Charging energy effects on a single-edge anyon braiding detector

TL;DR

This paper analyzes how charging energy from edge-to-edge capacitance affects the detection of anyon braiding in a single-edge interferometer for Laughlin states. Using a two-point Green's function approach with Dyson's equation to incorporate charging energy, the authors compute the current and finite-frequency cross-correlations, showing that the braiding signature $πλ$ persists but becomes entangled with the loop capacitance $c_E$. They reveal that zeros in the cross-correlation noise, which encode the braiding phase in the $c_E=0$ limit, shift when $c_E>0$, necessitating independent measurement of the loop capacitance; they propose a gate-coupled readout to determine $c_E$ and then extract $ν_λ$ from the remaining braiding-dependent terms. The work provides a practical protocol to robustly access anyonic statistics in realistic devices and suggests extensions to more general FQH states, including non-Abelian ones.

Abstract

We investigate the influence of capacitive coupling on the detection of anyon braiding in a single-edge interferometer realized in the fractional quantum Hall regime. In this setup, a quantum point contact bends a single edge into a loop, where tunneling occurs at the open end and is controlled by the QPC voltage. In contrast with previously studied two-edge geometries, the weak backscattering regime is dominated by the first-order perturbative term, allowing quantum transport quantities to factorize into a non-universal prefactor and a braiding-induced contribution that provides direct access to the universal statistical angle $πλ$. While previous analyses neglected edge-to-edge capacitance, we show that capacitive effects, which are known to play a crucial role in mesoscopic capacitors, modify both the current and the current cross-correlations. Using a two-point Green's function formalism augmented by Dyson's equation to include the charging energy, we quantify how the fluctuations of the cross-correlations depend simultaneously on $λ$ and on the capacitance of the loop. Our results indicate that a reliable extraction of the statistical angle requires a parallel measurement of the loop capacitance, which can be implemented via a charged gate coupled to the junction.

Figures (4)

• Figure 1: Schematic representation of the setup showing positions on the $x$ axis. On the left, the injection QPC is driven by a constant voltage and emits a tunneling current on the right-movers edge. Far after, that current is represented as a sequence of quasi-particles in a certain state $|in\rangle$. They travel the loop, influencing the tunneling process between $-d/2$ and $d/2$. Please note that the $x$ axis is curvilinear, $d$ corresponding to the length of the loop following the upper path. The distance separating the points $-d/2$ and $d/2$ by the QPC is approximately 0. The current is measured by a detector situated at a position $D>d/2$.
• Figure 2: Equation (\ref{['eqN']}) with a unique injection at $\tau_1=d-D$. $N(t)$ expresses the amount of quasi-particles having reached the detector at time $t$. At zeroth order in tunneling, the only relevant parameter is the charging energy. At $c_E=0$, the injected quasi-particle reaches the detector at once at time $t=d$, corresponding to the delay of traveling the loop. For nonzero capacitive coupling, some part of the signal is advanced. When $c_E$ goes to infinity, the quasi-particle reaches the detector at once at $t=0$. Everything happens like if the capacitive coupling had allowed to jump the loop.
• Figure 3: First order current obtained in Eq. \ref{['I_F']}, with a unique injection at $\tau_1=d-D$ and divided by the prefactor $\frac{e\nu\Gamma}{\pi a}e^{\nu\mathcal{G}(d)}$. Different values of $c_E$ are used. Other parameters used are $\nu_\lambda=1/3$ and $\kappa=\pi/5$ so $\sin(2\pi\nu_\lambda-\kappa)-\sin(-\kappa)\simeq1.6$. For $c_E=0$, we find back the result of Ronetti25. As $c_E$ increases, the signal takes longer time to vanish but peaks get thinner. Thus, there is almost no first-order current left at $c_E=20$.
• Figure 4: Spacing between zeros without charging energy (top) and with $c_E=2$ (bottom). The curves correspond to the prediction $\pi d/\int dt\, \sin(2\pi \nu_\lambda F(t))$ as a function of $\nu_\lambda$. Knowing $c_E$, one can measure $\Delta (\gamma d)$ and infer the corresponding value of $\nu_\lambda$. For low capacitance, multiple statistical angles can correspond to the same $\Delta (\gamma d)$.