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Time domain braiding of anyons revealed through a nonequilibrium fluctuation dissipation theorem

Ines Safi

TL;DR

By imposing a time-domain braiding constraint within the UNEPT framework, the authors derive a nonequilibrium fluctuation–dissipation theorem that encodes the anyonic braiding phase $\theta$ in transport observables. They propose two methods to extract $\theta$ (and, in appropriate regimes, the scaling dimension $\delta$): (i) a DC-noise-to-conductance KK integral, and (ii) a DC noise–AC current phase shift relation that yields $\tan\theta$ robustly. In a thermalized Tomonaga–Luttinger liquid with $\delta=\theta/\pi>1/2$, the quantum regime provides a direct link $\phi_{\omega} \approx -\theta$, enabling determination of $\delta$ via $\delta = 1 - \phi_{\omega}/\pi$. The framework remains valid for nonequilibrium initial states and general QPC geometries, offering a practical alternative to interferometry or cross-correlations for probing fractional statistics.

Abstract

We derive a novel fluctuation--dissipation theorem (FDT) valid for nonequilibrium initial states that imprint the braiding of anyons in the time domain. The derivation is carried out within the Unifying Nonequilibrium Perturbative Theory (UNEPT), which applies to both standard reservoir geometries and configurations with one or two quantum point contacts (QPCs) injecting dilute anyonic fluxes. Based on this FDT, we propose complementary methods to determine the time-domain braiding phase. The first method relates the DC backscattering noise to the integral of the current with respect to DC drives, while the second connects the AC current phase shift to the DC noise. The latter provides a particularly robust probe of the statistical angle $θ$, offering an intrinsic calibration and cancelling certain nonuniversal renormalization effects. Specializing to a thermalized Tomonaga--Luttinger liquid (TLL), we further show that in the quantum regime the phase shift enables a direct determination of the scaling dimension $δ$ when $δ> 1/2$. These results define experimentally accessible schemes to extract either $θ$ or $δ$ in minimal single-QPC setups, without relying on interferometry or cross-correlation measurements.

Time domain braiding of anyons revealed through a nonequilibrium fluctuation dissipation theorem

TL;DR

By imposing a time-domain braiding constraint within the UNEPT framework, the authors derive a nonequilibrium fluctuation–dissipation theorem that encodes the anyonic braiding phase in transport observables. They propose two methods to extract (and, in appropriate regimes, the scaling dimension ): (i) a DC-noise-to-conductance KK integral, and (ii) a DC noise–AC current phase shift relation that yields robustly. In a thermalized Tomonaga–Luttinger liquid with , the quantum regime provides a direct link , enabling determination of via . The framework remains valid for nonequilibrium initial states and general QPC geometries, offering a practical alternative to interferometry or cross-correlations for probing fractional statistics.

Abstract

We derive a novel fluctuation--dissipation theorem (FDT) valid for nonequilibrium initial states that imprint the braiding of anyons in the time domain. The derivation is carried out within the Unifying Nonequilibrium Perturbative Theory (UNEPT), which applies to both standard reservoir geometries and configurations with one or two quantum point contacts (QPCs) injecting dilute anyonic fluxes. Based on this FDT, we propose complementary methods to determine the time-domain braiding phase. The first method relates the DC backscattering noise to the integral of the current with respect to DC drives, while the second connects the AC current phase shift to the DC noise. The latter provides a particularly robust probe of the statistical angle , offering an intrinsic calibration and cancelling certain nonuniversal renormalization effects. Specializing to a thermalized Tomonaga--Luttinger liquid (TLL), we further show that in the quantum regime the phase shift enables a direct determination of the scaling dimension when . These results define experimentally accessible schemes to extract either or in minimal single-QPC setups, without relying on interferometry or cross-correlation measurements.

Paper Structure

This paper contains 14 sections, 37 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction.
  2. UNEPT: a short review
  3. Braiding links
  4. Braiding FDT
  5. Method 1: DC noise via DC conductance.
  6. Review: UNEPT admittance
  7. Method 2: DC noise versus AC current
  8. Initial thermal distribution and the TLL model.
  9. Discussion, Conclusion, and Perspectives.
  10. Admittance in the UNEPT
  11. Time domain braiding constraint
  12. Proof of the braiding FDT
  13. Renormalization effects
  14. AC current in the quantum regime

Figures (3)

  • Figure 1: A QPC transfers anyons with charge $e^*$ and braiding phase $\theta$ in the time domain. in a thermalized TLL at temperature $T$ with scaling dimension $\delta=\theta/\pi>1/2$, a small AC voltage at $\omega \gg k_B T/\hbar=\omega_{\rm{th}}$ (i.e., the quantum regime) induces a dephasing $\pi(1-\delta )$. The braiding FDT for dephasing (Eq.\ref{['eq:ratio_omegaJ0']}) holds more generally for low frequencies $\omega$ and non-equilibrium states such as the "anyon collider".
  • Figure 2: Phase shift in the TLL model with $\mathcal{R}=0.01$, at three values of $\delta$ and $\omega_{\rm{th}}\geq \omega_{\rm{min}}(\delta)$ given in units of the UV cutoff $\omega_c$ by $4. 10^{-4}$, $8.10^{-3}$, $10^{-4}$.
  • Figure 3: TLL model: $T$ or $\omega_{\rm{th}}$-dependence of $\phi_{\rm{\omega}}/\pi (-\delta+1)$ at a fixed $\omega=0.1\;\omega_c$. The minimum value is given by $\omega_{\rm{min}}({\delta})$, extremely small for $\delta=4/5$ and indicated by the vertical dotted line for $\delta=1/3$.