Session F17: Anyons in the Fractional Quantum Hall Effect

Quantum Point Contacts and Interferometry in Correlated Topological States of Two-Dimensional Semiconductors

Heterostructures of two-dimensional semiconductors can host a fascinating variety of correlated and topological states, ranging from excitonic insulators to Kondo lattices and the recently observed fractional quantum anomalous Hall seen in twisted homobilayers of MoTe2. These systems offer the potential to control the spatial variation of the electronic phase locally using patterned gates. One ccould use this, for example, to create quantum point contacts and interferometers for investigating boundaries between phases, excitation spectra, and exchange statistics of the excitations. We will report on our progress towards patterned gating, as well as improved contacts, in two-dimensional semiconductor systems.   

Strongly coupled edge states and frequency doubling in a graphene quantum Hall interferometer

Electronic interferometers using chiral, one-dimensional (1D) edge channels of the quantum Hall effect (QHE) can demonstrate a wealth of fundamental phenomena including quasiparticle exchange statistics. When multiple edge channels are involved, FP interferometers in past works have also exhibited Aharonov-Bohm (AB) interference frequency doubling, suggesting pairing of elementary charges to form 2e quasiparticles. Here, following our recent report we discuss measurements in a highly tunable graphene-based QHE FP interferometer that elucidates the connection between interferometer phase jump and AB frequency doubling. By tuning the electron density from the QHE filling factors ν < 2 to ν > 7, we observe continuously modulating periodic interference phase jumps leading to apparent AB oscillation frequency doubling. These results reveal that repulsive Coulomb coupling between the spin-split, copropagating edge channels can explain aspects of this novel strongly coupled regime, where a perfect anti-correlation exists between the two channels. These results expand the understanding of edge state coupling in multichannel QHE interferometers and provide an informative framework for future multi-edge interference experiments.   

Interference of fractionally charged quasiparticles in a graphene quantum Hall interferometer

Fabry-Pérot quantum Hall interferometers provide a direct means to measure the charge and exchange statistics of quasiparticles inherently present in quantum Hall states. In recent years, devices constructed out of GaAs/AlGaAs quantum wells have demonstrated this geometry's aptitude in measuring fractional exchange statistics in several abelian odd-denominator states. Alongside, several groups including our own have been working to develop similar devices in graphene van der Waals heterostructures. While offering exceptional tunability, these graphite-gated devices notably offer a high degree of screening and bulk-capacitive coupling, promoting the desired Aharonov-Bohm operating regime and enabling smaller functioning devices with increased oscillation coherence. While these advantages have enabled our group to observe inter-edge coupling effects in integer quantum Hall states (to be discussed in a separate talk from our group), here we will describe our progress towards observing fractional exchange statistics in graphene-based devices. Specifically, we report on observing clear Aharonov-Bohm oscillations when interfering the fractional edge in odd-denominator fillings while modulating the effective device area with electrostatic gating.   

1/3 phase shifts in graphene Fabry-Perot quantum Hall interferometers at integer bulk filling factors

The Fabry-Perot (FP) quantum Hall interferometer is considered a powerful tool for exploring anyon statistics in the fractional quantum Hall regime. Here, we present Aharonov-Bohm interferences in a high-mobility graphene FP quantum Hall interferometer. We observed phase shifts of 1/3 of the period at integer bulk filling factors νb = 1 and νb = 2 under various quantum point contacts settings. Our results challenge the interpretation in terms of anyon braiding of similar phase shifts recently observed in GaAs in the fractional quantum Hall regime.   

Quantum Hall interferometry at finite bias with multiple edge channels

In a quantum Hall interferometer, the dependence of the signal on source-drain voltage is controlled by details of the edge physics, such as the velocities of edge modes and the interaction between them and with screening layers. Such dependence of the signal has been seen in recent experiments at various integer and fractional filling factors, including ν=2 and ν=2/5 where two edge modes are present. Here we study theoretically the current-voltage curves for various values of the relative edge velocities, interaction strength and the temperature, in a model containing two edge modes. We consider separate cases where the inner mode or the outer mode is weakly backscattered at the tunneling contacts. When the inner mode is completely reflected and the outer mode is partially transmitted, we find striking features at low temperature related to resonance of excitation of the closed inner channel. However, these features disappear rapidly with increasing temperatures. We argue that the effect of a two-dimensional screening layer can be best captured by a model with an additional chiral edge channel, rather than a model coupled to an ideal two-dimensional conductor. We compare our predictions for ν=2/5 with the experimental observation in Ref. [1] including screening effects.

 

[1] J. Nakamura, S. Liang, G. C. Gardner, and M. J. Manfra, Fabry-Perot interferometry at the ν = 2/5 fractional quantum Hall state, arXiv:2304.12415 (2023).   

Testing non-Abelian and Abelian statistics by controlled single quasiparticle addition to quantum Hall interferometers

Fractionally charged quasiparticle excitations in two-dimensional electron systems have been proposed to obey anyonic Abelian statistics and in some cases anyonic non-Abelian statistics. Demonstration of these statistics has been difficult but could be made explicit with control of individual quasiparticle number within the interferometer, where these quasiparticles are encircled by the trajectories of interfering quasiparticles. Here we use a Fabry-Perot interferometer equipped with a small interior top gate that allows addition or subtraction of a single encircled quasiparticle/quasihole (e.g. charge e/4). At the 7/2 filling factor fractional quantum Hall state, which is anticipated to have non-Abelian excitations, such controlled addition of quasiparticles results in systematic pi phase changes in the resistance oscillations over the range of magnetic field sweeps. The oscillations themselves are due to change in the number of bulk quasiparticles induced by the changing magnetic field (different from those trapped by the top gate!) . This pi phase shift effect due to change in the number of charge e/4 quasiparticles is a specific signature of non-Abelian statistics at 7/2 (and 5/2), the even-odd effect. The method of central gate charge modulation is substantiated here by its demonstration at filling factor 2+1/3. These results at 7/3 demonstrate controlled phase change consistent with anyonic statistics, and at 7/2 provide further evidence for the non-Abelian nature of charge e/4 excitations at that filling factor.   

Dynamics and Entanglement Properties of Anyons in the Quantum Hall Bulk

Since the experimental detection of anyons, there has been a resurgence of interest in the field of fractional statistics. In this talk, we present a study of the dynamics of anyons in the quantum Hall bulk. Using the Husimi-Q distribution, we describe the scattering of two-particle anyon coherent states in the lowest Landau level in the presence of a saddle potential and their relative motion in an elliptical trap.  We compute the dynamically evolving mutual entanglement properties of anyons in such quadratic potentials and contrast them with fermions, bosons, and distinguishable particles to identify the signatures of quantum statistics in this measure. Finally, we analyze the general algebraic structure and symmetries in quadratic potentials in the lowest Landau level to better organize and understand the structure in the dynamics.   

Classical signatures in the dynamics of two-particle coherent states

It is known that the time evolution of a given set of quantum states can be formulated in terms of classical dynamics over their projected effective phase space. We investigate this effective phase space and the associated Lagrangian dynamics by considering two-particle coherent states in the symmetric (bosonic) and anti-symmetric (fermionic) configurations. By applying this formalism to 1D scattering problems as well as quadratic potentials in Hall physics regimes, we show that the exchange statistics of the particles in question leave distinct signatures in their classical trajectories. Further, the particles are seen to experience an effective “statistical force” that is a classical manifestation of their quantum mechanical exclusion properties. We also show that this formalism is easily extended to many-body problems, and thus offers a way to extract classical trajectories in these complex and possibly chaotic systems.   

Fractional quantum Hall Fabry-Pérot edge-state interferometry in graphene

Quantum Hall edge state Fabry-Pérot interferometers provide a useful platform in which to study phase-coherent transport in the quantum Hall regime. A seminal experiment used this type of device to demonstrate the anyon braiding statistics of quasiparticles in the fractional quantum Hall (FQH) regime in a GaAs heterostructure [1]. Due to the promising characteristics of its FQH states—particularly the large energy gaps observed at even-denominator—graphene is an attractive venue in which to replicate and extend these results. Prior experiments have shown Fabry-Perot interference of integer quantum Hall edge modes, but until now interference in the FQH regime has remained elusive [2 - 5].

Here, we report robust, high-visibility Aharonov-Bohm dominated edge state Fabry-Pérot interference in the FQH regime. Our devices consist of small-area (less than 1 μm²) all-van der Waals-gate-defined interferometers. We present our findings on the effects of Coulomb interactions, bulk-edge coupling, and the interplay between charge fluctuations and anyon statistics on the observed interference patterns.

[1] Nakamura et. al. Nature Physics 16, 931–936 (2020)

[2] Ronen et. al. Nature Nanotech. 16, 563–569 (2021)

[3] Deprez et. al. Nature Nanotech. 16, 555–562 (2021)

[4] Zhao et. al. Nano Lett. 2022, 22, 23, 9645–9651 (2022)

[5] Fu et. al. Nano Lett. 2023, 23, 2, 718–725 (2023)   

Time-domain braiding of anyons

Experimental evidence of anyon fractional statistics has been so far exclusively obtained in the DC regime [1-5], without possibility of a time-domain study. We demonstrate here the on-demand generation of subnanosecond single anyon current pulses. These pulses are artificial anyons whose fractional statistics can be continuously tuned by varying the fractional charge carried by each pulse [6-8]. In this work, we use artificial anyons as a probe to study the dynamics of the tunneling of bulk topological anyons in the time domain.

We implement a Hong-Ou-Mandel (HOM) experiment between two artificial anyons at a quantum point contact (QPC) in a fractional quantum Hall fluid at filling factor nu=1/3. The incoming artificial anyons and the tunneling topological anyons braid at the QPC, thus effectively probing the role of anyon braiding on the characteristic timescale of anyon tunneling. We measure as proposed in [6] a HOM dip in the current noise at the outputs of the QPC, which width depends on the characteristic timescale for tunneling. By comparing integer and fractionally charged pulses, we observe that anyon dynamics is controlled by the scaling dimension, contrasting with the electron case where without braiding, the timescale for tunneling is set by the temporal width of the current pulses.

This experiment provides a new route for studying the role of braiding on the dynamics and temporal correlations of topological excitations. It also opens the way to a new generation of experiments where anyons are emitted on-demand in a circuit.

 

[1] H. Bartolomei et al., Science 368, 173 (2020)

[2] J. Nakamura et al., Nature Physics 16, 931 (2020)

[3] M. Ruelle et al., Phys. Rev. X 13, 011031 (2023)

[4] P. Glidic et al., Phys. Rev. X 13, 011030 (2023)

[5] J.-Y. M. Lee et al., Nature 617, 277 (2023)

[6] T. Jonckheere et al., Phys. Rev. Lett., 130, 186203 (2023)

[7] J.Y. Lee et al., Phys. Rev. Lett. 125, 196802 (2020)

[8] C. Mora, arXiv:2212.05123 (2022)   

Studying the non-abelian even-denominator state in Bernal-stacked bilayer graphene by scanning tunneling microscope

Non-abelian anyons have drawn interest from physicists because of their potential application as qubits in topological quantum computation. The even-denominator fractional quantum Hall (FQH) states are candidates for achieving non-abelian anyons. In this talk, I will discuss our recent scanning tunneling microscope (STM) measurement of the FQH states in the N=1 Landau levels of bilayer graphene. By measuring thermodynamic gaps locally, we find energy gaps for the even-denominator FQH states to be around 30 K at 14 T, larger by one or two orders of magnitude than prior transport and penetration capacitance measurements in other related systems. We find supporting evidence for the even-denominator states to be non-abelian by observing its Levin-Halperin daughter states. We also measure the influence of local single-atom defects on the even-denominator state gaps and identify the intrinsic FQH gaps with sizes consistent with theoretical calculations. The large gap reported in this study makes bilayer graphene an ideal platform for the future realization of topological qubits based on non-abelian anyons.   

Non-abelian parton states from two body interactions on a lattice

Much attention has been given recently to a special class of multi-component fractional quantum Hall states that can be stabilized by solvable two-body interactions and feature excitations with non-abelian statistics. Owing to the simplicity of their parent hamiltonians, these parton states can be rigorously analyzed in terms of clustering conditions and zero-mode counting, even though the presence of higher Landau levels prompts one to resort to a second-quantized treatment, at least in part. The same simplicity might render these states attractive candidates to be realized in quantum simulators, provided that we connect them to appropriate lattice Hamiltonians; this talk will introduce a scheme for developing discrete Hamiltonians that faithfully represent the physics of continuum mixed Landau level fractional quantum Hall states on a lattice.   

Fusion Mechanism for Quasiparticles and Topological Order for Quantum Hall Fluids in the Lowest-Landau-Level

We study systems exhibiting the fractional quantum Hall effect using the properties of the ring of symmetric polynomials and concepts originating from the second quantization formalism. These techniques are then advanced to examine elementary fractionally-charged excitations for such fluids. In particular, rigorous formulation of  Laughlin's originally suggested fractionalization mechanism in the second quantization formalism led us to a derivation of the quasielectron wave function for systems with filling fractions 1/M,  and the observation that these quasiparticles are subject to a  particular fusion mechanism. Extending these ideas to more complex systems originating from Halperin's bilayer systems, we are able to rigorously study such excitations in non-Abelian fluids like  Moore-Read (Pfaffian) or Hafnian states, as well as other classes of states described by more complex polynomials. Furthermore, the approach based on second quantization allows us to generate non-local composite (generalized Read) operators that characterize topological or the so-called off-diagonal long-range order.  In particular, we rigorously find concrete forms of such operators for the families of Pfaffian and Hafnian quantum states.   

Anisotropy in GaAs/AlGaAs Heterostructures and its Effect on Interferometer Performance

Intrinsic and extrinsic anisotropies are present in high mobility GaAs/AlGaAs heterostructures used to study the Fractional Quantum Hall Effect (FQHE). These anisotropies effect magnetotransport behavior along orthogonal crystal axes. These transport differences have an effect on the performance of devices created with these heterostructures. We show how the anisotropy presents in bulk heterostructure samples and how the bulk transport correlates with interferometer device performance.