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

Inès Safi

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Inès Safi. Time domain braiding of anyons revealed through a nonequilibrium fluctuation dissipation theorem. 2025. hal-05413858

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

Inès Safi

Laboratoire de Physique des Solides, CNRS UMR 5802-University Paris-Saclay, 91405, Orsay Cedex-France

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 $\theta$, 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 $\delta$ when $\delta > 1/2$. These results define experimentally accessible schemes to extract either $\theta$ or $\delta$ in minimal single-QPC setups, without relying on interferometry or cross-correlation measurements.

**Introduction.** Two-dimensional systems can host ex-
otic quasiparticles that transcend the conventional bo-
son-fermion dichotomy. A striking realization occurs in
the fractional quantum Hall effect (FQHE), where quasi-
particles not only carry fractional charge $e^*$ [1] but also
obey unconventional exchange statistics—earning them
the name *any*-ons [2]. In Abelian FQHE states, ex-
changing two quasiparticles introduces a phase factor $e^{i\theta}$,
where $\theta$ is the statistical phase. For Laughlin states at
simple fractional filling $v = 1/(2k + 1)$ (with integer $k$),
one finds $e^* = ve$ and $\theta = v\pi$. More complex filling
factors $v_C$, such as the Jain series, remain only partially
understood, with low-energy effective theories predicting
multiple quasiparticle species carrying distinct charges
and statistical phases [3].

Quantum transport through Hall bars with one or
more quantum point contacts (QPCs) has long served as
a primary platform to probe quasiparticle properties. De-
termining their fractional charge is a crucial step toward
accessing their anyonic statistics. Despite challenges in
modeling edge states in the FQHE regime, this has been
successfully achieved [4–6] using the Poissonian fluctu-
ation-dissipation theorem (FDT) for DC shot noise [7],
generalized beyond bipartite systems and inversion sym-
metry within the *unifying nonequilibrium perturbative
theory* (UNEPT) [8, 9]. UNEPT has also enabled al-
ternative charge-determination methods based on time-
dependent transport, such as photoassisted or finite-
frequency current and noise[10] [8, 11, 12]. Their im-
plementation in experiments at $v_C$ [13, 14] indicates the
dominance of a single quasiparticle species. These meth-
ods are model-independent with respect to edge-state de-
scriptions—an essential feature at $v_C$, where the observed
DC current often deviates from Tomonaga-Luttinger liq-
uid (TLL) predictions. Such deviations may result from
edge reconstruction, spatially extended QPCs, or inter-
edge interactions leading to charge fractionalization [15–

[17], all of which can be described within the UNEPT.

Anyonic statistics has mainly been investigated along
two successful but limited DC approaches probing braid-
ing in space or time domain. Interferometric setups [18–
20], supported by theoretical analyses [3, 21, 22], are
highly sensitive to Coulomb interactions [23] and yield
ambiguities in determining $\theta$ [24, 25], particularly at
$v_C$ [19, 20]. Alternatively, cross-correlations in Han-
bury Brown–Twiss geometries [26–28], or configurations
with QPCs injecting diluted anyonic beams [22, 29, 30],
the so-called "anyon colliders" [31] have been realized in
Laughlin states [32, 33]. Yet such experiments do not
directly confirm the TLL modeling assumed in theory.
In particular, cross-correlations cannot disentangle the
braiding phase $\theta$ from nonuniversal parameters, such as
the scaling dimension $\delta$ [34] or a parameter describing
plasmonic-mode renormalization, and the definition of
the associated Fano factor is not unique. Experimen-
tal results for the Jain sequence even show discrepancies
with theory [35–38].

This motivates the search for more direct and unam-
biguous probes of anyonic statistics [33, 39–42], which
have thus far remained largely confined within the
Tomonaga–Luttinger liquid (TLL) framework. A notable
example is the proposal of Ref. [43] to use the width of
the Hong–Ou–Mandel dip for time-controlled anyons gen-
erated by voltage pulses as suggested in Ref. [74]. The
experimental realization in Ref. [44] did not provide con-
clusive evidence of TLL behavior, and its interpretation
has since been revisited in Refs. [45, 46].

In parallel, UNEPT has also provided routes to ex-
tract $\theta$ through finite-frequency noise or AC conductance
in systems with one or two QPC sources [11] where the
precise functional form of observables need not be spec-
ified: one compares two related functions to extract the
unknown Josephson-type frequency $\omega_{dc}$. Here, we adopt
a strategy closer to the determination of the fractional
charge through the poissonnian noise. We demonstrate that a single QPC constitutes the minimal setup required to probe $\theta$. Our proposal relies on two novel time-domain braiding FDTs in the DC and AC regimes, which relate the DC noise either to the DC current via integration or to the susceptibility—the imaginary part of the admittance—a quantity often overlooked in FQHE experiments. We focus on the AC current phase shift $\phi_{\omega}$ induced by a small AC modulation, as it is particularly robust against some types of nonuniversal renormalizations of the AC voltage and current.

Admittance, depending on both the DC drive frequency $\omega_{dc}$ and the AC phase frequency $\omega$, can be expressed via exact nonequilibrium FDTs [47] that have proven useful in various contexts [48–51]. Another class of exact relations, derived within the nonequilibrium bosonized edge theory, connects the admittance defined via chiral currents to the backscattering admittance [52, 53], demonstrating that it is as informative as the noise spectrum. Here we focus on the backscattering admittance at zero DC drive ($\omega_{dc} = 0$), leaving finite-drive and chiral-admittance studies to future work.

Our methods apply to any model within UNEPT that satisfies a time-domain braiding link (Eq. (6)), which forms an alternative to other expressions in Refs. [22, 33, 39, 54]. This extends to general filling factors $\nu_C$, provided a single quasiparticle species dominates, and also to anyon-collider geometries. Rather than starting from a specific TLL model with thermal states and inversion symmetry (reducing to electron-hole symmetry for non-interacting electrons), we treat the braiding constraint as a foundational input. This allows us to handle models with nonequilibrium states and without inversion symmetry, disentangling the role of $\theta$ as a braiding phase from its role as a scaling dimension $\delta$ in the TLL framework.

Extracting $\delta$ remains a key challenge [6, 55–58]. Traditional power-law methods are often unreliable as discussed in Ref [6]. They may also become out of reach in both the low- and high-bias limits for $\delta < 1/2$, rendering a reliable determination of $\delta$ feasible only at finite DC voltage and temperature [59] (see also Appendix E). This appears consistent with some notable experimental works in the FQHE [60–63], as well as in engineered quantum circuits simulating TLL physics [64–66]. The methods introduced here offer a powerful alternative: they remain valid even at vanishing DC bias and at both low and high AC frequencies, enabling the extraction of $\delta$ without invoking asymptotic scaling.

**UNEPT: a short review** While the UNEPT applies to a broad class of physical systems and observables, we focus here on its application to the FQHE. The Hamiltonian reads [8, 9, 12]:

$$ \mathcal{H}(t) = \mathcal{H}_0 + e^{-i\varphi(t)} A + e^{i\varphi(t)} A^{\dagger} \quad (1) $$

where $\varphi(t)$ encodes constant or time-dependent drives. The quasiparticle tunneling current operator is defined by:

$$ \hat{I}_{\text{tun}}(t) = \frac{ie^*}{\hbar} (e^{-i\varphi(t)} A - e^{i\varphi(t)} A^{\dagger}). \quad (2) $$

The term “tunneling” is employed here for convenience; it does not presuppose a bipartite decomposition of $\mathcal{H}_0$. In particular, interactions between the upper and lower edges are fully allowed within this formulation. Here $e^*$ can be viewed as a fractional charge or as a free parameter. The UNEPT is valid under three main conditions: (i) the amplitude of $A$ is small, and second-order perturbation theory yields finite results; (ii) the initial nonequilibrium density matrix $\rho_{neq}$ commutes with $\mathcal{H}_0$, i.e., $[\rho_{neq}, \mathcal{H}_0] = 0$, thus it is stationary, being diagonal with respect to many-body eigenstates; (iii) One has: $\langle A_{\mathcal{H}_0}(t)A_{\mathcal{H}_0}(0) \rangle = 0$, where $A_{\mathcal{H}_0}(t) = e^{i\mathcal{H}_0t}Ae^{-i\mathcal{H}_0t}$. This can be related to charge conservation and, for a certain class of $\hat{\rho}_{neq}$, to a negligible supercurrent, [9] satisfied in the FQHE. All second-order expressions are, therefore, determined by two correlators, which constitute the building blocks of time-dependent transport:

$$ \hbar^2 X_{\to}(t) = \langle A_{\mathcal{H}_0}^{\dagger}(t) A_{\mathcal{H}_0}(0) \rangle; \quad \hbar^2 X_{\leftarrow}(t) = \langle A_{\mathcal{H}_0}(0) A_{\mathcal{H}_0}^{\dagger}(t) \rangle \quad (3) $$

The DC regime is characterized by a drive frequency $\omega_{dc}$ [67] through a phase $\varphi(t) = \omega_{dc}t$. This frequency can satisfy a Josephson-type relation in terms of the applied DC voltage $V$ [8, 68, 69], $\omega_{dc} = e^*/\hbar V$, but it can take more complicated forms in general nonequilibrium setups—such as the “anyon collider” [11]—so here we treat $\omega_{dc}$ as a free parameter. The DC backscattering current average and noise read:

$$ I_{\text{tun}}(\omega_{\text{dc}}) = e^* [X_{\to}(\omega_{\text{dc}}) - X_{\leftarrow}(\omega_{\text{dc}})] \quad (4) $$

$$ S_{\text{tun}}(\omega_{\text{dc}}) = e^{*2} [X_{\to}(\omega_{\text{dc}}) + X_{\leftarrow}(\omega_{\text{dc}})] \quad (5) $$

$X_{\to}$ and $X_{\leftarrow}$ can be interpreted as transfer rates in opposite directions [8], but these relations remain valid even beyond bipartite systems, when $\mathcal{H}_0$ cannot be split into two independent terms, e.g. when upper and lower edges interact. Inverting these expressions, one can equivalently regard $I(\omega_{dc})$ and $S(\omega_{dc})$ as the fundamental building blocks of UNEPT, fully determining time-dependent transport (finite-frequency observables and arbitrary drives) and encoding the properties of both the model and $\hat{\rho}_{neq}$. In the absence of detailed balance or inversion symmetry, the DC backscattering current and noise are generally independent.

**Braiding links** We now impose a central link between the two correlators that introduce the phase $\theta$, $\langle A_{\mathcal{H}_0}^{\dagger}(t)A_{\mathcal{H}_0}(0) \rangle = e^{-2i\theta \text{sign}(t)}\langle A_{\mathcal{H}_0}(0)A_{\mathcal{H}_0}^{\dagger}(t) \rangle$, motivated in Appendix B, thus one has the link:

$$ X_{\to}(t) = e^{-2i\theta \text{sign}(t)} X_{\leftarrow}(t). \quad (6) $$
It is obeyed by different forms derived from the specific TLL model and having a dominant interpretation: injected diluted anyons form a braiding loop in the time-domain with thermally excited quasiparticle-quasihole pairs created by A. [22, 29, 39, 70] Here, such an interpretation is not always relevant, even though we adopt the same terminology to refer to the link in Eq.(6). We do not necessarily have thermal states, tunneling operator form for A, nor require diluted anyons. Regardless of the specific form of A, we choose to impose Eq. (6) by averaging with respect to $\rho_{neq}$. All UNEPT expressions in which $\tan \theta$ appears are grounded in this braiding link. We see indeed that the phase in Eq.(6) is doubled, introducing an indeterminacy of $\theta$ mod $\pi$, and giving the same phase for fermions and bosons.

**Braiding FDT** With Eq. (6), the two correlators are no longer independent: a single function suffices to determine all time-dependent transport. A convenient choice is the retarded correlator $X^R(t) = \theta(t) [X_-(t) - X_+(t)]$, which already provides the DC current [8, 9]:

$$I_{\text{tun}}(\omega_{\text{dc}}) = 2e^* \Re[X^R(\omega_{\text{dc}})]. \quad (7)$$

Now that Eq.(6) holds, we can demonstrate that $X^R$ also determines the DC backscattering shot-noise, by employing tricky manipulations of Fourier transforms given in Appendix C:

$$S_{\text{tun}}(\omega_{\text{dc}}) = -2e^* \cot \theta \Im[X^R(\omega_{\text{dc}})]. \quad (8)$$

This central relation constitutes a braiding nonequilibrium FDT: $X^R$ is a response function even for nonequilibrium initial states and for an arbitrary DC drive $\omega_{dc}$ in the UNEPT validity range. We can check that it holds for the TLL model, either in a two-terminal or a collider type-geometry. It opens several routes to extract the braiding phase $\theta$ or the scaling dimension $\delta$ when $\theta = \pi\delta$ from time-dependent transport. In the case of fermions, $\theta = \pi$, the divergence is compensated by the dependence of $X^R(\omega_{dc})$ on $\theta$. In this Letter we focus on two particularly accessible schemes, requiring only DC noise and its comparison with the current average: either under a finite DC drive, or at zero DC drive with weak AC phase modulation.

**Method 1: DC noise via DC conductance.** One can now use the Kramers–Krönig (KK) relation for the imaginary and real parts of $X^R$. In order to make a more direct connection with the second method and to ensure controlled convergence of the integral, we only focus on the odd and even components of I and S:

$$2\bar{I}_{\text{tun}}(\omega_{\text{dc}}) = I_{\text{tun}}(\omega_{\text{dc}}) - I_{\text{tun}}(-\omega_{\text{dc}}), \quad (9)$$

$$2\bar{S}_{\text{tun}}(\omega_{\text{dc}}) = S_{\text{tun}}(\omega_{\text{dc}}) + S_{\text{tun}}(-\omega_{\text{dc}}), \quad (10)$$

We further consider the differential DC noise and conductance obtained from Eqs.(7), (8) with respect to $\omega_{dc}$,

which permits a free relation between $\omega_{dc}$ and possibly DC drives:

$$\bar{G}_{\text{tun}}(\omega_{\text{dc}}) = \frac{\partial \bar{I}_{\text{tun}}(\omega_{\text{dc}})}{\partial \omega_{\text{dc}}}, \quad (11)$$

$$\frac{\partial \bar{S}_{\text{tun}}(\omega_{\text{dc}})}{\partial \omega_{\text{dc}}} = - \frac{2e^* \omega_{\text{dc}}}{\pi} \cot \theta \, P.V. \int_{0}^{\infty} d\omega' \frac{\bar{G}_{\text{tun}}(\omega_{\text{dc}} = \omega')}{\omega_{\text{dc}}^2 - \omega'^2}. \quad (12)$$

Notice that $\bar{G}$ has the dimension of electric charge. This constitutes a braiding non-equilibrium FDT, offering a route to extract $\theta$ if the fractional charge $e^*$ that rescales the backscattering current is already determined. One can verify that this relation indeed holds for a Tomonaga–Luttinger liquid (TLL) with a scaling dimension $\delta = \theta/\pi$, and that the fermionic limit $\delta = 1$ can be safely taken since the integral, that explicitly depends on $\delta$, vanishes at $\delta = 1$. The main experimental difficulty lies in performing a continuous measurement of the differential conductance. Nevertheless, the integrand is predominantly governed by frequencies $\omega'_{dc}$ close to $\omega_{dc}$ due to the kernel $1/(\omega_{dc}^2 - \omega'_{dc}^2)$. As a consequence, the range of validity of the effective model and the perturbative expansion, which may constrain $\omega_{dc}$ to a finite frequency window, also naturally bounds the main contributions of the integral to the same region.

**Review: UNEPT admittance** We now specify, for simplicity, to $\omega_{dc} = 0$, and replace the DC phase $\omega_{dc}t$ by a small sine phase modulation $\varphi(t) = \delta\varphi_{\omega} \sin(\omega t)$ in Eq.(1). We do not specify the way $\varphi(t)$ is related to possible source or gate voltages or time-dependent boundary conditions on the edges [15]. Thus, similarly to the DC regime (see Eq.(11)), we differentiate the AC current with respect to $\omega\varphi(\omega)$:

$$\begin{aligned} \frac{\delta I_{\text{tun},\omega}(t)}{\omega \delta \varphi_{\omega}} &= G_{\text{tun},\omega} \cos(\omega t) + B_{\text{tun},\omega} \sin(\omega t) \\ &= \sqrt{G_{\text{tun},\omega}^2 + B_{\text{tun},\omega}^2} \cos(\omega t + \phi_{\omega}). \end{aligned} \quad (13)$$

The (rescaled) associated admittance reads $Y_{\text{tun},\omega} = G_{\text{tun},\omega} + iB_{\text{tun},\omega}$, where the rescaled AC conductance $G_{\omega}$ and susceptibility $B_{\omega}$ can be extracted from the in-phase and quadrature components of the current, and provide the phase shift $\phi_{\omega}$:

$$\tan \phi_{\omega} = \frac{B_{\text{tun},\omega}}{G_{\text{tun},\omega}}. \quad (14)$$

Notice that these relations are nonperturbative, and therefore independent of the assumption of the weak backscattering regime. A key advantage of this ratio is that any rescaling of both the voltage and current at the level of the QPC by real or complex factors is cancelled (see Appendix D), and, in the perturbative limit, it remains independent of the QPC reflection coefficient $R$. It
is therefore more robust than the susceptance and naturally provides a built-in calibration with the AC conductance, which is particularly useful when the precise amplitude of the AC voltage at the QPC is not known. However, this cancellation no longer holds if the rescaling enters through a complex multiplicative factor on $Y_{\text{tun},\omega}$, as expected when measuring the chiral current at a distance $L$ from the QPC [52]. In that case, one must restrict to $\omega \ll v/L$, where $v$ is the plasmon velocity—a regime that is also preferable for complex filling factors $\nu_C$, to suppress non-universal interedge interactions or edge reconstruction effects.

Within UNEPT, $Y_{\text{tun},\omega}$ can be expressed to second order in $A$ through the retarded correlator $X^R$ (see Appendix A):

$$ \hbar\omega Y_{\text{tun},\omega} = e^{*2} [-X^R(\omega) + X^{R*}(-\omega) + 2i\Im X^R(0)] \quad (15) $$

In view of the DC current expression in Eq.(7), this determines the AC backscattering conductance [8, 9]:

$$ \omega G_{\text{tun},\omega} = \bar{I}_{\text{tun}}(\omega_{\text{dc}} = \omega), \quad (16) $$

where $\bar{I}$ is the odd part of the backscattering current in Eq. (9). The l. h. s. involves a finite frequency $\omega$ at zero DC frequency drive, while the r. h. s. involves zero frequency current at a finite DC frequency drive $\omega_{\text{dc}} = \omega$.

**Method 2: DC noise versus AC current** Let us now impose the braiding relation in the time-domain. Taking the imaginary part of Eq.(15), this yields a direct relation between susceptance and the even part of the DC noise, in view of Eq. (10) and Eq.(8), which is the third central relation of this letter:

$$ e^* \omega B_{\text{tun},\omega} = -\tan \theta \cdot [\bar{S}_{\text{tun}}(\omega_{\text{dc}} = \omega) - S_{\text{tun}}(\omega_{\text{dc}} = 0)] \quad (17) $$

Thus the susceptance at finite frequency $\omega$ and zero DC drive is related to the even part of the zero-frequency noise generated under a DC drive frequency $\omega_{\text{dc}} = \omega$. Therefore, for fixed $\omega$, measuring both $B_\omega$ and the even part of the DC noise at $\omega_{\text{dc}} = \omega$ and $\omega_{\text{dc}} = 0$ enables direct extraction of $\tan \theta$. Since $B_\omega$ and $G_{\text{tun},\omega}$ are linked through the KK relation, this is consistent with Eq.(12), with the same remark about the restricted domain of integration close to $\omega$.

Now, we can infer the phase shift $\phi_\omega$ in Eqs. (13) and (14) by dividing Eq. (16) by Eq. (17):

$$ \tan \phi_{\omega} = -\tan \theta \cdot \frac{\bar{S}_{\text{tun}}(\omega_{\text{dc}} = \omega) - S_{\text{tun}}(\omega_{\text{dc}} = 0)}{e^* \bar{I}_{\text{tun}}(\omega_{\text{dc}} = \omega)} \quad (18) $$

Comparing the ratio on the r. h. s. at a finite DC drive $\omega_{\text{dc}} = \omega$ to the phase shift $\phi_\omega$, rather measured at finite frequency $\omega$ and zero DC drive ($\omega_{\text{dc}} = 0$), provides $\tan \theta$. This gives $\theta$ with an indeterminacy of $\pi$, as is currently the case for all current methods, but which will be lifted in the case of a TLL.[24]

For initial non-equilibrium states, such as those arising from non-equilibrated edges, temperature gradients, non-thermalized QPCs, or in presence of one [33] or two QPC sources (such as the "anyon collider") [32], $S_{\text{tun}}(\omega_{\text{dc}})$ is super-Poissonian [8, 11] and $S_{\text{tun}}(0)$ reflects a non-equilibrium noise.

<figure><img src="image_2.png" /><figcaption>FIG. 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 $\omega \gg k_B T/\hbar = \omega_{\text{th}}$ (i.e., the quantum regime) induces a dephasing $\pi(1 - \delta)$. The braiding FDT for dephasing (Eq.(14)) holds more generally for low frequencies $\omega$ and non-equilibrium states such as the "anyon collider".</figcaption></figure>

**Initial thermal distribution and the TLL model.** Let us now address the case of an initial thermalized system at a temperature $T$ obeying inversion symmetry, so that $\bar{I} = I$, and let us denote by:

$$ \omega_{\text{th}} = \frac{k_B T}{\hbar} ; \quad \bar{\omega} = \frac{\omega}{\omega_{\text{th}}} $$

In particular, $\bar{S}_{\text{tun}}(\omega_{\text{dc}} = 0) = 2e^* \omega_{\text{th}} G_{\text{tun}}(0)$ is the equilibrium noise, where $G_{\text{tun}}(0) = G_{\text{tun},0}$ (see Eq.(11)) is also the linear AC conductance at $\omega = 0$. Then the relation in Eq.(18) becomes:

$$ \tan \phi_{\omega} = -\tan \theta \cdot \left[ \coth(\bar{\omega}/2) - \frac{2\omega_{\text{th}} G_{\text{tun}}(\omega_{\text{dc}} = 0)}{I_{\text{tun}}(\omega_{\text{dc}} = \omega)} \right] \quad (19) $$

For any pair $(\omega, \omega_{\text{th}})$ within the weak backscattering regime, one can extract $\tan \theta$ by measuring $\phi_\omega$, $G_{\text{tun}}(0)$ and $\bar{I}_{\text{tun}}(\omega_{\text{dc}} = \omega)$—without requiring any direct measurement of DC noise. In most experimental settings, the frequency $\omega$ is fixed. To validate the extracted value of $\tan \theta$ more robustly, it is therefore advantageous to plot both sides of Eq. (19) as functions of temperature. Interestingly, the indeterminacy of $\theta$ modulo $\pi$ is resolved within the TLL model.

More precisely, we adopt the DC current given by the TLL prediction [68] whose scaling behavior is controlled
by $\delta$ (see Appendix E). In that case, Eq. (19) yields:

$$ \tan \phi_{\omega} = - \frac{\tan \theta}{\sinh(\bar{\omega}/2)} \left[ \cosh(\bar{\omega}/2) - \frac{\Gamma(\delta)^2}{|\Gamma(\delta + i\frac{\bar{\omega}}{2\pi})|^2} \right], \quad (20) $$

with $\Gamma$ the Gamma function, so that $\phi_{\omega}$ also obeys the same scaling behavior, and its frequency dependence can be inferred from its T-dependence. Even though one expects that $\theta = \pi\delta$ mod $\pi$ [46], here we have retained $\tan \theta$ to explicitly track its role in the braiding FDT.

We recall that the braiding FDT, and thus the above relation, holds as well at arbitrarily low frequencies. But it is also interesting to analyze the quantum regime, $\omega \gg \omega_{th}$, which one might naively expect to reach at very low temperatures, where the equilibrium noise $S_{\text{tun}}(0) = 2e^*\omega_{th}G_{\text{tun}}(0)$ in Eqs. (18), (19) would vanish as $\omega_{th} \to 0$. [71] This would rather drive the system into the insulating regime [12, 59]. Importantly, the weak backscattering regime imposes an IR bound on $\omega_{th}$, $\omega_{\min}(\delta)$, which depends on $\delta$ and $\mathcal{R}$, such that $G_{\text{tun}}(0) \propto \mathcal{R}T^{2(\delta-1)} \ll \nu e^2/h$ (See Appendix E). Thus the quantum regime requires:

$$ \omega_{\min}(\delta) \le \omega_{th} \ll \omega \ll v/L, \quad (21) $$

where we have recalled the cutoff $v/L$ to avoid renormalization by plasmon propagation over the distance $L$ (assuming $v/L \ll \omega_c$). Once one gets $\bar{\omega} \gg 1$, we obtain from Eq. (19):

$$ \tan \phi_{\omega} = -\tan \theta \left[ 1 - \frac{\Gamma(\delta)^2}{\pi} \left( \frac{|\bar{\omega}|}{2\pi} \right)^{1-2\delta} \right]. \quad (22) $$

Thus, we have a very distinct behavior depending on $\delta$. For $\delta < 1/2$, the second term dominates and cannot be ignored. On the contrary, for $\delta > 1/2$, opening the frequency window in Eq. (21) is easier, and the first term dominates, while the second one induces only a small relative correction when evaluating $\phi_{\omega}$. Thus, one obtains a striking and simple identity:

$$ \phi_{\omega} \approx -\theta \mod \pi. \quad (23) $$

Let us now assume that $\theta = \pi\delta \mod \pi$ [46] so that the methods we have provided a direct access to $\delta$. In addition, and at least in the quantum regime, they resolve the indeterminacy of $\delta$ modulo 1. For that, we notice that for $\delta = 1$, the tunneling current is linear, $\langle I_{\text{tun}}(t) \rangle = G_{\text{tun}}(0) \dot{\varphi}(t)$, yielding $\phi_{\omega} = 0$. By requiring continuity of Eq. (23) as $\delta$ approaches unity, one obtains a complete determination of $\delta > 1/2$ through dephasing in the quantum regime, given by

$$ \delta = 1 - \frac{\phi_{\omega}}{\pi}. \quad (24) $$

This can be illustrated by plotting the ratio $\phi_{\omega}/[\pi(1-\delta)]$ from Eq. (20), which asymptotically approaches unity

<figure><img src="image_3.png" /><figcaption>FIG. 2: Phase shift in the TLL model with $\mathcal{R} = 0.01$, at three values of $\delta$ and $\omega_{th} \ge \omega_{\min}(\delta)$ given in units of the UV cutoff $\omega_c$ by $4.10^{-4}$, $8.10^{-3}$, $10^{-4}$.</figcaption></figure>

at $\delta = 3/4$. For $\delta = 2/3$, this remains valid with a small relative error $\Delta\delta/\delta \simeq 0.02$. By contrast, for $\delta = 1/3 < 1/2$, the second term in Eq. (20) deviates from Eq. (24), as shown in the figure, making the full braiding FDT of Eq. (19) essential to determine $\delta$. We refer to Appendix E, where the temperature dependence of the dephasing is plotted as well.

Even for $\delta > 1/2$, where Eq. (23) is a good approximation, it may be necessary to use Eq. (19) if only low frequencies are experimentally accessible due to practical constraints and the limited window defined in Eq. (21). For instance, if the AC voltage is imposed via the electrochemical potentials $\mu_{L,R}(\omega)$ of the left and right reservoirs, inelastic processes characterized by a time $\tau_{\text{in}}$ must efficiently establish these potentials, requiring $\omega\tau_{\text{in}} \ll 1$ [15].

*Discussion, Conclusion, and Perspectives.* We have derived fluctuation-dissipation theorems (FDTs) that directly encode the imprint of anyonic braiding in the time domain through a phase $\theta$, valid for nonequilibrium initial states within the UNEPT framework. Consequently, our results apply to both standard reservoir geometries and configurations involving one or two QPC sources injecting dilute anyonic fluxes. These FDTs provide robust and experimentally accessible probes of $\theta$ that circumvent the limitations of interferometric or cross-correlation methods.

The first FDT applies in the DC regime and connects the DC backscattering noise to a KK integral over the DC drives. The second, formulated in the AC regime, links the phase shift of the AC current at zero DC bias to the DC backscattering noise, providing a direct route to extract $\tan \theta$. This phase-shift method is particularly robust, as it naturally calibrates itself and suppresses families of nonuniversal renormalization effects. Nevertheless, when measuring chiral currents far from the QPC, sufficiently low frequencies should be used to preserve this cancellation.
As is the case in interferometric and cross-correlation approaches [24], our method inherits an indeterminacy of $\theta$ modulo $\pi$, since $\tan(\pi + \theta) = \tan\theta$. In a thermally induced Tomonaga–Luttinger liquid (TLL) with scaling dimension $\delta > 1/2$, and for $\delta = \theta/\pi$ mod $\pi$ [46], we have shown that the quantum regime, although experimentally demanding, lifts this indeterminacy, as $\delta = 1 - \phi_{\omega}/\pi$. These results are especially timely given the renewed experimental access to TLL physics in fractional quantum Hall systems [60–63] [72] and the emergence of graphene-based quantum Hall platforms free from edge reconstruction [73].

It is important to emphasize that, while the braid- ing link in Eq. (6) successfully captures statistical ef- fects in transport, it remains a weaker condition than its operator-level counterparts, Eqs. (B1)–(B2). The link does not, by itself, imply the existence of quasiparticle operators satisfying those algebraic relations—it instead constitutes a phenomenological encoding of time-domain braiding observable in quantum transport. This distinction is particularly relevant when comparing the FQHE with systems exhibiting fractionalization of charge but not necessarily of statistics. In the integer quantum Hall effect, for example, inter-edge interactions can induce charge fractionalization [15, 74–77], involving nonuniversal plasmonic modes distinct from Laughlin quasiparticles. Similarly, quantum circuits mapped onto TLLs with $\delta$ set by the impedance display effective charges $e^* = \delta e$ [64], even though only electrons are present—fractional whenever the circuit resistance is quantized [65]. Remarkably, in both cases the time-domain braiding relation Eq. (B2) remains valid with $\theta = \pi\delta$ mod $\pi$ [46] (see also alternative links in Refs. [79, 80]). This leads us to conclude that “particles” in quantum transport should be regarded as emergent constructs defined through measurable correlations rather than fundamental entities.

Beyond the FQHE, our framework can thus be applied to a wider class of systems exhibiting nontrivial correlations and emergent braiding in time. Extending these results to both finite DC and AC bias or tunable gate voltages—naturally incorporated within UNEPT—will open additional experimental regimes and provide cross-verification between frequency- and bias-dependent observables. Within this unified treatment, Ref. [58] analyzed the phase shift of photo-assisted current, claiming, in contrast to our present results, that an AC voltage cannot be used to extract $\delta$. Our findings will allow us to clarify and revisit this issue in a separate publication: the phase shift $\pi\delta$ in the second harmonic of the current obtained there follows as a particular limit of our more general braiding FDT.

Finally, the admittance formulation developed here provides a conceptually and experimentally simpler alternative to cross-correlations in “anyon collider” setups [29, 32, 36, 80], complementing our finite-frequency noise proposal of Ref. [11]. By translating anyonic braid-

ing into measurable phase shifts and current–noise relations, it opens the way to direct time-domain detection of fractional statistics with minimal experimental complexity.

**Acknowledgments.** This work was supported by the ANR grant ”QuSig4QuSense” (ANR-21-CE47-0012). The author thanks Benoît Douçot, Aleksander Latyshev, Frédéric Pierre and Bertrand Reulet for valuable discussions or suggestions for the manuscript.

## Appendix A: Admittance in the UNEPT

We derive here the rescaled admittance under an AC phase $\phi(t)$, defined in the time domain as

$$Y_{\text{tun}}(t', t) = \frac{\delta \langle I_{\text{tun}, \mathcal{H}}(t') \rangle}{\delta \dot{\phi}(t)}$$

where the subscript $\mathcal{H}$ denotes the Heisenberg picture with respect to $\mathcal{H}(t)$ in Eq. (1). In general, for finite DC and AC drives, together with a stationary non-equilibrium distribution $\rho_{\text{neq}}$, one must use the generalized non-equilibrium response theory [47], recently tested in Ref. [81]. For simplicity, by setting $\omega_{\text{dc}} = 0$ and taking the limit $V(t) = 0$, we obtain a generalized Kubo formula valid for non-equilibrium states (up to a renormalizing prefactor):

$$i\omega Y_{\text{tun},\omega} = \frac{i}{\hbar} \int_{0}^{\infty} (e^{i\omega t} - 1) \langle [I_{\text{tun},\mathcal{H}}(t), I_{\text{tun},\mathcal{H}}(0)] \rangle dt. \quad (\text{A1})$$

The same exact expression holds when one considers chiral currents instead of backscattering ones, [52, 82] whose relations have been used to analyze finite-frequency noise measurement in the FQHE [14].

Once $Y_{\text{tun},\omega}$ is expressed to second order in $A$, one has simply to substitute $I_{\text{tun},\mathcal{H}}(t)$ with $I_{\text{tun},\mathcal{H}_0}(t) = e^{i\mathcal{H}_0 t} I_{\text{tun}}(t) e^{-i\mathcal{H}_0 t}$, leading to Eq. (15) where $X^R(t) = \theta(t) [X_{\leftarrow}(t) - X_{\rightarrow}(t)]$ and the correlators are given in Eq. (3). The AC conductance and susceptibility, $G_{\text{tun},\omega}$ and $B_{\text{tun},\omega}$, are therefore related to $\bar{I}_{\text{tun}}$ and $\bar{S}_{\text{tun}}$ via Eqs. (16), (8).

## Appendix B: Time domain braiding constraint

Let us consider two oppositely chiral edges with bosonic fields labeled by their chirality $\sigma = \pm$ obeying $[\Phi_{\sigma}(x), \Phi_{\sigma'}(x')] = -i\pi\sigma\delta_{\sigma,\sigma'}\text{sign}(x - x')$ and determining quasiparticle fields: $\Psi_{\sigma}(x) = e^{-i\sigma kx + i\sqrt{\nu}\Phi_{\sigma}(x)}/\sqrt{2\pi a}$. Here, $k$ is the Fermi momentum, and $a = 2\pi v/\omega_c$ is a short-distance cutoff of the order of the magnetic length, with $v$ the plasmonic velocity, which introduces the frequency cutoff $\omega_c$. Using the Baker-Campbell-Hausdorff formula, one obtains the exchange relation for the quasiparticle fields: $\Psi_{\sigma}(x)\Psi_{\sigma'}^{\dagger}(x') =$
$e^{i\pi\nu\sigma} \text{sign}(x-x')\Psi_{\sigma}^{\dagger}(x')\Psi_{\sigma}(x)$, which reveals an exchange fractional phase $\theta = \pi\nu$.

If the Hamiltonian $\mathcal{H}_0$ in Eq.(1) is quadratic with respect to two bosonic fields $\Phi_{\sigma}$ one has $\Phi_{\sigma, \mathcal{H}_0}(x, t) = e^{i\mathcal{H}_0 t} \Phi_{\sigma}(x) e^{-i\mathcal{H}_0 t} = \Phi_{\sigma}(x - \sigma vt)$. Choosing spatial locality by setting $x = x'$, one obtains the braiding relation in the time domain:

$$ \Psi_{\sigma, \mathcal{H}_0}^{\dagger}(x, t) \Psi_{\sigma, \mathcal{H}_0}(x, t') = e^{-i\theta \text{sign}(t-t')} \Psi_{\sigma, \mathcal{H}_0}(x, t') \Psi_{\sigma, \mathcal{H}_0}^{\dagger}(x, t) \quad (B1) $$

Assuming that the QPC backscattering operator in Eq. (1) is given by $A = \Gamma_B \Psi_+^\dagger(0) \Psi_-(0)$, this leads to a similar relation for $A$ with a doubled phase:

$$ A_{\mathcal{H}_0}^{\dagger}(t) A_{\mathcal{H}_0}(0) = e^{-2i\theta \text{sign}(t)} A_{\mathcal{H}_0}(0) A_{\mathcal{H}_0}^{\dagger}(t) \quad (B2) $$

Averaging both sides over the initial distribution $\rho_{neq}$, one obtains Eq.(6). For complex filling factors $\nu_C$, various theoretical models predict the involvement of multiple species of quasiparticles. For each species, the exponential factor $\sqrt{\nu} \Phi$ in $\Psi$ is replaced by a combination of several bosonic fields whose weights determine their individual charges and statistical phases—quantities not always simply related to $\nu$. We will show in a separate publication that our method provides information on these phases which is not sufficient to determine them when multiple species contribute equally. Only when all phases coincide does our present approach apply as it does in the single-species case. The same caution applies to the analysis of cross-correlations in the "anyon collider", contrary to the generality claimed in Ref. [29]. Experiments beyond Laughlin states, whether probing fractional charge [13, 14] or cross-correlations [36, 62], suggest that a single quasiparticle species dominates.

The supplementary time domain braiding condition on $A$ and $\mathcal{H}_0$ in Eq. (6) naturally restricts the class of models to which the UNEPT can be applied, and whose determination presents a compelling challenge. [83]

Let us assume a more restrictive hypothesis of inversion symmetry, thus $X_-(t) = X_+(t) = X(t)$, or $X_+(\omega_{dc}) = X_-(\omega_{dc})$, so that $I$ is odd and $S$ is even in $\omega_{dc}$. A single function $X$ is involved in the time domain braiding link in Eq. (B2). Let us write $X(t) = e^{-g(z)}$ with $z = \tau_0 + it$. Then the most general solution to Eq.(6) reads $g(z) = \theta/\pi \cdot \log(z/\tau_0) + f(z)$ with $f$ analytic separately in the two half planes $Rez > 0$ and $Rez < 0$. This also applies to the case where one has an initial thermal distribution, which imposes the additional detailed balance equation so that $g(z) = \theta/\pi \cdot \log \sin(\pi T z) + f(z)$. The "anyon collider," within the TLL description of the edges, obeys such a form as we will see in a separate publication. However, identifying alternative Hamiltonian terms $\mathcal{H}_0$ and $A$ that give rise to such correlators remains an open problem.

Finally, in case one has an initial thermal distribution $\hat{\rho}_{neq} \rightarrow \hat{\rho}_{eq} \propto e^{-\beta \mathcal{H}_0}$ with $\beta = 1/T$, one has the detailed balance equation $X_-(\omega) = e^{\beta\omega} X_+(\omega)$.

## Appendix C: Proof of the braiding FDT

Here we show how to derive Eq.(8) once we adopt the braiding link in Eq.(6). For that, let us introduce the two transforms of $X_-$ in Eq.(3) over positive and negative times, for $\epsilon = \pm$:

$$ X_{-}^{\epsilon}(\omega) = \int dt \, \theta(\epsilon t) e^{i\omega t} X_{-}(t) $$

where $\theta$ refers here to Heaviside function. Now using Eq.(6), we can write Eq.(5) as:

$$ \begin{aligned} S_{\text{tun}}(\omega_{\text{dc}})/e^{*2} &= \sum_{\epsilon=\pm} (1+e^{2i\epsilon\theta}) X_{-\rightarrow}^{\epsilon}(\omega_{\text{dc}}) \\ &= (1+e^{2i\theta}) [X_{-\rightarrow}^{+}(\omega_{\text{dc}}) + e^{-2i\theta} X_{-\rightarrow}^{-}(\omega_{\text{dc}})] \end{aligned} $$

On the other hand, we can write, using $X^R(t) = \theta(t) [X_-(t) - X_+(t)]$ and $X^*_{-\rightarrow}(t) = X_-(t)$,

$$ \begin{aligned} 2i \Im X^R(\omega_{\text{dc}}) &= \sum_{\epsilon=\pm} \epsilon (1 - e^{2i\epsilon\theta}) X_{-\rightarrow}^{\epsilon}(\omega_{\text{dc}}) \\ &= (1 - e^{2i\theta}) [X_{-\rightarrow}^{+}(\omega_{\text{dc}}) + e^{-2i\theta} X_{-\rightarrow}^{-}(\omega_{\text{dc}})] \end{aligned} $$

Therefore

$$ S_{\text{tun}}(\omega)/e^{*2} = 2i \frac{1 + e^{2i\theta}}{1 - e^{2i\theta}} \Im X^R(\omega) = -2 \cot \theta \Im X^R $$

This provides Eq.(5).

## Appendix D: Renormalization effects

Assuming that non-universal renormalization effects of $Y_{\text{tun},\omega}$ arise from a time-domain convolution, they can be described by a frequency-dependent multiplicative factor $r_{\omega}$. If $r_{\omega}$ is purely real or purely imaginary, it cancels out in the ratio defining the phase shift, Eq. (14). However, this cancelation may fail when measuring chiral currents—i.e., the "chiral admittance"—due to the propagation of plasmonic modes from the QPC over a distance $L$ to the detection points.

For simple filling factors $\nu$, the nonequilibrium bosonized edge theory [52] yields a renormalization factor

$$ r_{\omega} = e^{2iL\omega/v}, \quad (D1) $$

which accounts equally for the propagation of the AC signal from the left reservoir to the QPC and that of the current toward the right reservoir. Consequently, one must restrict frequencies

$$ \omega \ll v/L, \quad (D2) $$

to avoid oscillatory corrections in $\phi_{\omega}$. In this regime, the quantum regime, $\omega_{\text{th}} \ll \omega$ also ensures thermal coherence along the propagation path, as $\omega_{\text{th}} \ll v/L$. For complex
<table><caption>TABLE I: The values of the IR bound in units of the frequency cutoff, $\omega_{min}(\delta)/\omega_c$.</caption><thead><tr><th><i>R</i></th><th>&delta; = 1/3,</th><th>&delta; = 2/3</th></tr></thead><tbody><tr><td>0.1</td><td>6 &times; 10<sup>-2</sup></td><td>1.2 &times; 10<sup>-2</sup></td></tr><tr><td>0.01</td><td>8 &times; 10<sup>-3</sup></td><td>4 &times; 10<sup>-4</sup></td></tr></tbody></table>

filling factors $\nu_C$, one expects a more intricate form of $r_\omega$, arising from coupling between several edge modes or from edge reconstruction. Nevertheless, finite-frequency measurements in Ref. [14], performed at $\nu_C = 2/3$, were successfully analyzed without invoking such oscillatory renormalization factors, suggesting that these effects are either weak or effectively averaged out under experimental conditions.

## Appendix E: AC current in the quantum regime

Interestingly, one can also consider the amplitude of the AC current, as given in Eq. (13). Using Eqs. (16) and (17), along with the Poissonian noise relation, we find:

$$ \frac{\delta I_{\text{tun},\omega}(t)}{\omega \delta \varphi_{\omega}} = \left[ \frac{\bar{I}_{\text{tun}}(\omega_{\text{dc}})}{\omega_{\text{dc}}} \right]_{\omega_{\text{dc}}=\omega} \frac{\cos(\omega t + \theta)}{|\cos \theta|} \quad (\text{E1}) $$

Thus, the amplitude of the AC current near equilibrium is governed both by the braiding phase $\theta$ and the odd part of the DC current evaluated at a nonequilibrium effective voltage, as implied by Eq. (21), so that one could extract $\theta$ via $|\cos \theta|^{-1}$. However, this is less advantageous than using dephasing; the precise amplitude of the AC voltage at the level of the QPC is not known exactly, especially in the quantum regime for which renormalization of the AC voltage is more pronounced.

## APPLICATION TO THE THERMALIZED TLL MODEL

The transmitted DC current through the QPC is given by $G_0\omega_{dc} - I(\omega_{dc})$, where we introduce the perfectly renormalized conductance, which has the dimension of a charge: $G_0 = \nu e^2/2\pi e^*$ so that $G_0 \omega_{dc} = \nu e^2 V/h$. We adopt the current (Eq.(9)) given by the TLL prediction, which is odd, thus $\bar{I}_{\text{tun}} = I_{\text{tun}}$:

$$ \bar{I}_{\text{tun}}(\omega_{\text{dc}} = \omega) = 2\omega_{\text{th}} G_{\text{tun}}(0) \sinh(\bar{\omega}/2) \frac{|\Gamma(\delta + i\frac{\bar{\omega}}{2\pi})|^2}{\Gamma(\delta)^2}, \quad (\text{E2}) $$

$$ G_{\text{tun}}(0) = \frac{e^* \mathcal{R} \Gamma(\delta)^2}{2\pi \Gamma(2\delta)} \left[ (2\pi)^2 \frac{\omega_{\text{th}}}{\omega_c} \right]^{2(\delta-1)} \quad (\text{E3}) $$

A counterintuitive feature appears for $\delta < 1/2$, where

$G_{\text{tun}}(\omega_J \gg \omega_{\text{th}}) < 0$, meaning that backscattering increases the DC conductance. By contrast, $G_{\text{tun}}(\omega_J \ll \omega_{\text{th}}) > 0$. Imposing the condition $G_{\text{tun}}(0) \le 0.1 \nu e^2/h$ defines a minimum temperature or thermal frequency $\omega_{\text{min}}(\delta)$:

$$ \omega_{\text{min}}(\delta) = \omega_c \frac{1}{(2\pi)^2} \left[ \frac{10 e^{*2} \Gamma(\delta)^2 \mathcal{R}}{\Gamma(2\delta) \nu e^2} \right]^{\frac{1}{2(1-\delta)}}. \quad (\text{E4}) $$

It decreases with decreasing $\mathcal{R}$ and increasing scaling dimension $\delta$. It also introduces a nontrivial correction through the second term in Eq. (22) especially when $\delta < 1/2$ or $\mathcal{R}$ is not sufficiently low [59]. Table I recalls the values of $\omega_{\text{min}}(\delta)$ [59] for $\delta = 1/3$ and $\delta = 2/3$ as well as $\mathcal{R} = 0.1$ and $\mathcal{R} = 0.01$, whenever $\nu = 1/3$ and $e^* = e/3$.

For $\nu = \delta = 1/3$ and $\mathcal{R} = 0.1$, the window for the quantum regime disappears, while it remains accessible if $\mathcal{R} = 0.01$. In such cases, however, the equilibrium noise exceeds the nonequilibrium noise evaluated at a DC voltage $\omega_{\text{dc}} = \omega \gg \omega_{\text{th}}$. It is also instructive to examine

<figure><img src="image_4.png" /><figcaption>FIG. 3: TLL model: $T$ or $\omega_{\text{th}}$-dependence of $\phi_\omega/\pi(-\delta+1)$ at a fixed $\omega = 0.1 \omega_c$. The minimum value is given by $\omega_{\text{min}}(\delta)$, extremely small for $\delta = 4/5$ and indicated by the vertical dotted line for $\delta = 1/3$.</figcaption></figure>

the temperature dependence of the phase shift $\phi_\omega$ at a fixed frequency, as shown in Fig. 3. The simple identity $\phi_\omega = \pi(-\delta+1)$ can be attained at very low temperatures and $\delta > 1/2$.

Let us also estimate the UV cutoff $\omega_c$, assumed to be below the gap to bulk quasiparticle excitations. Letting $a \simeq l_B \simeq 10$ nm for $B = 6 - 10$ Teslas, and $v \simeq 10^4 - 10^5 m/s$, one gets $\omega_c \simeq 160 - 1600$ GHz

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