Fusion channels of non-Abelian anyons from angular-momentum and density-profile measurements

TL;DR

This paper introduces a static measurement-based method to identify non-Abelian anyons by linking angular momentum and density profiles, demonstrated on the Moore-Read state, enabling new ways to characterize these exotic particles.

Contribution

The paper proposes a novel, interference-free approach to detect non-Abelian anyons using angular momentum and density measurements, applicable to experimental and theoretical studies.

Findings

Successfully identified different fusion channels of non-Abelian anyons

Linked angular momentum properties to anyonic statistics in the Moore-Read state

Demonstrated density-profile measurements can reveal non-Abelian features

Abstract

We present a method to characterize non-Abelian anyons that is based only on static measurements and that does not rely on any form of interference. For geometries where the anyonic statistics can be revealed by rigid rotations of the anyons, we link this property to the angular momentum of the initial state. We test our method on the paradigmatic example of the Moore-Read state, that is known to support excitations with non-Abelian statistics of Ising type. As an example, we reveal the presence of different fusion channels for two such excitations, a defining feature of non-Abelian anyons. This is obtained by measuring density-profile properties, like the mean square radius of the system or the depletion generated by the anyons. Our study paves the way to novel methods for characterizing non-Abelian anyons, both in the experimental and theoretical domains.

Figures

Tables

Table 1. Table 1: Quasihole braiding phase φ br MC superscript subscript 𝜑 br MC \varphi_{\text{br}}^{\mathrm{MC}} obtained numerically via equation ( 11 ) (third column, with the Monte Carlo statistical uncertainty) and its prediction φ br subscript 𝜑 br \varphi_{\text{br}} in equation ( 9 ) (fourth column), for M = 2 , 1 𝑀 2 1 M=2,1 and for different parities P N subscript 𝑃 𝑁 P_{N} of the particle number N = 150 𝑁 150 N=150 and N = 149 𝑁 149 N=149 . For the | η 1 | = | η 2 | subscript 𝜂 1 subscript 𝜂 2 |\eta_{1}|=|\eta_{2}| term in equation ( 11 ), we set η 1 = − η 2 subscript 𝜂 1 subscript 𝜂 2 \eta_{1}=-\eta_{2} , which is the optimal configuration for a finite-size system. For M = 2 𝑀 2 M=2 ( M = 1 𝑀 1 M=1 ) case, | η 1 | / l B subscript 𝜂 1 subscript 𝑙 𝐵 |\eta_{1}|/l_{B} is equal to 7.5 (6.5).

$M$	$P_N$	${\phi }_{\text{br}}^{\mathrm{MC}}$ $\left[2\pi \right]$	${\phi }_{\text{br}}$ $\left[2\pi \right]$
$2$ (fermions)	$0$	$0.05\pm 0.06$	$0$
	$1$	$0.49\pm 0.07$	$0.5$
$1$ (bosons)	$0$	$0.13\pm 0.04$	$0.125$
	$1$	$0.59\pm 0.04$	$0.625$

Equations

$${\eta }_{\mu }(t)={\eta }_{\mu }(0)e^{i\theta (t)},\theta (t)=\frac{t}{T}{\theta }_f,$$

$${\hat{H}}_2({\partial }_{z_j},{\partial }_{{\bar{z}}_j},z_j,{\bar{z}}_j;t)={\hat{H}}_1({\partial }_{z_j},{\partial }_{{\bar{z}}_j},z_j,{\bar{z}}_j;t=0)-\frac{{\theta }_f}{T}{\hat{L}}_z,$$

$$|{\Psi }_2(t)⟩=e^{-i{E}_{0}t/\hslash }\sum _{\alpha =1}^m{\gamma }_{\alpha }(t)|{\psi }_{\alpha }⟩,{\gamma }_{\alpha }(0)=⟨{\psi }_{\alpha }⟩{\Psi }_0.$$

$$i\hslash \frac{\mathrm{d}{\gamma }_{\alpha }(t)}{\mathrm{d}t}=-\frac{{\theta }_f}{T}\sum _{\beta =1}^m{\mathcal{L}}_{\alpha \beta }{\gamma }_{\beta }(t),$$

$$|{\Psi }_2(T)⟩=e^{-i{\hat{H}}_{2}T/\hslash }|{\Psi }_0⟩=e^{-i{E}_{0}T/\hslash }{e}^{i{\theta }_f\mathcal{L}/\hslash }|{\Psi }_0⟩,$$

$$|{\Psi }_1(T)⟩$$

$$=e^{-i{E}_{0}T/\hslash }{e}^{-i{\theta }_f{\hat{L}}_z/\hslash }{e}^{i{\theta }_f\mathcal{L}/\hslash }|{\Psi }_0⟩.$$

$$\Psi (\{z_j\})=\text{Pf}(W)\prod _{i<j}{\left(z_i-z_j\right)}^M{e}^{-{\sum }_i|z_i{|}^2/4l_B^2},$$

$$W_{ij}=\frac{({\eta }_1-z_i)({\eta }_2-z_j)+(i\leftrightarrow j)}{z_i-z_j}\forall i\ne j.$$

$$\frac{{\phi }_{\text{br}}}{2\pi }=\frac{1}{4M}-\frac{1}{8}+\frac{P_N}{2}.$$

$$\frac{{\phi }_{\text{br}}}{2\pi }=\frac{1}{\hslash }\left[⟨{\hat{L}}_z{⟩}_{|{\eta }_1|=|{\eta }_2|}-⟨{\hat{L}}_z{⟩}_{{\eta }_1={\eta }_2}\right].$$

$$\frac{{\phi }_{\text{br}}}{2\pi }=\frac{N}{2l_B^2}\left[⟨r^2{⟩}_{|{\eta }_1|=|{\eta }_2|}-⟨r^2{⟩}_{{\eta }_1={\eta }_2}\right].$$

$$\frac{{\phi }_{\text{br}}}{2\pi }=\frac{1}{2l_B^2}{\int }_{A_1,A_2}{r}^2\left[n_{|{\eta }_1|=|{\eta }_2|}(\vec{r})-n_{{\eta }_1={\eta }_2}(\vec{r})\right]d\vec{r}.$$

$$\frac{{\phi }_{\text{br}}}{2\pi }=\frac{1}{2l_B^2}\int d\vec{\rho }{\rho }^2\left[d_{2\text{QH}}(\vec{\rho })-2d_{1\text{QH}}(\vec{\rho })\right],$$

Full text

Introduction.— The standard classification of particles into bosons and fermions breaks down in two spatial dimensions, where exotic objects known as anyons can exist Wilczek (1982); Halperin (1984); Arovas et al. (1984); Leinaas and Myrheim (1977); Wu (1984); Stern (2008). The key concepts for defining the statistics of anyons are the adiabatic motion of one anyon around another, hereafter the braiding, and the adiabatic exchange of the anyons positions Tong (2016). Anyons can be characterized by merging two of them, and the properties of the new composite object depend on the fusion rules of the original anyons. When there is the possibility of fusing in more than one way, anyons can be non-Abelian Moore and Read (1991); Read and Green (2000); Ivanov (2001); Read and Rezayi (1999): they are the heart of topological quantum computation Nayak et al. (2008), and their experimental realization is thus highly desired. Several existing platforms are expected to host them as emergent quasi-particles, but the unambiguous experimental demonstration of their properties is still the matter of an intense debate Willett et al. (2009); Mourik et al. (2012).

In the last twenty years, several works addressed the problem of extracting the properties of the anyons hosted by the ground states of a given Hamiltonian. The simplest approach relies on explicitly following the ground-state evolution when anyons are exchanged Paredes et al. (2001); Tserkovnyak and Simon (2003); Baraban et al. (2009); Wu et al. (2014); Nielsen (2015). Within other approaches, the analytical study of paradigmatic wave functions has also clarified important issues about the statistics of excitations Nayak and Wilczek (1996); Bonderson et al. (2011); Ivanov (2001). On the experimental side, interferometric schemes have been proposed to compare the state before and after the adiabatic time evolution Halperin et al. (2011); Campagnano et al. (2012); Paredes et al. (2001); Das Sarma et al. (2005); Stern and Halperin (2006); Bonderson et al. (2006), but none of them has produced unambiguous results Camino et al. (2005); Rosenow and Halperin (2007).

We propose a method to characterize non-Abelian anyons: By considering geometries where the anyonic statistics can be revealed through rigid rotations of the anyons (see Fig. 1), we relate their statistical phase to the angular momentum and to the density profile of the system. This protocol allows one to identify the existence of different fusion channels, a defining property of non-Abelian anyons, with remarkable experimental simplicity in the context of ultracold atoms Pitaevskii and Stringari (2016); Cooper et al. (2019) and photons Carusotto and Ciuti (2013); Ozawa et al. (2019). Moreover, our study represents a powerful theoretical tool to inspect excitations with unknown statistics, going beyond the observation of multiple fusion channels. As a showcase study, we discuss our method for the case of the Moore-Read (MR) state Moore and Read (1991), and outline an experimental procedure for computing the statistical phases of its quasiholes.

Rigid rotations of the anyons.— We consider a two-dimensional (2D) system of $N$ particles (bosons or fermions) supporting anyonic excitations. The Hamiltonian $\hat{H}_{1}$ is a function of particle positions and momenta, as well as of time. We use the complex coordinate notation $z_{j}=x_{j}+iy_{j}$ for the position of the $j$-th particle. The time dependence of $\hat{H}_{1}(\partial_{z_{j}},\partial_{\bar{z}_{j}},z_{j},\bar{z}_{j};t)$ is only due to a set of parameters $\eta_{\mu}(t)$ defining the centers of some external local potentials $V_{\mathrm{ext}}(z,\eta_{\mu}(t))$. These potentials typically couple with the particle density, creating and pinning the anyons at positions $\eta_{\mu}(t)$ Paredes et al. (2001); Macaluso and Carusotto (2018); Wan et al. (2006, 2008); Prodan and Haldane (2009).

To reveal the anyonic statistics, one option is to braid the anyons through rigid rotations of the pinning-potential coordinates (see Fig. 1). These transformations are defined as

[TABLE]

where $\theta_{f}$ is the final rotation angle and $T$ is the time duration of the process.

Since we consider rigid rotations, we can study the problem in the reference frame $R_{2}$ co-rotating with the anyons, rather than using the laboratory reference frame $R_{1}$. We assume that $V_{\mathrm{ext}}(z,\eta_{\mu}(t))$ is a function of the distance $|z-\eta_{\mu}(t)|$ between particles and anyons, and that the remaining terms in $\hat{H}_{1}$ are rotationally invariant. Under these assumptions, the generator of the time evolution in $R_{2}$ in the time span $[0,T]$ reads Pitaevskii and Stringari (2016):

[TABLE]

which is manifestly time-independent. The first term on the right-hand side is the initial Hamiltonian in $R_{1}$, while the second one describes the effect of the rotation. Being interested in an adiabatic process, we consider $T\to\infty$. The rotation term is then a small contribution and can be treated perturbatively.

To describe the dynamics in $R_{2}$, we consider an initial state $\ket{\Psi_{0}}$ belonging to the $m$-fold degenerate ground-state manifold $\mathcal{H}_{E_{0}}$, spanned by the basis $\{\ket{\psi_{\alpha}}\}_{\alpha=1,\dots,m}$, with $\hat{H}_{1}(t=0)\ket{\psi_{\alpha}}=E_{0}\ket{\psi_{\alpha}}$ and $\braket{\psi_{\alpha}}{\psi_{\beta}}=\delta_{\alpha\beta}$. If the dynamics is slow enough, we can use the adiabatic theorem to state that the dynamics is restricted to $\mathcal{H}_{E_{0}}$ (an explicit proof is in Sup ), and make the following ansatz:

[TABLE]

By applying the Schrödinger equation, we recover the time-evolution equation of the $\gamma_{\alpha}$’s:

[TABLE]

where $\mathcal{L}_{\alpha\beta}=\braket{\psi_{\alpha}}{\hat{L}_{z}}{\psi_{\beta}}$ is the angular momentum restricted to $\mathcal{H}_{E_{0}}$. The solution reads

[TABLE]

in terms of the matrix exponential $\exp\left[i\theta_{f}\mathcal{L}/\hbar\right]$.

To find the state $\ket{\Psi_{1}(T)}$ in the laboratory frame, we need to rotate $\ket{\Psi_{2}(T)}$ by an angle $\theta_{f}$:

[TABLE]

The state in equation (6) is the exact result for an adiabatic braiding process performed through a rigid rotation of all anyons by an angle $\theta_{f}$. We recognize a dynamical phase proportional to $T$, that is unessential to the discussion of non-Abelian statistics and therefore neglected from now on. The remaining geometric contribution is the product of two unitary matrices: $\mathcal{B}$, with matrix elements $\mathcal{B}_{\alpha\beta}=\bra{\psi_{\alpha}}e^{-i\theta_{f}\hat{L}_{z}/\hbar}\ket{\psi_{\beta}}$, and $\mathcal{U}_{\mathrm{B}}\equiv e^{i\theta_{f}\mathcal{L}/\hbar}$, which is the Berry matrix of the adiabatic process under study, once one makes a suitable choice of the basis states for each angle $\theta(t)$ Sup .

To guarantee that the ground-state manifold is $\mathcal{H}_{E_{0}}$ at both times Nayak et al. (2008), the angle $\theta_{f}$ must be such that $\hat{H}_{1}(t)$ is the same at times $t=0$ and $T$. Depending on the anyon positions, this constraint can be satisfied even for rotation angles which are not multiple of $2\pi$ [see Fig. 1 b-c]. When $\theta_{f}=2\pi k$, with $k$ integer, $\mathcal{B}$ is trivially the identity matrix. In this case, $\mathcal{U}_{\mathrm{B}}$ encodes the full geometrical contribution to the time evolution, made up of both topological and non-topological parts. We stress that $\mathcal{U}_{\mathrm{B}}$ only depends on measurable properties of the ground-state manifold at the initial time, namely the angular-momentum matrix elements. Therefore no actual time evolution is needed to measure it, which constitutes an undeniable experimental advantage. The case of $\theta_{f}\neq 2\pi k$ is relevant in the theoretical context, where –in contrast with experimental studies– nothing precludes the extraction of $\mathcal{B}$ [see example in Ref. Sup ]. A comprehensive analysis of this case is left for a future work.

Moore-Read state and its quasihole excitations.— We now consider the MR state, which is described by the wave function Moore and Read (1991)

[TABLE]

where $l_{\text{B}}$ is the magnetic length. $\text{Pf}(W)$ denotes the Pfaffian of the $N\times N$ anti-symmetric matrix $W$, with $W_{ij}=1/(z_{i}-z_{j})$ for $i\neq j$. For even (odd) values of the positive integer $M$, this wave function represents a fermionic (bosonic) FQH state at filling $\nu=1/M$, which belongs to the lowest Landau level (LLL) Tong (2016). This state is the ground state for 2D charged particles, in the presence of a transverse magnetic field and of a specific three-body repulsion Greiter et al. (1991), and it is believed to be in the same universality class of the FQH state observed at filling $\nu=5/2$ Greiter et al. (1992); Morf (1998); Rezayi and Haldane (2000).

In the presence of properly designed external potentials, the ground state may also host a specific number of localized anyonic excitations Wan et al. (2006, 2008); Prodan and Haldane (2009). The quasihole (QH) excitations of the MR state obey non-Abelian statistics Moore and Read (1991); Nayak and Wilczek (1996); Nayak et al. (2008). In particular, they are Ising anyons with an additional Abelian contribution to their statistical phase, and they can fuse in two different ways. For a given set of coordinates $(\eta_{1},\dots,\eta_{2n})$ of $2n$ such QHs, there is a $2^{n-1}$-fold degenerate set of states Nayak and Wilczek (1996).

In the following, we will consider the case $2n=2$, for which the system is not degenerate. In this case, the MR wave function $\Psi^{2\text{QH}}$ has the same form as in Eq. (7); yet the anti-symmetric matrix $W$ depends on the even/odd parity $P_{N}=0,1$ of the particle number $N$. For $P_{N}=0$, it is $N\times N$ and reads

[TABLE]

For $P_{N}=1$, on the other hand, $W$ is a $(N+1)\times(N+1)$ matrix. The $N\times N$ upper-left block is defined as in Eq. (8), while the entries of the $(N+1)$-th row (column) are equal to $+1$ ($-1$) Sup .

The fusion channel of the two QHs depends on $P_{N}$ Bonderson et al. (2011). As a consequence, the braiding of two MR QHs induces a phase $\varphi_{\rm br}$ that depends on $P_{N}$:

[TABLE]

The dependence of the braiding phase $\varphi_{\text{br}}$ on $P_{N}$ is thus a direct indication of the non-Abelian statistics of QHs, because it indicates that the two QHs are in different fusion channels when $N$ is even or odd Nayak et al. (2008).

$\varphi_{\text{br}}$* from the mean square radius.—* As previously mentioned, for a $2\pi$-rotation of the QHs, $\mathcal{B}$ is the identity matrix. For the non-degenerate MR state with two QHs, the unitary transformation $\mathcal{U}(T)$ associated with this process reduces to the phase factor $\mathcal{U}_{\text{B}}=e^{i\varphi_{\text{B}}}$, where $\varphi_{\text{B}}=2\pi\mathcal{L}/\hbar$ is the Berry phase. In this case, $\mathcal{L}$ is the expectation value of the angular-momentum operator over the initial state, $\langle\hat{L}_{z}\rangle$.

The Berry phase $\varphi_{\text{B}}$ has a non-topological contribution, which can be interpreted as an Aharonov-Bohm phase Sup . Although this phase factor contains information on the QH fractional charge, we have to remove it to isolate the QH braiding phase $\varphi_{\text{br}}$. To this purpose we consider the difference between the Berry phases for two particular states [see Fig. 2 a and b]:

[TABLE]

The expectation value $\langle\hat{L}_{z}\rangle_{|\eta_{1}|=|\eta_{2}|}$ is taken on a state with QHs sufficiently far from each other, at positions $\eta_{1}$ and $\eta_{2}$ such that $|\eta_{1}|=|\eta_{2}|$. On the other hand, $\langle\hat{L}_{z}\rangle_{\eta_{1}=\eta_{2}}$ is measured on the state with the two QHs on top of each other at $\eta_{1}=\eta_{2}$ [for details, see Ref. Sup ].

The mean angular momentum of a state in the LLL is related to its mean square radius: $\langle\hat{L}_{z}\rangle/\hbar+N=N\langle r^{2}\rangle/2l^{2}_{B}$ Ho and Mueller (2002); Umucalılar et al. (2018). This simplifies equation (10) which reads

[TABLE]

Moreover, within the LLL approximation, the mean square radius of the cloud, and so $\varphi_{\text{br}}$, can be measured after time-of-flight expansion Read and Cooper (2003); Umucalılar et al. (2018).

To validate equation (11), we compute $\langle r^{2}\rangle$ through the Monte Carlo technique Sup . Numerical results –reported in Table 1 for both $M=2$ (fermionic case) and $M=1$ (bosonic case) and for different parities $P_{N}$ of the particle number $N$– are fully compatible with equation (9). This demonstrates that the existence of multiple fusion channels for the MR QHs can be experimentally probed without braiding them.

$\varphi_{\text{br}}$* from the quasihole density depletions.—* Although the protocol suggested in equation (11) is already close to the current experimental capabilities, it requires the ability to pin QHs with high precision and the knowledge of the particle number. Moreover, $\varphi_{\text{br}}$ is difficult to compute for large systems, since it is a $\mathcal{O}(1)$ number obtained as the difference between two $\mathcal{O}(N^{2})$ quantities. However, equation (11) can be recast in a form which does not depend neither on $N$ nor on the precise QH positions, as we prove in the following. Due to the incompressibility of the FQH states Tong (2016), the densities of the configurations under study only differ in the regions $A_{1}$ and $A_{2}$ surrounding the QHs [see red circles in Fig. 2 a and b]. Therefore, the integrals in equation (11) can be restricted to $A_{1}$ and $A_{2}$:

[TABLE]

In these regions, the densities in Eq. (12) can be expressed in terms of the density depletions $d_{1\text{QH}}$ and $d_{2\text{QH}}$ caused by a single QH and two overlapping QHs Sup . This allows us to write the braiding phase as

[TABLE]

in which $\vec{\rho}$ is the distance from a QH position, $d_{1\text{QH}}(\vec{\rho})=n_{\text{b}}-n_{|\eta_{1}|=|\eta_{2}|}(\vec{\rho}+\eta_{i})$ and $d_{2\text{QH}}(\vec{\rho})=n_{\text{b}}-n_{\eta_{1}=\eta_{2}}(\vec{\rho}+\eta_{i})$ are the aforementioned QH density depletions, w.r.t. the bulk density $n_{\text{b}}=1/2\pi Ml^{2}_{\text{B}}$ [see Fig. 2 c and e]. The integration region must be large enough to ensure an appropriate decay of the density oscillations induced by the QHs. At the same time, a cutoff $\rho<R_{\mathrm{max}}$ is needed to avoid spurious contributions coming from the density deformations generated at the cloud boundaries. The numerical validation of equation (13) is shown in Fig. 2 d, f for the different parities $P_{N}$, and for $M=2,1$. Residual deviations from the expected $\varphi_{\text{br}}$ are due to finite-size effects.

Equation (13) constitutes an operative way to measure $\varphi_{\text{br}}$, which depends only on local properties in the bulk region. As such, it is robust against edge modes, which are the typical low-energy excitations due to finite-temperature effects Macaluso and Carusotto (2017, 2018). Moreover, since $d_{1\text{QH}}(\rho)$ does not depend on $P_{N}$ [see Fig. 2 c and e], all the information on the fusion channels is encoded in $d_{2\text{QH}}(\rho)$, which is completely different for even and odd values of $N$. Although this dependence on $P_{N}$ was already known Prodan and Haldane (2009); Baraban (2010), the key result of our work is that the depletion profiles also contain quantitative information on the braiding phase. Note that this result holds for the QH excitations of any state in the LLL.

Experimental procedure.— While $d_{1\text{QH}}(\rho)$ can be indifferently measured in the ground state with either a single QH or two well-separated ones Wan et al. (2006, 2008), the characterization of two overlapping QHs involves more subtleties: First, the state in Eq. (8) with overlapping QHs may not be the ground state in the presence of a given external potential. For instance, for odd parity $P_{N}$, having two QHs close to each other might cost more energy than just exciting a low-energy fermionic excitation at the boundary Wen (1995); Milovanović and Read (1996); Wan et al. (2006). Furthermore, the presence of these fermionic edge modes may modify the relation between the QHs fusion channel and the particle number parity $P_{N}$ [see footnote [33] in Ref. Wan et al. (2008)].

We thus propose to proceed as follows for the measurement of $d_{2\text{QH}}(\rho)$: two QHs are created far apart, by cooling the system in the presence of pinning potentials. The two QHs are then slowly brought closer and fused Sup . According to the general theory of topological quantum computation Kitaev (2003); Nayak et al. (2008), the fusion channel cannot change during this process, so the system is adiabatically transported into the (possibly metastable) desired state, where the depletion profile $d_{2\text{QH}}(\rho)$ is measured. Note that unless special care is taken, we can argue that in an actual experiment the QH fusion channel will be randomly chosen at each repetition Sup . Nonetheless, the non-Abelian statistics of the QHs will be still visible in the bi-peaked probability for $\varphi_{\text{br}}$. A rigorous proof of this statement requires numerical experiments based on a model Hamiltonian and a particular cooling mechanism; we leave it for a future study.

Conclusions and Outlook.— In this work, we presented a scheme to assess the statistical properties of anyonic excitations which does not rely on any kind of interference. Our protocol is based on a mathematical link between statistics and angular-momentum measurements, derived by considering rigid rotations of the anyons. This relation further simplifies for states in the LLL, where anyonic statistics is encoded in the density profile. Having access to the anyonic statistics without performing any interference scheme is remarkable in itself; moreover, relating statistics to density measurements makes our protocol readily applicable to state-of-the-art experiments with ultracold atoms Cooper et al. (2019) and photons Ozawa et al. (2019).

Beyond the identification of the Moore-Read fusion channels, on which our scheme has been validated, the study of the two-anyons case opens several other perspectives. For example, our method can be employed to distinguish the Moore-Read and anti-Pfaffian states, whose quasiholes have different Abelian contributions to the braiding phase Levin et al. (2007); Lee et al. (2007); Son (2015); Simon (2018a); Feldman (2018); Simon (2018b). Moreover, it gives access to a key property in topological quantum computation Kitaev (2003); Nayak et al. (2008), namely the dependence of the braiding phase on the distance between the anyons Baraban et al. (2009).

Our method can also be useful for theoretical studies of states supporting anyons of unknown type. When one can compute the matrix elements of the angular-momentum and rotation operators in the ground-state manifold, our scheme gives access to all contributions to the time-evolution operator, for any rigid rotation of the anyons. We stress that in the case of non-Abelian anyons rigid rotations are sufficient to induce non-trivial mixing of the ground states Nayak et al. (2008), although only a subset of the possible anyonic exchanges is accessible in this way Sup . Therefore, we envision the possibility of a more precise theoretical characterization of the anyons, beyond the present identification of fusion channels.

Natural extensions of our analysis include other states in the LLL –like the Read-Rezayi state Read and Rezayi (1999)– or the p-wave superconductor, closely related to the Moore-Read state Read and Green (2000). An exciting question is whether the link between the anyonic statistics and the system density remains valid also for lattice systems Hafezi et al. (2007); Mazza et al. (2010); Regnault and Bernevig (2011); Wu et al. (2014); Hafezi et al. (2014); this is the subject of ongoing study Macaluso et al. (2019).