Fractional charge and fractional statistics in the quantum Hall effects

Whereas fractional charge can be easily defined for a single isolated quasiparticle or for a collection of localized quasiparticles, the concept of fractional statistics requires the consideration of two or more quasiparticles that are able to move around each other or to interchange positions. If one is confined to a suitable low-energy subspace, one may hope to describe the quantum mechanical state of such a system by an effective wave function ψ_eff that depends only on the coordinates of the quasiparticles, rather than of all the electrons in the system. The effective wave function should evolve in time according to a Schrödinger equation with some effective Hamiltonian H_eff. Fractional statistics will be a characteristic of the combination ψ_eff and H_eff.

As was first noted by Leinaas and Myrrheim, in 1977, in two dimensions it is possible to extend the formulation of quantum mechanics to a situation where the wave function of a set of identical particles is multiplied by a complex phase factor different from ±1, provided we may exclude from consideration points where two quasiparticles coincide precisely in space [3]. Specifically, one may require that if one interchanges the positions of two identical particles by moving their coordinates in a counterclockwise direction along a closed contour C that encloses N_C other quasiparticles of the same type, the wave function should be multiplied by a phase factor e^−iθ , where

$$\begin{equation}\theta =\left(1+2{N}_{C}\right){\theta }_{m},\end{equation} \tag{ 2 }$$

where the angle θ_m , defined modulo 2π, is a characteristic of the type of quasiparticle in question. (We use the index m to distinguish between different species of quasiparticles.) If the position of a single quasiparticle is moved along a closed loop enclosing N_C other identical quasiparticles, the wave function must be multiplied by ${\mathrm{e}}^{-2\mathrm{i}{N}_{C}{\theta }_{m}}$. For cases other than θ_m = 0 or π, this requires that the wave function be multivalued, or equivalently that it is defined on a multi-sheeted Riemann surface. Nevertheless, quantum mechanics can be generalized in a straightforward way to deal with this situation. In a case where θ_m ≠ 0 mod π, if the effective Hamiltonian H_eff can be written as a local function of the positions r_j and the momenta p_j = −iℏ∇_j , with the possible addition of long-range Coulomb forces that depend on position variables only, one says that the quasiparticles obey fractional statistics, with statistical angle θ_m . Such quasiparticles are often referred to as anyons [4].

To describe quasiparticles with fractional statistics, however, it is not actually necessary to employ multivalued wave functions. The multiple phase factors can be eliminated by implementation of a unitary transformation, essentially a singular gauge transformation [3]. Specifically, if ψ_eff is a multivalued wave function as described above, let us define a transformed wave function ${\psi }_{\mathrm{e}\mathrm{ff}}^{\prime }$ by

$$\begin{equation}{\psi }_{\mathrm{e}\mathrm{ff}}^{\prime }\left\{{r}_{j}\right\}={\psi }_{\mathrm{e}\mathrm{ff}}\left\{{\mathbf{r}}_{j}\right\}\prod\limits _{k{< }l}{\left(\frac{{z}_{k}-{z}_{l}}{\vert {z}_{k}-{z}_{l}\vert }\right)}^{{\theta }_{m}/2\pi },\end{equation} \tag{ 3 }$$

where z_j = x_j + iy_j is the position of particle j in complex coordinates. The transformed wave function ${\psi }_{\mathrm{e}\mathrm{ff}}^{\prime }$ will be single valued and will be invariant under interchange of two particles, as would be expected for particles obeying Bose–Einstein statistics. The price one has to pay however, is that the transformed Hamiltonian ${H}_{\mathrm{e}\mathrm{ff}}^{\prime }$ will not longer be local in space. To obtain ${H}_{\mathrm{e}\mathrm{ff}}^{\prime }$ from H_eff, one must replace the operators p_j by [p_j − a_j (r_j )], where a_j , known as a Chern–Simons vector potential, depends on the positions of all the other quasiparticles in the system. Specifically, one has

$$\begin{equation}{\mathbf{a}}_{j}\left({\mathbf{r}}_{j}\right)=\frac{{\theta }_{m}}{2\pi }\sum\limits _{k\ne j}\left(\frac{\hat{z}{\times}\left({\mathbf{r}}_{j}-{\mathbf{r}}_{k}\right)}{\vert {\mathbf{r}}_{j}-{\mathbf{r}}_{k}{\vert }^{2}}\right),\end{equation} \tag{ 4 }$$

where $\hat{z}$ is the unit vector normal to the plane. Thus, an alternate definition of fractional statistics is that if quasiparticles are described by a single-valued wave function ${\psi }_{\mathrm{e}\mathrm{ff}}^{\prime }$ that is invariant or changes sign under interchange of quasiparticle positions, the effective Hamiltonian must be non-local, containing a Chern–Simons vector potential of the form (4).

The definitions of fractional statistics must be extended in the case where there may be several kinds of quasiparticles present. Now we must introduce a new set of quantities θ_mm', which will be equal to one-half the phase acquired when a quasiparticle of type m is moved around a quasiparticle of type m', in a representation with multivalued wave functions and no Chern–Simons vector potential. When m ≠ m', the quantity θ_mm' is only defined mod π, but for identical particles, we require θ_mm = θ_m  mod 2π.

Again, as an alternative to the above definition, one can make a singular gauge transformation to a representation with single-valued wave functions, at the cost of introducing Chern–Simons vector potentials, analogous to (4), which may couple to the different species in different ways. In particular, to obtain the value of a_j seen by a particle of type m, we must include a sum of terms of the form (4), where the coefficient θ_m is replaced by θ_mm', if particle k is of type m':

$$\begin{equation}{\mathbf{a}}_{j}\left({\mathbf{r}}_{j}\right)=\sum\limits _{{m}^{\prime }}\frac{{\theta }_{m{m}^{\prime }}}{2\pi }\sum\limits _{k\ne j}\left(\frac{\hat{z}{\times}\left({\mathbf{r}}_{j}-{\mathbf{r}}_{k}\right)}{\vert {\mathbf{r}}_{j}-{\mathbf{r}}_{k}{\vert }^{2}}\right),\end{equation} \tag{ 5 }$$

It follows from the above definitions that in all cases, θ_m'm = θ_mm'. Also, if we combine two quasiparticles of type m and m' to form a new quasiparticle, of type M, the angle ${\theta }_{M{m}^{{\prime\prime}}}$ describing the mutual statistics between the hybrid quasiparticle and a third quasiparticle of type m'' will be equal to ${\theta }_{m{m}^{{\prime\prime}}}+{\theta }_{{m}^{\prime }{m}^{{\prime\prime}}}$. As a corollary, if we group together n identical quasiparticles of type m, the resulting clusters will have a self statistical angle of θ_M = n² θ_m .

As an example to illustrate a physical consequence of fractional statistics, let us consider a system containing either one or two identical anyons with charge q_m in an external magnetic field B. We shall assume that there is at most a short-range interaction between the anyons. We also assume a weak circularly symmetric parabolic electrostatic potential of the form

$$\begin{equation}{\Phi}\left(\mathbf{r}\right)=\frac{K}{2}{r}^{2},\end{equation} \tag{ 6 }$$

with q_m K > 0, in addition to a stronger short-range attractive potential that can trap a localized quasiparticle near the origin. The case of a single charged particle in a uniform magnetic field and a weak parabolic potential is exactly solvable. For a particle in the lowest Landau level, the energy eigenstates will consist of a series of circular orbits with

$$\begin{equation}\langle {r}^{2}\rangle =2\left(n+1\right)\hslash /\vert {q}_{m}B\vert ,\quad n=0,1,2,\dots ,\end{equation} \tag{ 7 }$$

and energy given by

$$\begin{equation}E={E}_{0}^{{\ast}}+\frac{{q}_{m}K}{2}\langle {r}^{2}\rangle ,\end{equation} \tag{ 8 }$$

where ${E}_{0}^{{\ast}}$ is a constant. The addition of a localized potential well near the origin will have negligible effect on the energies or eigenstates for large values of n.

Let us now consider a system with one quasiparticle, say quasiparticle 1, localized in the well near the origin, and the second quasiparticle sitting in a circular orbit of large radius. According to the Bohr–Sommerfeld rules, we should calculate the allowed radii by requiring that the action for the circular orbit should be equal to an integer multiple of 2π. Because of the Chern–Simons term due to the presence of particle 1, the action for quasiparticle 2 will be shifted by an amount

$$\begin{equation}\delta S=\oint {\mathbf{a}}_{2}\left({\mathbf{r}}_{2}\right)\cdot \mathrm{d}{\mathbf{r}}_{2}=2{\theta }_{m}.\end{equation} \tag{ 9 }$$

The result is that (7) will be replaced by

$$\begin{equation}\langle {r}^{2}\rangle =2\left(n+1-\sigma {\theta }_{m}/\pi \right)\hslash /\vert {q}_{m}B\vert ,\end{equation} \tag{ 10 }$$

where σ = sign(q_m B_z ). If θ_m ≠ 0 mod π, the set of allowed values for r², and hence for the energies for quasiparticle 2, will be different depending on whether quasiparticle 1 is present or not.

The above arguments can be generalized to the case where one has two indistinguishable quasiparticles in orbits that are not localized near the origin. In this case, one finds that the set of allowed energy levels will be sensitive to θ_m  mod 2π.

To make these ideas more concrete, let us return to the microscopic states for a system containing a given number N of identical quasiparticles. Let $\vert {\Psi}\left(\left\{{\mathbf{r}}_{j}\right\}\right)\left.\right\rangle$ be the many-electron state with quasiparticles localized at specified positions (r₁, ..., r_N ). The set of such states, which we here assume to be unique except for a phase, will form an (over-complete) basis for the set of states we are interested in. The set of allowed positions r_j may include restrictions, such as a minimum separation between two quasiparticles. We shall assume that any state in the Hilbert space of interest can be written as a superposition of basis states, in the form

$$\begin{equation}\vert {\Psi}\left.\right\rangle =\int \mathrm{d}{\mathbf{r}}_{1}\dots \mathrm{d}{\mathbf{r}}_{N}{\psi }_{\mathrm{e}\mathrm{ff}}\left(\left\{{\mathbf{r}}_{j}\right\}\right)\vert {\Psi}\left(\left\{{\mathbf{r}}_{j}\right\}\right)\left.\right\rangle .\end{equation} \tag{ 11 }$$

Once we have made a specific phase choice for the basis states $\vert {\Psi}\left(\left\{{\mathbf{r}}_{j}\right\}\right)\left.\right\rangle$, we can define a Berry connection,

$$\begin{equation}{\mathbf{\alpha }}_{k}\left(\left\{{\mathbf{r}}_{j}\right\}\right)=\mathrm{i}\left\langle \right.{\Psi}\left(\left\{{\mathbf{r}}_{j}\right\}\right)\vert {\nabla }_{k}\vert {\Psi}\left(\left\{{\mathbf{r}}_{j}\right\}\right)\left.\right\rangle ,\end{equation} \tag{ 12 }$$

where ∇_k is the gradient with respect to the position r_k . We may now consider a situation in which the positions of two quasiparticles, labeled k and l, are interchanged by moving them around a specified contour C in a counterclockwise fashion, until their final positions are interchanged from their initial positions, while the positions of all other quasiparticles are held fixed. The Berry phase for the process is given by

$$\begin{equation}{\theta }_{Ckl}=\int \mathrm{d}{\mathbf{r}}_{k}\cdot {\mathbf{\alpha }}_{k}+\int \mathrm{d}{\mathbf{r}}_{l}\cdot {\mathbf{\alpha }}_{l},\end{equation} \tag{ 13 }$$

where the integral is taken along the contour. Whereas the Berry connections α _j depend on the particular choices made for the phases of the basis states, the Berry phase θ_Ckl may be seen to be independent of those choices, up to an additive multiple of 2π. Thus the quantity ${\text{e}}^{\text{i}{\theta }_{Ckl}}$ is independent of the phase choice and is therefore gauge invariant. For a system of identical anyons of charge q_m in an external magnetic field, the value of θ_Ckl should be given by

$$\begin{equation}{\theta }_{Ckl}\sim \left(1+2{N}_{C}\right){\theta }_{m}+2\pi {q}_{m}{B}_{z}{A}_{C}/\hslash ,\end{equation} \tag{ 14 }$$

where A_C is the area enclosed by the contour C, θ_m is a constant characteristic of the type of quasiparticle under consideration, and N_C , as before, is the number of additional quasiparticles enclosed. Equation (14) is supposed to be exact when the contour C is large and quasiparticles k and l stay far from all other quasiparticles, but there can be corrections if these conditions are violated. Nevertheless, the implication of (14) is that with a suitable choice of gauge, α _j may be written in the form

$$\begin{equation}{\mathbf{\alpha }}_{j}={q}_{m}\mathbf{A}\left({\mathbf{r}}_{j}\right)+{\mathbf{a}}_{j}\left({\mathbf{r}}_{j}\right),\end{equation} \tag{ 15 }$$

where A is the vector potential due to the applied magnetic field B, and a_j is just the Chern–Simons vector potential given by (4). The discussion may be readily extended to the case where there are several types of quasiparticle present, in which case the first term in (14) should be replaced by $\left({\theta }_{m}+2{\sum }_{m}^{\prime }{N}_{C}^{{m}^{\prime }}{\theta }_{m{m}^{\prime }}\right)$, where ${N}_{C}^{{m}^{\prime }}$ is the number of quasiparticles of type m' enclosed by the contour, and the definition of a_j must be extended, as described in equation (5).

Next, we must examine the time evolution of a state $\vert {\Psi}\left.\right\rangle$ of the form (11). It is convenient for this purpose to use path integral approach. Then the state at time t can be related to the state at time 0 by a unitary transformation of the form

$$\begin{equation}{\psi }_{\mathrm{e}\mathrm{ff}}\left(\left\{{\mathbf{r}}_{j}\right\},t\right)=\int \mathrm{d}\left\{{\mathbf{r}}_{k}^{\prime }\right\}K\left(\left\{{\mathbf{r}}_{j}\right\},\left\{{\mathbf{r}}_{k}^{\prime }\right\}\right){\psi }_{\mathrm{e}\mathrm{ff}}\left(\left\{{\mathbf{r}}_{j}^{\prime }\right\},0\right),\end{equation} \tag{ 16 }$$

where the kernel K is given by the sum of e^−iS over all paths connecting the initial and final configurations of positions, with S being the action associated with the path. To a good approximation, we may evaluate S as

$$\begin{equation}S=\int \mathrm{d}{t}^{{\prime\prime}}U\left(\left\{{\mathbf{r}}_{j}^{{\prime\prime}}\right\}\right)+\sum\limits _{j}\int \mathrm{d}{\mathbf{r}}_{j}^{{\prime\prime}}\cdot {\mathbf{\alpha }}_{j}\left({\mathbf{r}}_{j}^{{\prime\prime}}\right),\end{equation} \tag{ 17 }$$

where $U\left(\left\{{\mathbf{r}}_{j}^{{\prime\prime}}\right\}\right)$ is the expectation value of the microscopic Hamiltonian H in the basis state $\vert {\Psi}\left\{{\mathbf{r}}_{j}^{{\prime\prime}}\right\}\left.\right\rangle$, and the integral is taken along the path from the initial to the final configuration. This expression coincides with the formula for the action of a collection of particles with charge q_m subject to an applied magnetic field and a Chern–Simons vector potential, in the presence of a position-dependent potential energy U, in the limit where the effective mass of the particle is taken to zero, i.e., in the limit where the particles are all in the lowest Landau level.

More generally, U should be replaced by an operator that may include terms that are slightly off-diagonal in the position variables, which would lead to additional momentum-dependent terms in the Hamiltonian, including, perhaps, short-range momentum-dependent interactions between the quasiparticles. Matrix elements of the microscopic Hamiltonian that mix states in the low-energy subspace we are considering with states outside that subspace may be taken into account via perturbation theory as corrections to the matrix elements of U. In a similar fashion, the effects of mixing between Landau levels due to interactions in a system of particles with nonzero mass may be included in a model that is projected onto a single Landau level by including suitable corrections to the interactions within the Landau level. As long as corrections to the interaction terms remain short-ranged in space, they can be distinguished from the Chern–Simons interaction, and will not affect the behavior of well-separated quasiparticles. Thus, the values of the statistical angles θ_mm' remain well-defined and unchanged.

The interplay of Landau-level mixing with long-range forces can change the apparent values of θ_mm , as discussed in [5, 6]. However, the deviations decay as a power of the distance between quasiparticles.

In our previous discussions, we have assumed that if the locations and types of all quasiparticle are specified, there will be a unique low-energy state of the Hamiltonian corresponding to this specification. However, a very different situation is believed to occur in some special quantized Hall states. For these states, in a situation with N localized quasiparticles, there should be a number of nearly-degenerate low-energy eigenstates which grows exponentially with N. The energy differences between these states should fall off exponentially with the separation between quasiparticles, and they are frequently treated as negligible in theoretical discussions.

Now, if a set of quasiparticles are slowly moved around each other or interchanged, in such a way that the set of final positions for each quasiparticle type is identical to the initial set, the final state of the system will be related to the initial state by a unitary transformation in the Hilbert space of low-energy eigenstates. Furthermore, if the braiding process is fast compared to the 'exponentially small' energy splittings of the Hilbert space, the unitary transformation will depend on the topology of the braiding, but will be independent of all other details of the paths that are taken. For processes that involve multiple interchanges of quasiparticles, the resulting unitary transformation will generally depend on the order in which the interchanges have been performed. Hence, the quasiparticles are said to obey 'non-Abelian statistics'.

Because a full discussion of various types of non-Abelian statistics and the ways in which they may be manifest in quantum Hall systems is complicated, we shall defer that discussion until later sections of this paper, and shall first concentrate on states with Abelian fractional statistics.

In Laughlin's landmark 1983 paper, which described his trial wave function for the fractional Hall ground state at ν = 1/3 and related fractions, he also proposed wave functions for the elementary quasiparticles, often denoted as quasielectrons and quasiholes [7]. The added electric charges associated with these proposed wave functions were, indeed, fractions of an electron charge, viz., q_c = ±e/3. Since the trial wave functions are not exact eigenstates of the Hamiltonian for a realistic model with Coulomb interactions, one might be tempted to question the exactness of the charge quantization based on them. However, Laughlin presented a more general argument that quasiparticles with fractional charge must be a feature of any FQH state.

Consider a two-dimensional electron system in a FQH state with filling factor ν on a large disk of radius R. Let us puncture the disk with a hole of diameter a at the center of the disk, and let us thread an infinite solenoid with radius less than a through the hole. In two dimensions, the scattering cross section of a barrier of radius a will vanish [8] in the limit a → 0, proportional to 1/ln²|a|. Thus, in the limit a → 0, the solenoid will have no effect on electrons in the system when there is no flux through the solenoid.

Now, start with a situation where the system is initially in its ground state and there is no flux in the solenoid, and gradually increase the flux until the solenoid contains precisely one flux quantum, pointing in the same direction as the uniform magnetic field. (Note: in our discussions of quantum Hall systems, throughout this paper, we shall assume that the applied magnetic field B points along the negative z axis, unless otherwise specified, and B = |B| > 0.) The time-dependent flux will generate an azimuthal electric field, which will drive electrons in toward the origin, due to the non-vanishing Hall conductance. A simple calculation shows that the total charge accumulated near the origin will be equal to −νe. This extra charge will have come from the edge of the system, where there is necessarily a reservoir of low-energy conducting states [9]. Since there is a finite energy gap in the bulk of the system, we expect, according to the adiabatic theorem, that the final state will again be an energy eigenstate of the system. (Although, in principle, the adiabatic theorem could break down at an instant where the added energy of the system due to the charge at the origin crosses the energy for adding the charge back to a state at the edge of the system, the matrix element for such a transfer will be exponentially small, if the radius R is very large compared to the magnetic length. In addition, for a system with circular symmetry, the matrix element will be identically zero by angular momentum conservation.)

Although the Hamiltonian with the added flux quantum is mathematically different from the original Hamiltonian, we can make a gauge transformation that eliminates the vector potential due to the solenoid, multiplying the wave functions by a position-dependent phase factor and restoring the Hamiltonian to its original form. Thus, the original Hamiltonian must have an eigenstate with the same energy and charge distribution as the one we have found for the state with an added flux quantum.

Of course, we can generate a quasiparticle with charge +νe by repeating the above procedure with solenoid flux in the opposite direction. There is no guarantee, however, that quasiparticles with charge ±νe have the lowest energy or the smallest charge of any possible quasiparticle in the system. In particular if ν = p/q, where p and q are integers with no common divisor, one can always construct a quasiparticle with charge q_m = ±e/q. Since p and q have no common divisor, there will necessarily exist integers n and n' such that nq − n'p = 1. Then a combination of n' quasiparticles of charge νe and n electrons will have total charge −e/q.

These arguments do not require that e/q is necessarily the smallest charge for a quasiparticle in the system. For example, the various competing models [10] proposed to explain the even-denominator quantized Hall state observed at ν = 5/2 have quasiparticles with charge ±e/4. Levin and Stern have argued [11], in fact, that for any FQH state with even denominator q, there must exist quasiparticles with charge q_m = ±e/2q.

The prediction that quasiparticles in FQH states should obey fractional statistics was made in 1984, by Halperin [12], and slightly later, by Arovas et al [13]. The analysis of Halperin was based on the behavior of effective wave functions for collections of quasiparticles, similar to the discussion in subsection 2.2.1, above. By contrast the analysis of Arovas, et al, made use of the definition presented in subsection 2.2.3, specifically, by calculating the Berry phase acquired on interchanging the positions of two quasiholes in the ν = 1/3 state, using Laughlin's trial wave function for the quasiholes.

The analysis of [12] was motivated by the following set of observations. Laughlin's wave functions for the FQH states at ν = 1/m involve a factor of ${\prod }_{j{< }k}{\left({z}_{j}-{z}_{k}\right)}^{m}$, in addition to a Gaussian factor which assures that the electrons have the correct density. These trial wave functions minimize the kinetic energy, as all particles lie in the lowest Landau level, and they are efficient at minimizing the potential energy, at least in the case of short-range repulsive interactions, as the wave functions vanish rapidly when two electrons come close together. Moreover, if m is an odd integer, the wave function is antisymmetric under the interchange of two particles, as required by Fermi statistics. If one were to replace the exponent m in this product by a non-integer exponent γ, and if one multiplied the exponent in the Gaussian factor by a constant s⁻¹, one would have a wave function that describes a collection of anyons in the lowest Landau level for particles of charge ±e/s. The exponent γ would be related to the statistical angle θ_m of the anyons by

$$\begin{equation}\gamma =2n{\pm}{\theta }_{m}/\pi ,\end{equation} \tag{ 18 }$$

where the sign in (18) depends on the sign of the anyon charge and the direction of the applied magnetic field. With our choice of B_z < 0, if one assumes θ_m = π/3 for quasielectrons in the ν = 1/3 state, and one choses n = 1, one finds that with the negative sign in (18) the density of e/3 quasielectrons is just such as to increase the filling factor to ν = 2/5. If one assumes again θ_m = 1/3 for the quasiholes, uses the positive sign in (18), and chooses n = 1, the density of quasiholes is such as to decrease the filling factor to ν = 2/7. FQH states were, indeed, observed experimentally at both these filling factors.

Other fractions could be generated using larger values of n. As noted in [12], this procedure could be repeated, so that starting from a given FQH state with ν = p/q, by adding quasiparticles of charge ±e/q and appropriate statistical angle, one could generate daughter states corresponding to fractions with larger values of p and q. An iterative formula was developed for predicting the statistical angle θ_m at each new fraction, and it was shown that in this manner one could generate a unique FQH state for any fraction p/q with odd denominator. (Of course, there would be no guarantee that the resulting FQH state would actually be the lowest energy state for a system with any particular form of the electron–electron interaction.)

The form of the effective wave function and the choice of statistical angle were further justified in [12] by comparison with microscopic trial wave functions that had been introduced earlier to describe a collection of quasielectrons at ν = 1/3 and the ground state at ν = 2/5 [14]. Of course, these trial wave functions are only approximate descriptions of the true energy eigenstates for a realistic Hamiltonian, as is also the case for the trial wave function used in [13] for a pair of quasiholes. However, the statistical angle should be a topologically protected quantity, meaning that its value cannot change under any deformation of the Hamiltonian that does not cause the energy gap to collapse and does not provoke a first-order phase transition.

In 1989, Jainendra Jain proposed the 'composite fermion' approach of generating trial wave functions for FQH states, which has proved to give energies and wave-functions that are generally much more accurate than those obtained by previous methods, particularly for states with large denominator [15]. However, the statistical angles calculated for quasiparticles at a given odd-denominator fraction ν turn out to be the same as the ones predicted for the same fraction in [12]. A general description of all possible FQH states with Abelian statistics, including but not limited to the Jain states, has been given by Wen [16]. The description includes predictions for the charges and statistical angles of quasiparticles, as well as other topological quantum numbers, such as the 'shift parameter', for these states.

As was seen in the case of fractional charge, the necessity that quasiparticles in an FQH state obey fractional statistics can actually be demonstrated by an argument that does not make any specific assumptions about the form of the ground state at a given fraction ν. Consider a gedanken experiment similar to that described in subsection 2.2.2, where an FQH system with ν = 1/3 is subject to a weak parabolic potential of form (6). If we add a single quasihole to the system, with positive charge q_m = e/3, it will have a series of equally spaced energy levels, with energies given by (7) and (8). If the quasihole is placed in orbit with radius r_n , there will be an electric current around the orbit of magnitude I_n = q_m v_d/2πr_n , where v_d = Kr_n /B is the classical drift velocity in the perpendicular electric and magnetic fields.

Now let us place a thin solenoid at the origin and slowly change the flux through the solenoid from zero to one flux quantum antiparallel to the applied magnetic field. This will produce an additional quasihole at the origin and will modify the orbit of the circulating quasihole. The electromotive force generated by the time-dependent flux will do work given by 2πI/|e|, which will increase the energy of the circulating quasihole by expanding its orbit in the parabolic confining potential. By equating the work done with the change in radius, we see that the new orbit radius will be related to the old one by replacing (7) with (10) and choosing θ_m = π/3. Thus the set of allowed orbits for the circulating quasiparticle is different from the set before the flux was turned on. Since the solenoid flux can be removed from the Hamiltonian by a gauge transformation, the change in the set of allowed radii is entirely due to the presence of a new quasihole at the origin, and not to any change in the Hamiltonian itself. Thus we see that the quasihole must have a statistical angle equal to π/3 mod π in this case. More complicated arguments can be used to demonstrate the necessary occurrence of fractional statistics for other FQH states. We shall address one such argument, employing a Mach–Zehnder interferometer, in section 7.

The appearance of fractional statistics in FQH states is closely related to the fact that the ground states of these systems are degenerate when studied on a torus or another compact manifold with genus ⩾1. It was noted early on that for a translationally invariant system containing N_e interacting electrons in the lowest Landau level in a finite rectangle with periodic boundary conditions containing N_Φ quanta of magnetic flux, every eigenstate of the Hamiltonian must be at least q-fold degenerate, where q is the denominator of the fraction N_e/N_Φ reduced to lowest terms [17, 18]. This degeneracy will generally be split in the presence of disorder, but in the case of an FQH state, where the ground states are separated from all other eigenstates by a finite energy gap, the splittings between the ground states will fall off exponentially with the size of the system, and will therefore be negligible for a sufficiently large system [19]. More generally, it can be shown that any system that supports excitations with fractional statistics must be degenerate on a large torus [20]. Predictions for the ground state degeneracies of Abelian and non-Abelian FQH states on a torus and on manifolds of higher genus may be found in various places in the literature [21–24]. Although these questions are significant theoretically, we shall not discuss them further in the present review, as we are focused on phenomena that can be studied experimentally.

In our previous discussions, we have focused on the properties of localized quasiparticles, or collections of quasiparticles, that are far from any edges of the sample. However, fractionally charged quasiparticles can also exist in delocalized states along the edges of a sample, or at a boundary between two quantized Hall states with different Hall conductivity. As was originally noted in [9], there must be gapless modes at a boundary between a gapped quantum Hall liquid and a vacuum. In the case of integer quantized Hall states, these edge modes may be understood as orbits for electrons at the Fermi level, which propagate along the edge in a particular direction. For FQH states, the edge modes may be similarly interpreted as orbits for quasiparticles of various types.

Though the charge on an edge will be conserved if there are no contacts to the edge and the edge is far from all other edges of the sample, charges can tunnel between two opposite edges of a sample if there is a narrow constriction which brings them close together. When the tunneling strength is small, transfer of charge from one edge to another can occur in discrete units which may be interpreted as the charge of a tunneling quasiparticle. In a geometry with two or more constrictions, there can be interference features that one can attribute to the difference in phase accumulated by a quasiparticle in tunneling from one edge to the opposite edge via the possible paths involving tunneling at different constrictions. For quasiparticles with fractional statistics, the accumulated phase will be sensitive to the number of other quasiparticles enclosed by the difference in paths. Therefore, interference experiments can provide a means for observing effects of the fractional statistics.

Although fractional statistics as well as fractional charge may be measured, in principle, with experiments on quasiparticles far from any sample edges, as illustrated by the gedanken experiments described above, in practice studies of fractional statistics have always employed interferometers with tunneling between edges. As described below, a few experiments have succeeded in measuring fractional charge accumulation in localized regions far from any edge of the sample, but even here, the majority of experiments have involved edge modes and have measured the charges of quasiparticles tunneling from one edge to another across a constriction.

Since quantum Hall edges are interacting gapless systems, it is not possible to define quasiparticle charge in the same way as in the bulk. In particular, charges propagating in a one-dimensional metal may break up into multiple pieces, giving rise to such phenomena as spin-charge separation and charge fractionalization [25–27]. These phenomena would be sensitive to details of the edge. However, charges tunneling from one edge to another through a gapped quantum Hall state should be quantized, and can be measured, at least in the dilute limit, by noise experiments. A more detailed discussion of edge modes in FQH states will be given below in the section on non-Abelian statistics.

This technique combines a dc bias V with an ac bias V_ac ∼  cos(ωt) in the geometry of figure 1. A charge q, emitted from the source, acquires a time-dependent phase ϕ(t) = q∫dtV_ac(t)/ℏ. This can be interpreted as a shift in the energy of tunneling quasiparticles. Without an ac bias, the available energy is q_m V. The absorption of n quanta of the ac field shifts the energy to q_m V + nℏω. The observed shot noise [65–67] is then a weighted sum of the noises at dc voltages V + nℏω/q_m ,

$$\begin{equation}S=\sum\limits _{n=-\infty }^{\infty }{w}_{n}{S}_{\mathrm{d}\mathrm{c}}\left(V+\frac{n\hslash \omega }{{q}_{m}}\right),\end{equation} \tag{ 22 }$$

where the weight w_n reflects the probability to absorb n quanta. The low-temperature noise is singular at V = ℏω/q_m . This was used [35] to verify q_m = e/3 at ν = 1/3 and q_m = e/5 at ν = 2/5.

In a related experiment [36], high frequency noise measurements at f ≈ 7 GHz with dc bias show q = 0.34e at ν = 4/3 and q = 0.38e at ν = 2/3. See [68] and references therein for related theoretical ideas.

The earliest version [41, 43] of charging spectroscopy involved tunneling through an antidot inside a constriction between two FQH edges (figure 4). This setting is related to that of interferometry, addressed in the next section. The electron gas is depleted inside the antidot and hence an FQH edge forms around it. Quasiparticles travel through the constriction by tunneling in and out of that edge. The size of the antidot is controlled by the depleting gate voltage. The technique probes how the conductance through the antidot depends on the gate voltage and the magnetic field.

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The dependence can be understood from the picture of quasiparticle orbits, introduced in section 2.2.2. Changes of the magnetic field or the gate voltage result in an orbit periodically crossing the chemical potential. When this happens, a resonance is seen in the transmission through the constriction. The period in voltage corresponds to adding or subtracting a quasiparticle from the antidot. The quasiparticle charge can be found from the observed period in the voltage and the geometric capacitance. The latter can be approximately extracted from the nominal area of the antidot, and can be checked with the magnetic field periodicity of the conductance. The calibration may be further checked by comparing with measurements in an integer quantized state.

To understand the magnetic field periodicity, it is necessary to take into account fractional statistics as well as fractional charge. The allowed orbits are determined by the Bohr–Sommerfeld quantization rule in terms of the phase, accumulated by an anyon on a closed orbit. The phase has two contributions. First, there is an AB phase for a particle making a circle around the antidot. For an e/3 particle in the Laughlin state at ν = 1/3, the AB phase is ϕ_AB = 2πΦ/3Φ₀, where Φ is the magnetic flux through the antidot and Φ₀ = hc/e is the magnetic flux quantum. In addition, each localized quasihole in the dot contributes the statistical phase −2π/3. Since a new quasihole is created or destroyed every time the flux changes by Φ₀ at a fixed charge density, the total phase accumulated on an orbit is periodic with the period of a flux quantum. Thus, in relating the magnetic field period to the area of the antidot, one must take into account the fractional statistics as well as the fractional charge.

The experimental results [41] are consistent with the theoretically predicted charge e/3 at ν = 1/3. Yet, it was argued that the fractional charge is not the only way to understand the data [42]. One could instead start with a picture of single-electron orbits around the antidot and assume that electron correlations ensure that only 1/3 of them are populated. Resonant transmission would still be observed when an orbit crosses the chemical potential. This predicts the same periodicity as the quasiparticle picture. Besides, electrostatic effects may not be captured by the above single-particle picture (see the next section).

An additional drawback of this technique is that in order for tunneling to occur, the antidot must be physically close to an edge of the sample. Thus one might question whether the results are necessarily reflective of the properties of excitations in the bulk. One might also question whether the gate period obtained from a transport measurement necessarily reflects the period for charge occupancy of the antidot.

It should be noted that a similar technique [46] showed excitations of charge 2e/3 at ν = 2/3. Very recently, charge e/3 was reported in an antidot tunneling experiment [47] in graphene at ν = ±1/3.

Difficulties in the interpretation of the antidot data necessitated a different strategy. Thus, later experiments used a different approach [44, 45]: a single electron transistor (SET), which is sensitive to variations in the local electrostatic potential, was placed on the surface of the heterostructure that embeds the FQH liquid, or on a scanning tip just above the surface. The SET was used to detect potential jumps when a quasiparticle or quasihole enters or leaves a local potential well created by fluctuations in the doping density, which were found to have a physical scale on the order of 200 nm, large compared to the magnetic length.

The SET technique was first developed [69] for 2D heterostructures outside the quantum Hall regime and allowed the spatial resolution of 100 nm. It was extended to studies of the integer quantum Hall effect in [70, 71].

In a subsequent development, fractional charges in FQH liquids were reported in [44, 45]. As was explained above, the entry of new quasiparticles into a well can be controlled with the gate voltage, and the quasiparticle charge, relative to that of an electron in an integer quantized state, can be extracted from the spacing of the jumps as a function of the voltage. The experimental results [44] are consistent with charge-e/3 excitations at ν = 1/3 and ν = 2/3. The absolute value of the quasiparticle charge could also be extracted, with lesser accuracy, from the SET measurements and were consistent with the value e/3. In the second Landau level, comparison between measurements at ν = 5/2 and ν = 7/3 obtained the ratio [45] q_m,5/2/q_m,7/3 = 3/4 in agreement with the theoretical expectation that the charges should be e/4 and e/3 in the two cases. For a detailed theoretical discussion at ν = 5/2, see [72].

We shall first consider the ideal case described above. We assume that the electron density drops rather sharply to zero at the boundary, and there is just a single edge mode propagating along the boundary, as indicated in figure 5. A quasiparticle propagating on the interferometer boundary may then be described by a Hamiltonian of the form

$$\begin{align}& \hat{H}={\hat{H}}_{\text{edge}}\\ & \qquad +\;\;\left[{{\Gamma}}_{1}\enspace \mathrm{exp}\left(\mathrm{i}{\phi }_{1}\right){\hat{T}}_{1}+{{\Gamma}}_{2}\enspace \mathrm{exp}\left(\mathrm{i}{\phi }_{2}\right){\hat{T}}_{2}+\mathrm{h}.\mathrm{c}.\right],\end{align} \tag{ 23 }$$

where ${\hat{H}}_{\text{edge}}$ describes charge propagation on the upper and lower edges, the operators ${\hat{T}}_{1,2}$ move a quasiparticle of charge q_m from the lower edge to the upper edge at the two constrictions, and Γ_1,2 exp(iϕ_1,2) are the associated tunneling amplitudes. We shall focus on the limit of weak tunneling, where the current between the lower and upper edges is much less than the incoming current νe² V/h, where V is the voltage difference between the lower and upper edges. The quasiparticle tunneling rate can then be extracted from Fermi's golden rule,

$$\begin{equation}p=\left[{{\Gamma}}_{1}^{2}+{{\Gamma}}_{2}^{2}\right]{r}_{0}\left(V,T\right)+2{{\Gamma}}_{1}{{\Gamma}}_{2}\enspace \mathrm{cos}\left({\phi }_{1}-{\phi }_{2}\right){r}_{1}\left(V,T\right),\end{equation} \tag{ 24 }$$

where r₀ and r₁ depend on V and the temperature T. Hence the tunneling current between the lower and upper edges of the interferometer will be given by

$$\begin{align}{I}_{t}=& {q}_{m}\left[{{\Gamma}}_{1}^{2}+{{\Gamma}}_{2}^{2}\right]{r}_{0}\left(V,T\right)\\ & +2{q}_{m}{{\Gamma}}_{1}{{\Gamma}}_{2}\enspace \mathrm{cos}\left({\phi }_{1}-{\phi }_{2}\right){r}_{1}\left(V,T\right).\end{align} \tag{ 25 }$$

To realize the experimental configuration illustrated in figure 5, one can connect a current source at voltage V to the lower left corner, and attach grounded contacts to the lower right and upper left corners. The back-scattered current I_t will then be equal to the current flowing into the upper left contact.

The phase difference θ = ϕ₁ − ϕ₂ = α − ϕ_AB − ϕ_s combines two key pieces of information: the quasiparticle charge q_m through the AB phase ϕ_AB = −2πq_m Φ/eΦ₀ and the statistical phase ϕ_s accumulated by a quasiparticle on the trajectory around the anyons, trapped inside the interferometer. Here Φ = BA is the total magnetic flux through the area A enclosed by the paths of the edge states between the constrictions. We ignore any additional slow dependence of the matrix elements r_0,1 on the magnetic field. The constant α will be set to zero without the loss of generality.

Although the above equations can be justified relatively easily in the case of a single edge mode in an ideal system with a sharp edge, there are subtleties involved in applications to a real system, with a continuously varying electron density and at least some disorder near the edge. It has been argued that in this case there will still be a discrete set of propagating modes at the boundary, which will be embedded in a region of weakly localized states but will still have a well-define phase rotation e^iθ along the edge [82]. Since the edge state will have a finite width, at least as large as the magnetic length, there will clearly be some ambiguity as to the physical area A enclosed by the state. However, it is assumed that this ambiguity can be resolved in such a way that ϕ_AB is precisely given by the expression above.

To get access to the information encoded in θ, an experimentalist needs to look for oscillations in the current as one varies the magnetic field and/or the area of the interferometer. The area may be varied by applying a voltage V_g to external gates along the sides of the device. We assume that the area does not depend on the magnetic field. This assumption is not crucial for the interpretation of the data in the incompressible regime. We will lift it in the discussion of the non-ideal compressible case.

If no quasiparticles enter or leave the interference region, contours of constant θ should lie along lines in the plane of |B| and V_g with slope

$$\begin{equation}\frac{\mathrm{d}V}{\mathrm{d}B}=-\frac{A}{B\enspace \left(\partial A/\partial {V}_{\mathrm{g}}\right)}.\end{equation} \tag{ 26 }$$

In the case of ν = 1/3, the spacing ΔB between successive conductivity maxima at fixed V_g should equal 3Φ₀/A, while the spacing ΔV_g at fixed B should correspond to an area change that contains one electron. At certain values of the parameters, however, it may be favorable for a quasiparticle to enter or leave a localized impurity state in the interferometer, at which point we would expect a jump in phase by an amount equal to ±2θ_m , caused by a change in the value of ϕ_s. For a quasiparticle in the Laughlin state at ν = 1/3, it is predicted that 2θ_m = 2π/3.

Behavior of this type was observed in recent experiments [76] by Nakamura, et al, as shown in figure 6. It is possible, of course, to ask whether the reported phase jumps could have been caused by some effect other than fractional statistics, such as Coulomb interactions between localized quasiparticles and the conducting states at the interferometer edges, which could cause a jump in the area enclosed by the interfering trajectories. (See discussion in subsection 4.1.2.) However, the sample in these experiments had nearby conducting planes designed to screen Coulomb interactions as much as possible. Moreover, it would be peculiar if phase jumps caused by residual Coulomb interactions would all have the same size for quasiparticle states localized at different impurity positions in the sample, and that these phase jumps just happened to be close to the value predicted by theory. The alternate possibilities should be further checked and hopefully ruled out by additional experiments, but assuming that the interpretation is correct, the results of [76] provide as direct a demonstration as one could imagine of fractional statistics and a measurement of the statistical phase of quasiparticles in the ν = 1/3 FQH state.

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In order to prepare a sample where one could enter the ideal incompressible regime and still see AB oscillations, the authors of [76] had to overcome major difficulties. The challenge comes from conflicting demands on the interferometer size imposed by weak Coulomb interaction and strong phase coherence. The interaction can be suppressed in a large interferometer, but phase coherence is favored by a small device size. A key improvement, described in [40, 76] came from introducing ancillary wells that screen Coulomb forces in the heterostructure.

It is important to note that the simple results shown in figure 6 were only observed over a limited range of magnetic field. This is to be expected because outside a certain range, the Fermi level will no longer be inside the energy gap of the ideal FQH state. In that case, we can expect that the sample would fall into the compressible regime described above, where there will be a large number of quasiparticles or quasiholes inside the interferometer, with only a small energy barrier to add or subtract an additional quasiparticle [83].

We present here a brief summary of our current theoretical expectations for the behavior of a Fabry–Perot quantum Hall interferometer in the compressible situation, which will apply to the integer quantum Hall regime as well as to FQH systems. Although these theoretical predictions have been confirmed in a variety of experiments in the integer regime, the reader should be warned that there has been little success so far in observing oscillations of the predicted type in FQH states.

Consider the example from the integer quantized Hall regime, illustrated in figure 7. The bulk of the system is in a state with a quantized Hall conductance given by ν = 3. The LL filling factor f near the center of the sample could be anywhere in the range 2.5 < f < 3.5; deviations from the ideal value of f = 3 are accommodated by a finite density of localized electrons in regions with f > 3 and localized holes for f < 3. Near edges of the sample, the electron density drops to zero, and f drops off accordingly. In our somewhat simplified model, we assume that there are quantized Hall strips with quantum numbers ν = 2, 1 and 0 in the edge region, with a single propagating edge mode separating each region. The locations of the propagating modes should fall roughly where the electron density is such that the local Landau-level filling factor is 2.5, 1.5, or 0.5. For situation illustrated in the figure, the density in the constriction is supposed to be slightly less than f = 2. The edge state separating ν = 2 and ν = 3 is totally reflected outside the constriction, the edge state separating ν = 0 and ν = 1 is totally transmitted, while the edge state separating ν = 1 and ν = 2 is partially transmitted. It is this last mode that is relevant in an interference experiment.

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The situation in figure 7 can readily be extended to FQH states. For example, if we interpret the labeled filling factors as effective fillings for composite fermions, with two flux quanta attached to each electron, the Hall states become quantized states with ν = 1/3, 2/5 and 3/7. More generally, we shall assume that there is a single partially-transmitted edge state, which separates inner and outer regions with quantum numbers ν_in and ν_out, with ν_in > ν_out. We shall assume that the tunneling processes occur at one well-defined point within each constriction, and we shall define the interference area A_I as the area enclosed by the interfering edge state between these points. We also define q_in and q_out as the charges of the fundamental quasiparticles in quantized Hall states with ν_in and ν_out. For the fractional case, we shall confine our discussion to the situation where ν_in corresponds to a Jain state in the bottom half of the lowest LL, so we may write

$$\begin{equation}{\nu }_{\mathrm{i}\mathrm{n}}=\frac{p}{2ps+1},\qquad {q}_{\mathrm{i}\mathrm{n}}=\frac{e}{2ps+1},\end{equation} \tag{ 27 }$$

where p and s are positive integers. The values of ν_out and q_out are obtained by replacing p by p − 1 in the above formulas. The situation in figure 7 corresponds to p = 2.

Next we define N_L as the net number of quasiparticles of charge q_in inside the area A_I. The number N_L takes into account the total excess charge in area, including charges in regions with ν > ν_in as well as positive or negative quasiparticles localized at density inhomogeneities within the ν_in region. Specifically, it is related to the total electric charge Q inside the interference area A_I by

$$\begin{equation}Q={N}_{\mathrm{L}}{q}_{\mathrm{i}\mathrm{n}}-{A}_{\mathrm{I}}{\nu }_{\mathrm{i}\mathrm{n}}eB/{{\Phi}}_{0}.\end{equation} \tag{ 28 }$$

If we assume that the dominant tunneling processes at the constrictions involve quasiparticles with charge q_in, then the interference phase seen by the tunneling particles will be given by

$$\begin{equation}\theta =-2{N}_{\mathrm{L}}{\theta }_{\mathrm{i}\mathrm{n}}+2\pi B{A}_{\mathrm{I}}{q}_{\mathrm{i}\mathrm{n}}/e{{\Phi}}_{0},\end{equation} \tag{ 29 }$$

where θ_in is the statistical phase associated with the quasiparticles of charge q_in, given by [16]

$$\begin{equation}{\theta }_{\mathrm{i}\mathrm{n}}=\pi \left[1-\frac{2s}{2ps+1}\right].\end{equation} \tag{ 30 }$$

We remind the reader that our sign conventions assume that the field points along the −z direction, and B = |B| is the magnitude of the field. The phase θ_in would have had the opposite sign if the magnetic field had been chosen to be in the positive z direction. Note that the statistical phase is the same for a particle and its antiparticle.

A key assumption is that N_L is restricted to take on integer values (positive or negative), because the localized states inside the interfering edge are isolated from the states outside and from the edge itself [82]. This does not mean that N_L is frozen in time, only that it is constant on the time scale for a quasiparticle in the edge state to move along the length of the interferometer. We assume that on the longer laboratory time scale, charges can hop readily from one localized state to another and that occupations will take on an equilibrium distribution determined by the temperature, the magnetic field, and any voltages applied to the gates and the current contacts. From this point of view, the entire region inside interfering edge state, as well as the region surrounding the edge state, should be considered as compressible in most cases [79, 80].

In contrast, the charge on the edge state can vary rapidly, because it is connected directly to the edge states outside the interferometer, and we consider here a situation where the backscattering probabilities at the constrictions are small. Thus, the edge charge is not quantized, and the area A_I, related to it by equation (31) below, may be considered to be a continuous variable.

We now define an energy function E(N_L, A_I), which describes the free energy of the system after all other variables have been integrated out [82]. We assume that the time-average interference current measured in an experiment is proportional to the thermodynamic average of Re(e^iθ ), weighted by the factor ${\mathrm{e}}^{-E\left({N}_{\mathrm{L}},{A}_{\mathrm{I}}\right)/T}$.

It is convenient to introduce another variable δn_L, so that we can write

$$\begin{equation}\delta {n}_{\mathrm{I}}\equiv -\left({\nu }_{\mathrm{i}\mathrm{n}}-{\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}}\right)B\left({A}_{\mathrm{I}}-\bar{A}\right)/{{\Phi}}_{0},\end{equation} \tag{ 31 }$$

$$\begin{equation}\delta {n}_{\mathrm{L}}\equiv {N}_{\mathrm{L}}\frac{{q}_{\mathrm{i}\mathrm{n}}}{e}-{\nu }_{\mathrm{i}\mathrm{n}}\frac{B\bar{A}}{{{\Phi}}_{0}}-\bar{Q},\end{equation} \tag{ 32 }$$

where $\bar{A}$ and $\bar{Q}$ are quantities chosen such that δn_L and δn_I would be zero if we were to minimize E without the constraint that N_L be an integer. The values of $\bar{A}$ and $\bar{Q}$ should be smooth monotonic functions of any applied gate voltages, with perhaps a weak smooth dependence on B. The variable δn_I describes charge fluctuations on the interfering edge, while δn_L is determined by fluctuations in the interior. Then for small fluctuations in the variables, we can expand E in the form

$$\begin{equation}E=\frac{{K}_{\mathrm{L}}}{2}\delta {n}_{\mathrm{L}}^{2}+\frac{{K}_{\mathrm{I}}}{2}\delta {n}_{\mathrm{I}}^{2}+{K}_{\mathrm{I}\mathrm{L}}\delta {n}_{\mathrm{L}}\delta {n}_{\mathrm{I}},\end{equation} \tag{ 33 }$$

where the constants K_L, K_I and K_IL depend on the geometry and are largely determined by the Coulomb interactions between charges. Equation (33) contains only the effect of long-range Coulomb forces and no contribution from the quasiparticle gap since the random potential creates an essentially continuous spectrum for anyons.

At T = 0, there will be no thermal fluctuations, and the phase factor e^iθ will exhibit jumps at discrete values of the parameters, where N_L increases or decreases by one. At finite temperatures, fluctuations become important, and one rapidly enters a regime where one or two Fourier components are dominant in a plot of the thermal expectation value ⟨e^iθ ⟩ as a function of the parameters B and V_g. The allowed contributions have the form [82]

$$\begin{equation}{D}_{m}\enspace \mathrm{exp}\left\{2\pi \mathrm{i}\left[m\left(\frac{B\bar{A}}{{{\Phi}}_{0}}\right)-\frac{\bar{Q}\left({q}_{\mathrm{i}\mathrm{n}}-me\right)}{e{\nu }_{\mathrm{i}\mathrm{n}}}\right]\right\},\end{equation} \tag{ 34 }$$

where m are integers restricted to values of form

$$\begin{equation}m=-\frac{{\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}}e}{{q}_{\mathrm{o}\mathrm{u}\mathrm{t}}}+g\frac{{\nu }_{\mathrm{i}\mathrm{n}}e}{{q}_{\mathrm{i}\mathrm{n}}},\end{equation} \tag{ 35 }$$

where g is an integer. The amplitudes D_m fall off exponentially with temperature, |D_m | ∝ exp(−2π² T/E_m ), so typically only the component with largest E_m is visible. An explicit expression for E_m in terms of the parameters of the model is given by equations (20) and (27) of [82]. According to those formulas, the largest value of E_m occurs when g is the closest integer to −Δθ/2π, where

$$\begin{equation*}{\Delta}\theta =-2{\theta }_{\mathrm{i}\mathrm{n}}+\frac{2\pi {q}_{\mathrm{i}\mathrm{n}}^{2}}{{e}^{2}\left({\nu }_{\mathrm{i}\mathrm{n}}-{\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}}\right)}\frac{{K}_{\mathrm{I}\mathrm{L}}}{{K}_{\mathrm{I}}}\end{equation*}$$

$$\begin{equation}=2\pi \left(\frac{2s{q}_{\mathrm{i}\mathrm{n}}}{e}-1+\frac{{q}_{\mathrm{i}\mathrm{n}}}{{q}_{\mathrm{o}\mathrm{u}\mathrm{t}}}\frac{{K}_{\mathrm{I}\mathrm{L}}}{{K}_{\mathrm{I}}}\right)\end{equation} \tag{ 36 }$$

is the jump in interferometer phase that would occur if N_L is increased by one at T = 0. The favored value of g corresponds to the Fourier component of the interference oscillations that is least sensitive to thermal fluctuations in N_L and A_I at higher temperatures.

The case g = 1 has been termed the AB regime, while the case g = 0 has been termed the CD regime. For integer quantum Hall states, where s = 0, the AB regime occurs when K_IL/K_I < 1/2, so that the coupling between edge and bulk is relatively weak, while the CD regime occurs for 1/2 < K_IL/K_I < 3/2, where the coupling is relatively strong. For a fractional state of the form (27), there will again be a CD-like regime with g = 0, and at least in principle, an AB regime with g = 1. However, the value of K_IL/K_I separating the two regimes will be <1/2, and the AB regime may be difficult to access. In fact, for the Laughlin states, with p = 1 and s ⩾ 1, one is in the CD-like regime, with g = 0, even for K_IL = 0, so the traditional CD designation is actually a misnomer in this case. [To reach the AB regime at ν_in = 1/3, one would actually need an attractive interaction between the edge mode and localized charges, with −7/2 < (K_IL/K_I) < −1/2.] For integer states and for Jain states of the form (27), the dominant term in the AB region has m = 1, while in the g = 0 CD-like region, it has m = 1 − (ν_in e/q_in) = 1 − p.

According to (34), if the gate voltage is held fixed, and if one can assume that $\bar{A}$ and $\bar{Q}$ are insensitive to the magnetic field, then the oscillations in conductance should have a period in the magnetic field given by ΔB = Φ₀/|m|A. In the AB regime, where m = 1, the flux period is Φ₀ for all the states under consideration. By contrast, in the CD regime, the period depends on the state, and it will be a submultiple of Φ₀ for states where there are two or more fully transmitted edge states, such as ν_in = 3 or ν_in = 3/7.

If one fixes B and varies V_g, one will generally see an oscillating conductance with a period that will depend on $\mathrm{d}\bar{A}/\mathrm{d}{V}_{\mathrm{g}}$ and $\mathrm{d}\bar{Q}/\mathrm{d}{V}_{\mathrm{g}}$. A color plot of the conductance oscillations as a function of B and V_g will lead to a series of parallel stripes, similar to those seen in figure 6. It was argued in [82] that at least for integer quantized Hall states, lines of equal phase should have a negative slope, similar to the stripes in figure 6, in the AB regime, but they should have a positive slope in the CD regime, provided there is at least one fully transmitted edge mode. Fabry–Perot experiments in the integer quantized regime have seen both types of behavior, depending on the details of the sample [38, 84]. Also, in certain samples, the two types of stripes were seen to coexist, leading to a checkerboard pattern of diamond shapes in the color plot. However, the situation is more complicated in the FQH case. If we define the measured phase $\tilde {\theta }$ as arg(⟨e^iθ ⟩), then following equation (34), $\partial \tilde {\theta }/\partial B$ will again have the same sign as m. However, for FQH states, the sign of $\partial \tilde {\theta }/\partial {V}_{\mathrm{g}}$ can depend on microscopic details.

For the case of ν_in = 1/3 and ν_out = 0, where there are no fully-transmitted edge modes, one has m = 0 in the CD-like regime, as noted above. Then the conductance will not show oscillations as the magnetic field is varied, and stripes of equal phase will be horizontal in the color plot. Behavior of this type was indeed observed in the experiments reported in [76] for magnetic fields on outside of the range shown in figure 6. We remark that this result differs from the original prediction of [83] that there should be a flux period of Φ₀ in this region, as one would expect in the AB compressible regime; however, that prediction has now been corrected.

Note that in the compressible domain, FQH states in a higher Landau level will have different flux periods than the corresponding states in the spin-polarized lowest Landau level. For example [82], for a state at ν = 7/3, which we assume to consist of a Laughlin liquid at ν = 1/3 on top of an integer quantized Hall state with ν = 2, we would have ν_in = 7/3 and ν_out = 2, so the allowed values of m in (34) will be equal to −2 + 7g. As at ν_in = 1/3, we expect that the dominant term should have g = 0. Experimental results at ν = 7/3 by Willett and collaborators [85] showed a flux period Φ₀/2, consistent with predictions for the compressible regime with m = −2. However, the dependence on gate voltage was not reported in this reference.

In another experiment at ν = 7/3, An and collaborators [86] reported a gate period with phase jumps, appearing in the form of telegraph noise, which was consistent, at least qualitatively, with what one might expect for a state in the incompressible regime. However, the flux period was not reported at this filling fraction, nor was there a reported calibration of the amount of charge entering the interferometer in one gate period.

Motivated by the experiments of [46], Schreier et al [87] have analyzed interference effects to be expected in a geometry where there is tunneling through an antidot inside a constriction. In particular, they considered a situation where there are two edge states around the antidot, and they found that the system was likely to be in an AB regime for an FQH state for the same geometry where one would observe CD behavior in the integer case. They advanced this as an explanation for the different behaviors observed in [46] between bulk filling factors ν = 2/3 and ν = 2.

It should be cautioned that our discussion of the Fabry–Perot interferometer ignored the possible effects of tunneling between different edge modes along the perimeter of the interferometer. While this has been justified by experiments in the integer regime in many cases, it may be more questionable for FQH states, particularly when there are edge modes propagating in two directions. Inter-mode scattering may contribute to decoherence effects, which may be a reason why interference oscillations have proved much more difficult to observe for FQH states than for integer states.

The analyses which led to the results described above, for both the AB and CD regimes in the FQH case with a compressible bulk, certainly made use of the property of fractional statistics. More generally, if one accepts that the interfering particles have fractional charge, then one needs to invoke fractional statistics to avoid flux periods which are integer multiples of Φ₀. However, the flux periods predicted above for the Jain states are identical to the ones predicted for integer states in both the AB and CD regimes, where the tunneling particles are electrons. Consequently, it might be hard to rule out the possibility that the interfering particles in an experiment [88] are electrons rather than fractionally charged quasiparticles. For this reason, observations of the predicted flux period in either regime might not be accepted as a convincing direct observation of fractional statistics.

Measurements using the Fabry–Perot geometry can be used to measure the charges of quasiparticles in various quantized Hall states in either the CD or AB regime. Using expression (34), if the values of $\mathrm{d}\bar{A}/\mathrm{d}{V}_{\mathrm{g}}$ and $\mathrm{d}\bar{Q}/\mathrm{d}{V}_{\mathrm{g}}$ are known, one can predict the oscillation period ΔV_g when B is held fixed. In the CD regime, this period corresponds to the addition of a charge equal to q_out to the interior of the interferometer.

Importantly, although (34) was derived in the regime of weak backscattering, the same result obtains, for a given partially-transmitted edge state, in the regime of strong backscattering, where the partially transmitted edge state is almost totally reflected at the constrictions, and there is only weak forward scattering. (See figure 8(a).) In this limit, the area enclosed by the interfering edge forms an isolated droplet of material in a quantum Hall state with quantum number ν_in, embedded in a region with quantum number ν_out. Charge can then enter or leave the droplet only in units of q_out, and the total charge in the droplet must be an integral multiple of this unit. If V_g is varied, periodic oscillations will occur in the amplitude for forward tunneling through the constriction, as the quantized charges enter or leave the droplet.

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In typical experiments, the filling factor f_c in the constrictions and the filling factor in the bulk are varied simultaneously by changing the magnetic field, while the overall electron density and gate voltages are held constant. A region of weak forward scattering should occur when f_c is slightly above a rational value ν_c that corresponds to a well-established quantized Hall state. (This will obtain when the magnetic field is slightly lower than the value at which f_c = ν_c.) In this case, we have ν_out = ν_c, so the charges measured in the CD regime will be that of the elementary quasiparticles in the region of the constriction. Figure 8(a) shows a case where ν_c = 2 and f_c ≈ 2.2. By contrast, the constrictions shown in figure 7 are in a regime of weak backscattering, where f_c is slightly below ν_c, meaning that the magnetic field is slightly higher than the value where f_c = ν_c. (In figure 7, we have ν_c = 2 and f_c < 2.) In such cases, we have ν_c = ν_in, so if ν_c is a fraction, the charge q_out measured in the experiment will differ from the elementary charge in the constriction. Note that the partially reflected edge states are not the same in figures 7 and 8(a).

In a well-made constriction, as parameters are varied, there will be intervals where f_c is sufficiently close to some quantized value ν_c that there is neither appreciable forward nor back scattering. (See figure 8(b).) In this regime, the measured Hall resistance of the device and the two-terminal conductance will sit on a quantized Hall plateau, where no interference oscillations will be seen.

Quasiparticle charge measurements in the CD regime obtained in [38] were consistent with the expected anyon charges e/3 and e/5 at ν = 1/3 and ν = 2/5 respectively. Charge ≈e/3 was reported for ν = 1/3, 2/3, 4/3, and 5/3 in [39]. A recent Coulomb blockade experiment [89] reveals charges e/3 at ν = 1/3 and 2/3.

As mentioned above, the color plots of conductance as a function of B and V_g presented in [76] showed a series of horizontal stripes, for fields outside the range of incompressible bulk, which is what one predicts for the CD regime when the bulk is in a compressible state on the ν = 1/3 plateau and the filling in the constrictions is less than that of the bulk. Moreover, the observed gate period ΔV_g is consistent with the predicted period in the CD regime, since q_out = e.

Quasiparticle charge can also be obtained from AB oscillations in the incompressible bulk regime shown in figure 6, as was done in [40]. Assuming the interference area A is known, if one can neglect Coulomb coupling between the edge and the bulk, the charge of the interfering particle can be extracted from the magnetic-field period in an interval where no localized quasiparticles enter or leave, by use of the equation q_m AΔB = eΦ₀. Alternatively, if the dependence of A on gate voltage is known, the charge may be extracted from the gate period using q_m BΔV_g = eΦ₀/(∂A/∂V_g ). The authors of [40] used the second method to extract the value of q_m at ν = 1/3, assuming that value of (∂A/∂V_g) was unchanged from the value at ν = 1, and they obtained the value q_m = 0.29e, in good agreement with the expected value e/3. On the other hand, measurements of the same type at ν = 2/3 obtained a result of 0.93e, suggesting that the tunneling charges in that case might be electrons rather than fractionally charged quasiparticles.

Using the model defined by equation (33), we can address the effects of the Coulomb interactions, omitted above and in section 4.1, on interferometer experiments in the incompressible region. If we continue to assume that the background parameters $\bar{A}$ and $\bar{Q}$ are insensitive to the magnetic field, then modifications of the interference area A_I are controlled by the coupling constants K_IL and K_I. In this case one finds that the jump in the interferometer phase on entry of a quasiparticle will be given by equation (36) while the magnetic field period, between jumps, will be renormalized to

$$\begin{equation}{\Delta}B=\frac{e{{\Phi}}_{0}}{{q}_{\mathrm{i}\mathrm{n}}\bar{A}}{\left[1-\frac{{K}_{\mathrm{I}\mathrm{L}}}{{K}_{\mathrm{I}}}\frac{{\nu }_{\mathrm{i}\mathrm{n}}}{\left({\nu }_{\mathrm{i}\mathrm{n}}-{\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}}\right)}\right]}^{-1}.\end{equation} \tag{ 37 }$$

The slope of lines of equal phase on the plane of B and V_g should not be affected by a non-zero K_IL. The distinct jumps predicted by (36) should be visible in the incompressible regime at temperatures much higher than in the compressible regime, in so far as the energy to create a quasiparticle is typically much higher than the scales of charging energies, K_I and K_L.

The claim in [40, 76] that Coulomb coupling may be neglected in their sample is supported by the fact that the interference stripes they observe at the integer filling ν = 1 are consistent with what one would expect in the incompressible regime on neglecting the correction proportional to K_IL/K_I in (37), or in the AB regime if the bulk is compressible.

Note that in the incompressible region, one finds only a gradual transition between the regimes of weak and strong Coulomb interaction, as the predictions for Δθ and ΔB vary continuously as a function of K_IL/K_I. This is in contrast to the compressible region, where the transition between AB and CD-like regimes is marked by simultaneous manifestation of two distinct periodicities, rather than a single intermediate period.

We conclude this section by mentioning puzzling behavior [90, 91] observed in a geometry with a ν = 1/3 channel going around a ν = 2/5 island, where the transport data were interpreted as showing a magnetic-field period of 5Φ₀ and a period of in the interferometer charge of 2e. The explanation advanced by the experimenters supposed that the enclosed ν = 2/5 region was in a compressible state, where e/5 quasiparticles could readily enter or leave, so as to keep the electron-density and area fixed as the magnetic field was varied. However, according to the analysis presented above, in a compressible region, regardless of whether one was in the AB regime or the CD regime, any observed flux periods should be Φ₀ or a submultiple of it, not a period larger than Φ₀. (See [82, 92–94] for further discussions.)

A possible resolution of the puzzle might be obtained if the quantum dots in these experiments were actually measured in a magnetic field interval where the interior state was essentially incompressible, as in the central magnetic-field region of [40, 76]. In that case, if the interfering quasiparticles have charge e/5, one would naturally expect to find a flux period of 5Φ₀ and a gate period corresponding to the addition of two electrons. It should be noted, however, that the varying gate employed in these experiments was not a side gate but rather a back gate, separated from the sample by the thickness of a sapphire substrate, which may complicate the analysis. In any case, a more detailed analysis, and perhaps further experiments, are needed to resolve these issues.

In our discussion of the proposed ν = 5/2 and ν = 12/5 orders we will ignore the two filled spin-resolved Landau levels. We will think of electrons in the partially-filled level in terms of composite fermions that combine an electron and two flux quanta [151]. Composite fermions move in zero effective magnetic field. Thus, one might expect that they form a gapless Fermi-liquid-like state. Gapless states are indeed observed [151] at ν = 1/2 and ν = 3/2 in GaAs. The gap at ν = 5/2 can be explained by Cooper pairing [22] of composite fermions. However, multiple ways exist to build a Cooper pair. In an isotropic system, one can have pairing in various angular momentum channels l. In an anisotropic system, where l is not a good quantum number, we can instead talk about the winding number of the phase of the order parameter as the fermion momentum moves around the Fermi surface. In general, for a spinless single-component Fermi surface, only odd values of l are allowed. However, in the presence of electron–electron interaction it is possible for a Fermi system to spontaneously divide itself into several components with independent Fermi surfaces, and in that case pairing with even values of l is allowed. For example, it has been proposed that for a wide quantum well at total filling ν = 1/2, electrons might organize themselves into two parallel sheets with 1/4 filling in each [152]. The bulk-edge correspondence gives a convenient principle for classifying the various states.

For a half-filled Landau level, the charge mode is described by the Lagrangian density (57) with ν = 1/2. Operators that create and annihilate an electron charge are proportional to Φ_± = exp(±2iϕ). One can check that operators Φ_± are bosonic and must be multiplied by a neutral fermion to produce a legitimate electron operator. This means that the edge theory should contain one or more gapless Fermi modes. Since a complex fermion is a combination of two Majorana fermions, we can assume without loss of generality that all fermion modes are Majorana. We can also assume that all Majorana modes are co-propagating since contra-propagating modes can be gapped out by electron tunneling between edge modes. The net number C of the Majorana modes is often referred to as a Chern number, because it has the form of a Chern index in the analysis presented in [108]. The Chern number is positive for downstream modes and negative for upstream modes. The Lagrangian density is

$$\begin{equation}{L}_{\mathrm{P}\mathrm{f}}=-\frac{2}{4\pi }{\partial }_{x}{\phi }_{\mathrm{c}}\left({\partial }_{t}+v{\partial }_{x}\right){\phi }_{\mathrm{c}}+\sum\limits _{k=1}^{\vert C\vert }\mathrm{i}{\psi }_{k}\left({\partial }_{t}+{v}_{n}\enspace \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\enspace C{\partial }_{x}\right){\psi }_{k}.\end{equation} \tag{ 60 }$$

There is no a priori reason for the velocities of the Majorana fermions to be the same, but edge disorder makes them equal in the long-scale limit [153–155].

No Majorana modes are present at C = 0. In that special case [14, 156, 157], known as the K = 8 state, electrons are gapped on the edge. For subtleties in the 113 state at C = −2, see [158]. Subtleties that emerge at C = −4 are addressed in [155].

The states with even Chern numbers are Abelian and the states with odd Chern numbers are non-Abelian. The topological order depends only on C mod 16. Thus, there are 8 Abelian and 8 non-Abelian possibilities known together as the 16-fold way [10, 108]. Orders with a large Chern number are seen as unlikely, and the bulk of research has focused on the orders listed in table 1. The states with the Chern numbers C and −(C + 2) are related by the particle–hole conjugation. The PH-Pfaffian order at C = −1 is unique in being its own conjugate [159, 160].

It appears that multiple orders of the 16-fold way are realized in nature. Numerical work has brought a preponderance of evidence in favor of the non-Abelian Pfaffian and anti-Pfaffian liquids at ν = 5/2 in GaAs without impurities [165–167]; see, e.g., [168–171]. Experiment appears consistent with a different non-Abelian PH-Pfaffian liquid [132, 160]. Some data were interpreted in terms of the Abelian 113 and 331 states [172–174]. Recent theoretical work suggests a complicated phase diagram in realistic disordered systems in which all topological orders with −3 ⩽ C ⩽ 1 are present [175–177] (see also [178–181] for the role of Landau level mixing). On the other hand, some theoretical proposals question [182–184] the existence of an energy gap at ν = 5/2.

Half-integer FQH plateaus have also been found in several systems beyond single-layer GaAs. The SU(2)₂ order was predicted in graphene [185, 186]. The 331 order [14] is believed [187–189] to be present in GaAs bilayer [152, 190] at the filling factor 1/2. Recent experiments on single-layer graphene have demonstrated the existence of gapped QH states in the N = 3 Landau level, corresponding to ν = 21/2, 23/2, 25/2 and 27/2, which have been attributed to a state with C = 3 or to its particle–hole conjugate with C = −5 [186]. Besides GaAs and graphene [186, 191–195], half-integer plateaus have been observed [196–198] in ZnO and WSe₂.

We finish the discussion of the 16-fold way by describing the quasiparticle types, fusion rules, and topological spins for each order [10]. (See table 1.) All non-Abelian orders possess excitations with three topological charges: 1, ψ, and σ. 1 and ψ carry half-integer electrical charges ne/2. σ carriers charge e/4 + ne/2. The fusion rules for the topological charges are given by equation (39). Electrical charges of the excitations, of course, add up in fusion. The topological spins of the excitations are determined by their topological charge t and their electrical charge ne/4 as

$$\begin{equation}{\theta }_{\left(t,n\right)}={\theta }_{t}\enspace \mathrm{exp}\left(\mathrm{i}\pi {n}^{2}/8\right),\end{equation} \tag{ 61 }$$

where θ₁ = 1, θ_ψ = −1, and θ_σ = exp(iπC/8).

The K = 8 state is effectively a Laughlin-like liquid of charge 2e bosons at Landau-level filling 1/8, which gives rise to Abelian anyons labeled by their electrical charges ne/4, with the topological spins exp(iπn²/8). In the remaining Abelian states, there are four topological charges 1, ψ, σ and μ. The electrical charges of 1- and ψ-excitations are ne/2, while σ and μ carry electrical charges e/4 + ne/2. The fusion rules depend on the parity of C/2. For odd C/2, σ × σ = μ × μ = ψ and σ × μ = 1. For even C/2, σ × σ = μ × μ = 1 and σ × μ = ψ. In all cases, ψ × ψ = 1, σ × ψ = μ, and μ × ψ = σ. The topological spins are given by equation (61) with θ₁ = 1, θ_ψ = −1, θ_σ = θ_μ = exp(iπC/8).

A key difference between Abelian and non-Abelian states is the existence of one charge-e/4 particle σ for the non-Abelian orders and two charge-e/4 particles σ and μ for the Abelian orders. This leads to subtleties in the interpretation of experiment since two quasiparticle types in Abelian states may have the same experimental consequences as two fusion channels for non-Abelian anyons [199].

Evidence exists for a ν = 1/4 plateau in wide GaAs quantum wells [200–202]. The 16-fold way was extended to that filling factor in [203].

The thermal conductance of a Majorana mode is determined by its central charge c = 1/2. A more general class of CFTs is known with c = (2k − 2)/(k + 2), where k is an arbitrary positive integer. The case k = 2 reduces to the Ising CFT, while the CFTs with k > 2 are known as parafermion theories [114]. They were used by Read and Rezayi to generate a family of FQH states [150] at the filling factors k/(k + 2). Anyon types [204] are distinguished by their electrical charge and their topological charge, which comes from the list of the primary fields in the parafermion CFT. There are k(k + 1)/2 primary fields ${{\Phi}}_{m}^{j}$ with j = 0, 1/2, ..., k/2, $\left(j-m\right)\in \mathbb{Z}$. Two identifications are made: (j, m) ≡ (j, m + k) and $\left(j,m\right)\equiv \left(\frac{k}{2}-j,m+\frac{k}{2}\right)$. This allows choosing j > 0 and −j < m ⩽ j. The topological spin of ${{\Phi}}_{m}^{j}$ is

$$\begin{equation}{\theta }_{m}^{j}=\mathrm{exp}\left(2\pi \mathrm{i}\left[\frac{j\left(j+1\right)}{k+2}-\frac{{m}^{2}}{k}\right]\right).\end{equation} \tag{ 62 }$$

The fusion channels are given by

$$\begin{equation}{{\Phi}}_{m}^{j}{\times}{{\Phi}}_{{m}^{\prime }}^{{j}^{\prime }}=\sum\limits _{{j}^{{\prime\prime}}=\vert j-{j}^{\prime }\vert }^{\mathrm{min}\left(j+{j}^{\prime },k-j-{j}^{\prime }\right)}{{\Phi}}_{m+{m}^{\prime }}^{{j}^{{\prime\prime}}}.\end{equation} \tag{ 63 }$$

Electrons carry the topological charge ${{\Phi}}_{1-k/2}^{k/2}$. The topological spin of an anyon of electrical charge se is the product of the neutral contribution (62) and exp(πi[k + 2]s²/k). The allowed combinations of the topological and electrical charges make the braiding phase (54) with an electron ${\phi }_{a{\times}e}^{ae}$ trivial. The lowest quasiparticle charge is e/(k + 2).

The state at k = 4 corresponds to the observed filling factor 8/3 = 2 + 2/3. There is some numerical evidence for a Read–Rezayi state at that filling factor [133], but experiment suggests that it hosts a Laughlin-like state (see [144] for a review). No plateau has been seen [148] at ν = 13/5 = 2 + 3/5, which would correspond to k = 3. A plateau is known [146] at the particle–hole conjugate filling factor 12/5. Apparently, Landau level mixing effects [140, 141] are responsible for the difference between ν = 12/5 and ν = 13/5. Numerics suggests a non-Abelian state [135–137] at ν = 12/5 that is the particle–hole conjugate of the k = 3 Read–Rezayi state. Such state is interesting from the point of view of quantum computing since it allows universal topological computation [24], impossible with the topological orders of the 16-fold way.

Note that a generalization of the Read–Rezayi states applies [150] to the filling factors ν = k/(Mk + 2) with an odd M. A negative-flux version [205] of the states was proposed at ν = k/(3k − 2), k > 2.

Another non-Abelian candidate at ν = 12/5 is a Bonderson–Slingerland state [138, 139], whose fractional statistics is closely related with that in the Ising topological order.