Charging energy effects on a single-edge anyon braiding detector

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.

💡 Research Summary

In this work the authors address a crucial missing ingredient in the theory of single‑edge anyon interferometry: the capacitive coupling (charging energy) of the loop formed by a quantum point contact (QPC) in a fractional quantum Hall (FQH) bar. The device under study consists of a single chiral edge that is bent by a QPC into a closed loop of perimeter d. Tunneling occurs at the open end of the loop and is controlled by the QPC voltage. Unlike the conventional two‑edge Fabry‑Perot or anyon‑collider geometries, the weak‑backscattering regime of this single‑edge configuration is dominated by the first‑order perturbative term in the tunneling amplitude Γ. Consequently, transport observables (current and current cross‑correlations) factorize into a non‑universal prefactor and a universal “braiding contribution” that directly encodes the statistical angle πλ (λ = ν for Laughlin states).

Previous theoretical treatments ignored the edge‑to‑edge capacitance that inevitably appears because the loop encloses a finite area and is coupled to the surrounding gates, an effect well known from mesoscopic capacitors. The authors therefore extend the standard bosonized Luttinger‑Liquid description of the edge by adding a charging Hamiltonian H_E = (1/2C) Q², where Q is the excess charge stored on the loop. In bosonized form this term becomes a bilinear coupling of the bosonic fields at the two junction points x = ±d/2, characterized by a charging energy c_E = e²ν/(2πC d²).

The central technical achievement is the derivation of the full Keldysh Green’s functions in the presence of H_E. Starting from the bare Green’s functions of the chiral boson, the authors set up Dyson equations with a self‑energy that is a delta‑function localized at the junction ends. Solving these equations yields modified retarded, advanced, and Keldysh propagators. The retarded propagator for points on opposite sides of the junction acquires an extra factor F(t) that satisfies an integral equation involving the charging energy. This factor encodes both a time delay (proportional to the loop length d) and an exponential‑like damping governed by c_E.

With the dressed propagators in hand, the authors compute the time‑dependent edge current I(x,t) and its zero‑frequency cross‑correlation S_{12}. The current is expanded perturbatively in Γ: the zeroth‑order term reflects the incoming anyon stream prepared in a time‑ordered state |in⟩, while the first‑order term (∝ Γ²) captures tunneling across the junction. The cross‑correlation, which is the key observable for extracting the braiding phase, contains contributions up to Γ⁴. Importantly, the authors find that the variance of S_{12} depends not only on the statistical angle λ but also on the loop capacitance C. For a fixed λ, increasing C (i.e., reducing the charging energy) suppresses the amplitude of the braiding‑induced fluctuations, whereas a small C enhances them. Thus, without an independent measurement of C, the extraction of πλ from noise data would be ambiguous.

To make the protocol experimentally viable, the paper proposes adding a metallic gate capacitively coupled to the loop. By measuring the gate‑induced charge response (for example, via a separate charge sensor or by monitoring the gate‑current as a function of gate voltage), the loop capacitance can be determined in situ. This parallel measurement allows one to calibrate the non‑universal prefactor and isolate the universal braiding contribution.

The manuscript concludes by emphasizing that the Dyson‑based Green’s‑function framework is not limited to pure capacitive coupling; it can be extended to include inductive elements, non‑linear capacitance, or environmental impedance, thereby providing a versatile toolbox for realistic anyon interferometry. Overall, the paper delivers a comprehensive theoretical analysis that bridges the gap between idealized anyon braiding detectors and the practical constraints imposed by charging effects, offering clear guidance for future experiments aiming to unambiguously measure fractional statistics.