## Looking for intermediate statistics

Elementary particles in three dimensions are either bosons or fermions, depending on their spin. In two dimensions, it is in principle possible to have particles that lie somewhere in between, but detecting the statistics of these so-called anyons directly is tricky. Bartolomei *et al.* built a collider of anyons in a two-dimensional electron gas of GaAs/AlGaAs (see the Perspective by Feldman). Two beams of anyons collided at a beam splitter and then exited the device at two outputs. The researchers studied the correlations of current fluctuations at the outputs, which revealed signatures of anyonic statistics.

## Abstract

Two-dimensional systems can host exotic particles called anyons whose quantum statistics are neither bosonic nor fermionic. For example, the elementary excitations of the fractional quantum Hall effect at filling factor ν = 1/*m* (where *m* is an odd integer) have been predicted to obey Abelian fractional statistics, with a phase ϕ associated with the exchange of two particles equal to π/*m*. However, despite numerous experimental attempts, clear signatures of fractional statistics have remained elusive. We experimentally demonstrate Abelian fractional statistics at filling factor ν = ⅓ by measuring the current correlations resulting from the collision between anyons at a beamsplitter. By analyzing their dependence on the anyon current impinging on the splitter and comparing with recent theoretical models, we extract ϕ = π/3, in agreement with predictions.

In three-dimensional space, elementary excitations fall into two categories depending on the phase ϕ accumulated by the many-body wave function while exchanging two particles. This phase governs the statistics of an ensemble of particles: Bosonic particles, for which ϕ = 0, tend to bunch together, whereas fermions (ϕ = π) antibunch and follow Pauli’s exclusion principle. In two-dimensional systems, other values of ϕ can be realized (*1*, *2*), defining types of elementary excitations called anyons (*3*) that obey fractional or anyonic statistics with intermediate levels of bunching or exclusion. The fractional quantum Hall effect (*4*, *5*), obtained by applying a strong magnetic field perpendicular to a two-dimensional electron gas, is one of the physical systems predicted to host anyons. For a filling factor ν of the first Landau level belonging to the Laughlin series (*5*)—that is, ν = 1/*m*, where *m* is an odd integer—the exchange phase is predicted to be given by ϕ = π/*m *(*6*, *7*) interpolating between the bosonic and fermionic limits.

Direct experimental evidence of fractional statistics has remained elusive. To date, most efforts have focused on the implementation of single-particle interferometers (*8*, *9*), where the output current is expected to be directly sensitive to the exchange phase ϕ. However, despite many experimental attempts (*10*–*15*), clear signatures are still lacking because the observed modulations of the current result not only from the variation of the exchange phase but also from Coulomb blockade and Aharonov-Bohm interference (*16*). In the case of non-Abelian anyons (*17*), where the exchange of quasiparticles is described by topological unitary transformations, recent heat conduction measurements showed evidence of a non-Abelian state (*18*, *19*), although these results give only indirect evidence of the underlying quantum statistics.

Here, we measured the fluctuations or noise of the electrical current generated by the collision of anyons on a beamsplitter (*20*), thereby demonstrating that the elementary excitations of the fractional quantum Hall effect at filling factor ν = ⅓ obey fractional statistics with ϕ = π/3. The measurement of the current noise generated by a single scatterer of fractional quasiparticles (*21*, *22*) has already shown that they carry a fractional charge *e** = *e*/3. Shortly after these seminal works, it was theoretically predicted (*20*, *23*–*26*) that in conductors comprising several scatterers, noise measurements would exhibit two-particle interference effects where exchange statistics play a central role, and would thus be sensitive to the exchange phase ϕ. In this context, current-current correlation measurements in collider geometries are of particular interest, as they have been extensively used to probe the quantum statistics of particles colliding on a beamsplitter. In a seminal two-particle collision experiment, Hong *et al.* (*27*) demonstrated that photons tend to bunch together in the same splitter output, as expected from their bosonic statistics. In contrast, collision experiments implemented in quantum conductors (*28*–*30*) have shown a suppression of the cross-correlations between the output current fluctuations caused by the antibunching of electrons, as expected from their fermionic statistics. This behavior can also be understood as a consequence of the Pauli exclusion principle that forbids two fermions from occupying the same quantum state at the splitter output. This exclusion principle can be generalized to fractional statistics (*31*, *32*) by introducing an exclusion quasiprobability *p *(*20*) interpolating between the fermionic and bosonic limits. In a classical description of a two-particle collision (Fig. 1A) (*33*), *p* accounts for the effects of quantum statistics on the probability *K* of finding two quasiparticles in the same output arm of the beamsplitter: *K* = *T*(1 – *T*)(1 – *p*), where *T* is the single-particle backscattering probability (Fig. 1A). The fermionic case is *p* = 1, leading to perfect antibunching, *K* = 0. Contrary to fermions, the bunching of bosons enhances *K*, meaning that 1 – *p* > 1 and *p* < 0.

To implement collision experiments in quantum conductors, it is necessary to combine a beamsplitter for quasiparticles, a way to guide them ballistically, and two sources to emit them. The two first ingredients can be easily implemented in two-dimensional electron gases in the quantum Hall regime. Quantum point contacts (QPCs) can be used as tunable beamsplitters and, at high magnetic field, charge transport is guided along the chiral edge channels. By combining these elements, single-particle (*34*) and two-particle (*35*) electronic interferometers have been realized, and fermionic antibunching resulting from the collision between two indistinguishable electrons has been observed (*30*). Investigating the anyonic case requires replacing the conventional electron sources (such as biased ohmic contacts) by sources of fractional anyonic quasiparticles. As suggested in (*20*) and as sketched in Fig. 1B, this implies using three QPCs. Two input QPCs labeled QPC1 and QPC2 are biased by dc voltages *V*_{1} and *V*_{2} and tuned in the weak backscattering regime to generate diluted beams of fractional quasiparticles. Indeed, it is known that in the fractional quantum Hall regime, the partitioning of a dc electrical current *I*^{0} with a small backscattering probability *T* ≪ 1 occurs through the random transfer of quasiparticles of fractional charge *q* = *e** (*24*). As experimentally observed, the proportionality of the current noise (*21*, *22*) with the input current *I*^{0}, the transmission *T*, and the fractional charge *e** shows that this random transfer follows a Poissonian law. QPC1 and QPC2 can thus be used as Poissonian sources of anyons, which then collide on a third quantum point contact labeled cQPC; cQPC is used as a beamsplitter in the collision experiment. The fractional statistics of the colliding quasiparticles can be revealed by measuring the cross-correlations between the electrical currents at the output of the beamsplitter.

The sample (Fig. 1C) is a two-dimensional electron gas (GaAs/AlGaAs). The magnetic field is set to *B* = 13 T, corresponding to a filling factor ν = ⅓ for a charge density *n*_{s} = 1.09 × 10^{15} m^{–2}. At this field and at very low electronic temperature *T*_{el} = 30 mK, ballistic charge transport occurs along the edges of the sample without backscattering (*33*). As discussed above, the two quasiparticle sources comprise two quantum point contacts with transmissions *T*_{1} and *T*_{2} (*T*_{1}, *T*_{2} ≪ 1). We apply the voltages *V*_{1} and *V*_{2} to ohmic contacts 1 and 2 in order to generate the input currents toward QPC1 and QPC2,*h* is the Planck constant. QPC1 and QPC2 randomly generate the quasiparticle currents *I*_{1} and *I*_{2} propagating toward the third quantum contact cQPC, which is used as a beamsplitter with transmission *T*. *I*_{1} and *I*_{2} are extracted from the measurement of the voltages at output contacts 5 and 6 with*T* is extracted from the measurement of the small ac current *I*_{ac} flowing in contact 3 when a small ac voltage *V*_{ac} is applied on contact 8:*I*_{3} and δ*I*_{4}, defined as*I*_{3}(*t*) = *I*_{3}(*t*) – 〈*I*_{3}(*t*)〉. They are measured as voltage fluctuations across the quantized Hall resistance *R*_{K} = 3*h*/*e*^{2}, which converts the currents *I*_{3/4} into the voltages *V*_{3/4} = (3*h*/*e*^{2})*I*_{3/4}. *V*_{3} and *V*_{4} are then amplified by a combination of homemade cryogenic and room-temperature low-noise amplifiers. The output voltage cross-correlations, 〈δ*V*_{3}δ*V*_{4}〉, are then measured by integrating *f* centered on the frequency *f*_{0} = 1.11 MHz, set by the *LC* tank circuits connected to contacts 3 and 4. The bandwidth δ*f* = 20 kHz is set by the quality factor, *Q* ≈ 55, of the *LC* resonators in parallel with *R*_{K}. The output voltage cross-correlations are directly proportional to the input current correlations *33*) for calibration of the conversion factor γ].

We focus first on the case *I*_{1} = *I*_{2} by setting the two input QPCs to equal transmission *T*_{S} = *T*_{1} = *T*_{2} and by applying the same voltage *V*_{1} = *V*_{2} at the input ohmic contacts 1 and 2. Defining the total input current *I*_{+} = *I*_{1} + *I*_{2} and the current difference *I*_{–} = *I*_{1} – *I*_{2}, this setting corresponds to a vanishing current difference between the splitter input arms, *I*_{–} = 0. In this regime, the differences between fermionic and anyonic statistics are emphasized. Indeed, a fermionic behavior (*p* = 1) results in a full suppression of the output cross-correlations *p* ≠ 1), negative cross-correlation *20*, *33*) and in the case *I*_{–} = 0, *K* and hence contains the information on the exclusion quasiprobability *p*:*I*_{+}, allowing us to define a generalized Fano factor*P* = –(1 – *p*)*T*_{S}, which shows that *P* is the relevant parameter carrying the information on statistics. The classical calculation thus provides valuable insight into the connection between the measurement of current cross-correlations in a collision experiment and quantum statistics. However, the accurate prediction of *36*). Within this framework, the current cross-correlations, resulting from the collisions between anyons randomly emitted by Poissonian sources, are predicted to directly reflect the braiding statistics of anyons. In the case *I*_{–} = 0 and for anyons with an exchange phase ϕ = π/*m* (with *m* ≥ 3), the cross-correlations *I*_{+}, with a generalized Fano factor explicitly related to ϕ (*20*):*P* is directly related to the exchange phase ϕ = π/*m* , *P* = –2/(*m* – 2) and is independent of *T*_{S} (in the limit *T*_{1} = *T*_{2} = *T*_{S} ≪ 1). In particular, for the filling factor ν = ⅓ (ϕ = π/3), the prediction is *P* = –2, which strongly differs from the fermionic behavior (*P* = 0). In light of the classical model (Eq. 5), the prediction *P* = –2 suggests *p* < 0. This corresponds to a bunching behavior, which is expected at ν = ⅓ because the exchange phase, ϕ = π/3 (*6*, *7*), is closer to the bosonic value (ϕ = 0) than to the fermionic one (ϕ = π). Our main result is the measurement of *P* = –2 at ν = ⅓, providing an experimental demonstration of anyonic statistics with an exchange phase ϕ = π/3 in agreement with predictions.

To enforce the Poissonian emission of fractional quasiparticles from QPC1 and QPC2, we set *T*_{S} = 0.05, in the weak backscattering regime. We then measured the cross-correlations *I*_{+} for different transmissions *T* of the beamsplitter: *T* = 0.2, *T* = 0.27, and *T* = 0.5 (*37*) (Fig. 2). In the three cases, for *I*_{+} ≥ 50 pA, negative cross-correlations varying linearly with the current *I*_{+} were observed. We extracted the slope α of the variation of *T*. The observed *T* dependence agrees with the binomial law *T*(1 – *T*) (dashed line) extending Eq. 7 for transmissions beyond the weak-backscattering regime (*32*). The generalized Fano factor can be extracted from the fit of α with the dependence α = *PT*(1 – *T*), giving *P* = –2.1 ± 0.1 and demonstrating the fractional statistics at ν = ⅓ with the predicted exchange phase ϕ = π/3. In striking contrast, we observed *P* ≈ 0 at filling factor ν = 2 (fig. S5), corresponding to the expected fermionic behavior for an integer filling factor.

The fermionic behavior can be restored at ν = ⅓ by increasing the transmissions of the input QPCs *T*_{1} and *T*_{2}, thereby deviating from the weak-backscattering regime suitable for the emission of anyons. For *T*_{S} = 1 (black points in Fig. 3A), we observed fermionic behavior: *I*_{+}. For intermediate values of *T*_{S}, the current-voltage characteristics of the input QPCs were strongly nonlinear (Fig. 3B). *T*_{S} decreased when *I*_{+} increased, eventually restoring the weak-backscattering limit at large bias. The measurements of *T*_{S} = 0.14 and *T*_{S} = 0.2 (for *I*_{+} = 0) plotted in Fig. 3A reflect this evolution. At low current *I*_{+}, fermionic behavior was observed: *I*_{+} was recovered, with a generalized Fano factor almost constant. *P* slightly increased from *P* = –2.00 ± 0.15 for *T*_{S} = 0.04 to *P* = –1.94 ± 0.12 for *T*_{S} = 0.14, and to *P* = –1.73 ± 0.10 for *T*_{S} = 0.2. As expected, the domain where the fermionic behavior was observed (*T*_{S} increased; it varied from |*I*_{+}| ≤ 200 pA at *T*_{S} = 0.14 to |*I*_{+}| ≤ 400 pA at *T*_{S} = 0.2. These data confirm that *P* = –2 is observed only in the regime of anyon emission and that regular fermionic behavior *P* ≈ 0 takes place away from the weak-backscattering limit.

Finally, we checked in more detail the agreement between our measurements and the quantum description of anyon collisions (*20*) by investigating the dependence of the Fano factor *P* on the ratio *I*_{–}/*I*_{+}. Contrary to the previous experiments where *I*_{–} = 0 was imposed by *V*_{1} = *V*_{2} and *T*_{1} = *T*_{2}, we instead modified the ratio *I*_{–}/*I*_{+} by varying the values of the input voltages *V*_{1} ≠ *V*_{2}. Figure 4A presents the evolution of *I*_{+} for four different values of the ratio *I*_{–}/*I*_{+} and *T*_{S} = 0.05. We observed in the four cases a linear evolution with *I*_{+}, with a slope *P* that decreases when *I*_{–}/*I*_{+} increases. The different values of *P* extracted from a linear fit of the data (dashed lines in Fig. 4A) are plotted in Fig. 4B. For *I*_{–}/*I*_{+} ≤ 0.2, *P* is constant with *P* ≈ –2. *P* then decreases linearly toward *P* ≈ –3 for *I*_{–}/*I*_{+} ≈ 1. These experimental results can be compared with the calculation of (*20*) (dashed line). The excellent agreement between our experimental results and the calculations further supports the quantum description of anyons with ϕ = π/3.

Our measurement of the Fano factor *P* = –2 demonstrates the anyonic statistics of the charge carriers with an exchange phase ϕ = π/3 in accordance with the predictions for the Laughlin state ν = ⅓. Interestingly, the prediction *P* = –2 for ϕ = π/3 is valid when edge reconstruction effects can be neglected. Although neutral modes have been observed even at ν = ⅓ (*38*), the agreement with the prediction for a simple edge structure suggests that their effect can be neglected in our experiment (*33*). Collision experiments similar to ours could be used to characterize the elementary excitations of other fractional quantum Hall phases with different fractional statistics or even more exotic cases where non-Abelian statistics (*17*) are predicted. Additionally, combining collision experiments with the triggered emission of fractional quasiparticles (*39*, *40*) would allow one to perform on-demand braiding of single anyons in a quantum conductor.

## Supplementary Materials

science.sciencemag.org/content/368/6487/173/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S8

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

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**Acknowledgments:**

**Funding:**Supported by the ANR grant “1shot reloaded” (ANR-14-CE32-0017), ERC consolidator grant “EQuO” (no. 648236), and the French RENATECH network.

**Author contributions:**A.M. and G.F. designed the sample; Y.J. fabricated the sample on GaAs/AlGaAs heterostructures grown by A.C. and U.G.; Y.J. and Q.D. designed and fabricated the low-frequency cryogenic amplifiers; M.K., H.B., and R.B. conducted the measurements; H.B., M.K., R.B., A.M., J.-M.B., E.B., B.P., and G.F. participated in the analysis and the writing of the manuscript with input from Y.J. and U.G.; G.F. supervised the project.

**Competing interests:**The authors declare no competing interests.

**Data and materials availability:**All data are available in (

*41*).