Selective braiding of different anyons in the even-denominator fractional quantum Hall effect

Abstract

Even-denominator quantum Hall states can host several types of anyons with distinct exchange statistics. Depending on the anyon type, exchanging two quasiparticles can impart a phase to the many-body wave function or even transform it into a different state. Here, we realize a gate-tunable Fabry-Pérot interferometer with an embedded antidot that provides local control over the number of anyons within the interference loop. By independently tuning the magnetic field, carrier densities across the device, and the antidot potential, we access regimes in which localized anyons form reproducibly and measure the associated statistical phases $e^{i θ_\mathrm{braid}}$. We resolve braiding phases of $θ_{\mathrm{braid}}=π$ and $θ_{\mathrm{braid}}=\fracπ{2}$, which we attribute to $e/2$ quasiparticles encircling either $e/2$ or $e/4$ quasiparticles, respectively. We further observe switching between different anyon occupancies of the antidot over time, directly resolving individual anyon tunnelling events into the interference loop. Similar behavior occurs at filling factor one third. Our work addresses one of the two key challenges in observing non-Abelian braiding, which requires control of both localized and interfering anyon types.

Figures (4)

• Figure 1: Fabry-Pérot interferometer with an embedded, gate-defined antidot. (a) False-color scanning electron microscopy image of the bilayer graphene-based Fabry--Pérot interferometer with an embedded antidot. A current $I_\mathrm{SD}$ enters through an ohmic contact (yellow), propagates along edge modes, and is partially transmitted through two quantum point contacts (QPCs) formed by the left and right split gates (LSG and RSG). The diagonal resistance is measured as $R_\mathrm{D} = (V_D^{+} - V_D^{-})/I_\mathrm{SD}$ in a single-ground configuration. Scale bar: $3~\upmu \mathrm{m}$. (b) Magnified view of the interference region near the antidot gate (ADG), defined by a central graphite island of area $0.1~\upmu \mathrm{m^2}$ within the lithographically defined interference area of approximately $1~\upmu \mathrm{m^2}$. The left and right air bridges (LBG and RBG), suspended approximately $200~\mathrm{nm}$ above the QPC regions, locally tune the electrostatic potential. Scale bar: $0.3~\upmu \mathrm{m}$. The QPC architecture with a $100~\mathrm{nm}$ gap is shown in the upper-right panel. (c) $\Delta$$R_\mathrm{D}$ at $\nu=-\frac{1}{2}$ displayed in a $B$-$V_\text{PG}$ plane at constant filling ($\alpha_c$), showing clear AB oscillations. (d) Corresponding 2D-FFT analysis. (e-g) Histograms of $R_\mathrm{D}$ at $\nu = -\frac{1}{2}$ as a function of $V_\mathrm{PG}$, constructed from repeated $V_\mathrm{PG}$ sweeps under three different antidot gate voltages $V_\mathrm{ADG}$. The bin size was $50-100~\Omega$, determined by using 150 bins across the interference range. (e) $V_\mathrm{ADG}=0.4~\mathrm{V}$, (f) $V_\mathrm{ADG}=0.235~\mathrm{V}$, (g) $V_\mathrm{ADG}=0.487~\mathrm{V}$. The white and red curves are cosine fits to the histogram peaks, demonstrating phase shifts of approximately $\pi$ in (f) and $\frac{\pi}{2}$ in (g).
• Figure 2: Antidot-tuned interference in integer and fractional quantum Hall states. (a-c) $R_\mathrm{D}$ displayed in the $V_\mathrm{ADG}$–$V_\mathrm{PG}$ plane at $\nu = -1$ under $B = 4~\mathrm{T}$ (a), $\nu = -\frac{1}{2}$ under $B = 8~\mathrm{T}$ (b), and $\nu = -\frac{1}{3}$ under $B = 8~\mathrm{T}$ (c). The indicated periods $\Delta V_\mathrm{ADG}$ were obtained from a 1D-FFT analysis; see SI4. (d, e). Magnified views of $R_\mathrm{D}$ data from the dashed regions in (a) and (c), respectively. Vertical lines mark $V_\mathrm{ADG}$ values where phase discontinuities occur. (f, g) Schematic depiction of the interference area $\delta A$ changing continuously with $V_\mathrm{ADG}$ until a quantized electron or quasiparticle enters the interferometer. (h, i) The interference phase $\theta$ of Eq. \ref{['eqn.theta']} for the scenario illustrated in (f, g), agreeing qualitatively with the data in (d, e).
• Figure 3: Tunable anyon dynamics at $\nu=-\frac{1}{3}$. (a) Time evolution of $R_\mathrm{D}$ at fixed magnetic field and gate voltages over 10 minutes. The excitation timescale $t_\mathrm{ex} \approx 11~\mathrm{s}$ was extracted from the autocorrelations of a 4000 seconds scan. (b, c) $R_\mathrm{D}$ as a function of $V_\mathrm{PG}$, scanned 100 times at a rate of one sweep every 90 seconds, for $V_\mathrm{ADG}=0.457~\mathrm{V}$ and $0.46~\mathrm{V}$, both realizing $\nu_\mathrm{AD}=0$. (d, e) Histograms of $R_\mathrm{D}$ as a function of $V_\mathrm{PG}$, constructed from the data in (b, c). (f) The excited state probability $P_\mathrm{ex}$ and the braiding phase $\theta_\mathrm{braid}$ as a function of $V_\mathrm{ADG}$, extracted from the histograms (see SI7 and SI9).
• Figure 4: Selective anyon dynamics at $\nu=-\frac{1}{2}$. (a, b) $R_\mathrm{D}$ as a function of $V_\mathrm{PG}$, scanned 100 times at a rate of one sweep every 90 seconds, for $V_\mathrm{ADG}=0.235~\mathrm{V}$, realizing $\nu_\mathrm{AD}=-\frac{1}{2}$, and $0.487~\mathrm{V}$, realizing $\nu_\mathrm{AD}=0$. (c, d) Line scans of the consecutive $V_\mathrm{PG}$ sweeps indicated by dashed lines in (a, b). (e) The excited state probability $P_\mathrm{ex}$ and the braiding phase $\theta_\mathrm{braid}$ as a function of $V_\mathrm{ADG}$, extracted from the histograms (see SI8 and SI9).