the fractional quantum Hall effect

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.

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

$$\begin{aligned} \Psi ^{\rm LQP} = e^{ -\sum _j |z_j|^2/4} \left( \prod \limits _l 2 \frac{\partial }{\partial z_l} \right) \prod \limits _{j<k} (z_j - z_k)^m, \end{aligned}$$

(9)

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]

$$\begin{aligned}&\Psi ^{\rm CF-QP} = \exp \left[ -\frac{1}{4} \sum \limits _j |z_j|^2 \right] \times \nonumber \\&\mathcal{P}_{\rm LLL} \left| \begin{array}{ccc} \bar{z}_1 &{} \bar{z}_2 &{} \ldots \\ 1 &{} 1 &{} \ldots \\ z_1 &{} z_2 &{} \ldots \\ \vdots &{} \vdots &{}\ldots \\ z_1^{N-2} &{} z_2^{N-2} &{} \ldots \end{array} \right| \prod \limits _{j<k} (z_j - z_k)^{2p} \end{aligned}$$

(10)

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 ${\eta }_2$

$$\begin{aligned} e^{ -{1\over 4}\sum \nolimits _j |z_j|^2} \prod \limits _l \left( 2 \frac{\partial }{\partial z_l} -\bar{\eta }_1\right) \left( 2 \frac{\partial }{\partial z_l} -\bar{\eta }_2\right) \prod \limits _{j<k} (z_j - z_k)^m, \end{aligned}$$

(11)

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,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 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.

• 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.

• 3 Composite fermions and their vorticity are well defined over a broader region than their fractional $e^*$ and $\theta ^*$. We illustrate with some examples:

  • 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.

  • 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].

  • 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 $n/(2pn\pm 1)$.

  • 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.

• 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]).