# Unexpected non-universality of the time braiding phase of anyons tied by the scaling dimension

Aleksander Latyshev, Inès Safi

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Aleksander Latyshev, Inès Safi. Unexpected non-universality of the time braiding phase of anyons tied by the scaling dimension. 2025. hal-05413852

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# Unexpected nonuniversality of the time braiding phase of anyons tied by the scaling dimension

Aleksander Latyshev¹ and Inès Safi¹

¹Laboratoire de Physique des Solides (UMR 5802), CNRS-Université Paris-Saclay, Bâtiment 510, 91405 Orsay, France

We use a braiding nonequilibrium fluctuation dissipation relation linking the DC noise to the response function inferred from the braiding constraint in the time-domain with a phase $\theta$ within the UNEPT (Unified Non equilibrium Perturbative Theory). By applying the Kramers-Krönig relations, we obtain an integral equation connecting DC current and noise that involves $\theta$. By specifying to thermal states so that noise is Poissonian, we find an analytical solution for the DC current via the Wiener-Hopf technique. It reveals that the time-braiding phase is determined by the scaling dimension $\delta$. This questions the universality of $\theta$ that can reflect the microscopic edge dynamics, in contrast to the topologically protected braiding phase in the space domain.

## Introduction
Anyons, quasiparticles exhibiting exchange phases intermediate between bosons and fermions, are expected to emerge in the fractional quantum Hall effect (FQHE) [1, 2]. Their braiding statistics manifest through quantum correlations and transport signatures, whose probing remains a central challenge. Two main lines of investigation have focused on braiding either in the spatial or temporal domain, both believed to yield the same universal topologically protected phase $\theta$.

The first approach is realized in interferometric setups, where interference between paths enclosing a magnetic flux reveals the statistical phase [3–7]. Such schemes are however notoriously sensitive to Coulomb interactions [8] and device-specific parameters, which complicates their interpretation and limits their universality [9, 10].

The second line is based on current cross-correlations involving two or more quantum point contacts (QPCs), such as Hanbury-Brown and Twiss setups [11–13] or the so-called anyon collider [14] geometry [15, 16], recently achieved in pioneering experiments [17, 18]. These setups have been interpreted as probing braiding in the time domain [19–24].

Yet, such experiments do not directly confirm the Tomonaga-Luttinger liquid (TLL) model with a nonuniversal scaling dimension $\delta$[25]. In particular, cross-correlations cannot disentangle the time-braiding phase $\theta$ from $\delta$ or from parameters describing plasmonic-mode renormalization. Experimental results beyond Laughlin states even show quantitative discrepancies with theoretical expectations [26–29].

This motivates the search for more direct and unambiguous probes of anyonic statistics [18, 20, 21, 30, 31], which so far have remained largely confined within the TLL framework. Some of these approaches are based on a single QPC [30, 32]. In particular, based on Ref. [33], which analyzes the width of the Hong-Ou-Mandel (HOM) dip for time-controlled anyons generated by voltage pulses (as originally suggested in Ref. [34]), the experimental work of Ref. [35] reports the disentanglement of the determination of the TLL parameter $\delta = 2/3$ from that of the braiding phase in time $\theta = \pi/3$, which, unlike $\delta$, is expected to be universal. However, that work did not provide conclusive evidence of TLL behavior, and its interpretation has since

been revisited in Refs. [36, 37].

Recent advances in dynamical transport, formulated within a unified nonequilibrium perturbative theory (UNEPT) [38–40], have suggested alternative routes for accessing the anyonic phase through noise and admittance measurements, without relying on specific TLL models [39, 41]. In particular, Ref. [41] analyzes both single- and multi-QPC configurations by imposing the constraint of time-domain braiding with a statistical phase $\theta$ within the UNEPT framework. Its key insight is the formulation of a *braiding fluctuation-dissipation theorem* (FDT), which relates the imaginary part of the response function to the DC shot noise even in nonequilibrium regimes, thereby encoding fundamental information about the phase $\theta$. This braiding FDT admits equivalent forms connecting the DC shot noise either to the differential conductance or to the admittance, thus providing model-independent methods for extracting $\theta$. This conceptual advance indicates that the anyonic braiding phase in the time domain can, in principle, be inferred using a single quantum point contact (QPC), without requiring anyon collider geometries although the same relations remain valid there as well.

In the present letter, we employ the braiding FDT to derive an integral equation relating the DC noise to the DC current. We then focus on a single QPC initialized in thermal equilibrium, where the DC noise is Poissonian. This leads to an integral equation for the DC current, which we solve using the Wiener-Hopf (W-H) technique [42, 43], uncovering deep analogies between the time-domain braiding phase and the analytic properties of the response function. Building upon this framework, we demonstrate that the time braiding phase $\theta$ is intimately connected to the scaling dimension $\delta$, and therefore shares its nonuniversality. This suggests that the phase $\theta$ may differ from the spatial braiding phase, which is defined independently of the system's dynamics.

At the first step, we consider the setup illustrated in Fig. 1, described by the time-dependent Hamiltonian

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

where the Josephson-type frequency $\omega_{dc} = e^*(V_u - V_d)/\hbar$ might account for the DC bias between the upper and lower edge states in a two-terminal geometry, or, in the case of an
<figure><img src="image_1.png" /><figcaption>FIG. 1. A QPC in the quantum Hall regime at an integer or fractional filling factor $\nu$. We focus here on edges such that each harbors only a single chiral mode. While $V_u$ and $V_d$ denote the reservoir voltages.</figcaption></figure>

anyon collider, is more intricately related to the DC drives applied to the source QPCs. It can therefore be generally regarded as a free parameter of the UNEPT. The tunneling operator

$$A = \xi \psi_u^\dagger(0) \psi_d(0), \quad (2)$$

introduces a small coupling between the channels, characterized by the tunneling amplitude $\xi$, which serves as the perturbative parameter in our analysis. Here we assume a single dominant tunneling process of a quasiparticle species with upper and down fields $\psi_{u(d)}$.

We now impose an exchange link in the time domain acting on the quasiparticle tunneling operator $A$. This construction is motivated by the case of chiral propagation, where the exchange phase in space can be identified with that in time, as discussed in Appendix. The generic forms to express time braiding in term of $A$ rather compares its correlator in nonequilibrium to that at equilibrium, and has been interpreted as the loops formed by injected anyons with quasiparticle-hole pairs created at the QPC by $A$ [19, 21]. This interpretation is not essential here, as we do not specify a reference given by an initial thermal state. Instead, we propose an alternative form that does not rely on the explicit identification of $A$ nor the nonequilibrium density matrix $\rho_{\text{neq}}$:[41] $\langle A_i^\dagger(t) A_i(0) \rangle = e^{-2i\theta \text{sign}(t)} \langle A_i(0) A_i^\dagger(t) \rangle$, which can be written as:

$$X_{\to}(t) = e^{-2i\theta \text{sign}(t)} X_{\leftarrow}(t), \quad (3)$$

where we have introduced the two correlators:

$$\begin{aligned} X_{\to}(t) &= \langle A_i^\dagger(t) A_i(0) \rangle / \hbar^2 \\ X_{\leftarrow}(t) &= \langle A_i(0) A_i^\dagger(t) \rangle / \hbar^2. \end{aligned} \quad (4)$$

Notice that contrary to all previous studies, we do not impose inversion symmetry, thus $X_{\to}(t) \neq X_{\leftarrow}(-t)$. For the most general Hamiltonian in Eq.(1) underlying UNEPT, which might include intra or inter edge interactions with arbitrary range and strength, obeying some minimal conditions (see End Matter) these are two building blocks for time dependent transport [44]. Actually, the DC backscattering current average and noise can be expressed as:

$$I_{\text{tun}}(\omega_{\text{dc}}) = e^* [X_{\to}(\omega_{\text{dc}}) - X_{\leftarrow}(\omega_{\text{dc}})], \quad (5)$$

$$S_{\text{tun}}(\omega_{\text{dc}}) = e^{*2} [X_{\to}(\omega_{\text{dc}}) + X_{\leftarrow}(\omega_{\text{dc}})]. \quad (6)$$

These two correlators are, in general, independent observables as long as neither inversion symmetry-as assumed in the TLL model-nor detailed balance-valid for an initial equilibrium distribution-is imposed. However, they become related through Eq. (3), which naturally leads to a unique building block that can be defined as

$$X^R(t) = \theta(t) [X_{\to}(t) - X_{\leftarrow}(t)],$$

where $\theta(t)$ is the Heaviside function. Actually, taking the Fourier image of the expression above

$$X^R(\omega) = i \int \frac{d\omega'}{2\pi} \frac{X_{\to}(\omega') - X_{\leftarrow}(\omega')}{\omega - \omega' + i0}, \quad (7)$$

therefore by using the expressions for the DC current [38, 44]:

$$I_{\text{tun}}(\omega_{\text{dc}}) = 2e^* \text{Re}[X^R(\omega = \omega_{\text{dc}})]. \quad (8)$$

Now that Eq.(3) holds, we can demonstrate that $X^R$ also determines the DC backscattering shot-noise, by employing some manipulations of Fourier transforms: [41]

$$\begin{aligned} S_{\text{tun}}(\omega_{\text{dc}})/e^{*2} &= \sum_{\epsilon=\pm} (1 + e^{2i\epsilon\theta}) X_{\to}^{\epsilon}(\omega_{\text{dc}}) \quad (9) \\ &= (1 + e^{2i\theta}) [X_{\to}^{+}(\omega_{\text{dc}}) + e^{-2i\theta} X_{\to}^{-}(\omega_{\text{dc}})] \end{aligned} \quad (10)$$

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

$$\begin{aligned} 2i \operatorname{Im} X^R(\omega_{\text{dc}}) &= \sum_{\epsilon=\pm} \epsilon (1 - e^{2i\epsilon\theta}) X_{\to}^{\epsilon}(\omega_{\text{dc}}) \quad (11) \\ &= (1 - e^{2i\theta}) [X_{\to}^{+}(\omega_{\text{dc}}) + e^{-2i\theta} X_{\to}^{-}(\omega_{\text{dc}})] \end{aligned} \quad (12)$$

This provides:

$$S_{\text{tun}}(\omega_{\text{dc}}) = -2e^{*2} \cot \theta \operatorname{Im}[X^R(\omega_{\text{dc}})]. \quad (13)$$

This fundamental relation constitutes a braiding nonequilibrium FDT: $X^R$ is a response function even for nonequilibrium initial states and for arbitrary DC frequencies $\omega_{\text{dc}}$ 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 time-braiding phase $\theta$ up to the usual indeterminacy mod $\pi$ from time-dependent transport. In the case of fermions, $\theta = \pi$, the divergence is compensated by the dependence of $X^R(\omega_{\text{dc}})$ on $\theta$.[41]

**Integral equation for time-braiding**

Applying the Kramers-Krönig relations to the real and imaginary parts of $X^R$, we subtract the zero-frequency component $S_{\text{tun}}(0)$ to ensure convergence of the integral. Since $I_{\text{tun}}(0) = 0$, the corresponding subtraction for the current is redundant. The resulting compact form for both equations reads

$$\begin{aligned} S_{\text{tun}}(\omega_{\text{dc}}) - S_{\text{tun}}(0) \\ = e^* \omega_{\text{dc}} \cot \theta \text{ P.V.} \int \frac{d\omega'_{\text{dc}}}{\pi} \frac{I_{\text{tun}}(\omega'_{\text{dc}})}{\omega'_{\text{dc}}(\omega_{\text{dc}} - \omega'_{\text{dc}})}, \end{aligned} \quad (14)$$
$$e^* I_{\text{tun}}(\omega_{\text{dc}}) = -\omega_{\text{dc}} \tan \theta \text{ P.V.} \int \frac{d\omega'_{\text{dc}}}{\pi} \frac{S_{\text{tun}}(\omega'_{\text{dc}})}{\omega'_{\text{dc}}(\omega_{\text{dc}} - \omega'_{\text{dc}})} \quad (15)$$

Here P.V. refers to the Principal Value. Whenever the integrals converge, these two relations provide practical means to extract $\tan \theta$ from measurements of either the DC current or the DC noise over a sufficiently broad frequency range $\omega'_{dc}$, typically centered around the external DC frequency $\omega_{dc}$ [41]. Although the integrand may depend on $\theta$, the prefactor $\tan \theta$ is thereby isolated. These relations apply to arbitrary stationary nonequilibrium distributions, including the “anyon collider” geometry and setups with a temperature bias. In what follows, we focus on the regime where the DC current and noise obey the Poisson relation, reflecting the detailed balance between the correlators in Eq. (4).

**Thermal states.** For thermal initial states, the detailed balance equation (see End Matter)

$$X_{\to}(\omega) = e^{-\omega/\omega_{th}} X_{\leftarrow}(\omega) \quad (16)$$

where $\omega_{th} = k_B T / \hbar$. This yields the generalized fluctuation-dissipation identity from Eqs (5) and (6) [38, 39, 45]

$$S_{\text{tun}}(\omega_{\text{dc}}) = e^* \coth\left(\frac{\omega_{\text{dc}}}{2\omega_{\text{th}}}\right) I_{\text{tun}}(\omega_{\text{dc}}). \quad (17)$$

In that case, the equilibrium noise is given by $S_{\text{tun}}(0) = 2e^*\omega_{th}G_{\text{tun}}(0)$ where the rescaled linear conductance is defined as $G_{\text{tun}}(0) = dI_{\text{tun}}(\omega_{\text{dc}})/d\omega_{\text{dc}}|_{\omega_{\text{dc}}\to 0}$ (thus has the dimension of an electric charge).

Letting $J(0) = G_{\text{tun}}(0)/e^*$, and starting from Eq. (14), we introduce the dimensionless functions:

$$J(\omega) = I_{\text{tun}}(\omega_{\text{dc}} = \omega)/(e^*\omega); \quad F(x) = J(2\omega_{th}x)/J(0), \quad (18)$$

where the scaled variable $x = \omega / 2\omega_{th}$. At small frequencies, the principal-value integral vanishes since $J(\omega)$ is even, leaving only the leading contribution proportional to $J(0)$. By using the Sokhotski-Plemelj identity, we arrive at the final form of the equation

$$(\coth x + i \cot \theta) F(x) = \frac{1}{x} - \cot \theta \int \frac{F(y)}{x - y + i0^+} \frac{dy}{\pi}. \quad (19)$$

The time-braiding constraint of the model is thus encoded in this functional integral equation.

**Wiener-Hopf method** To find the general solution of the Eq. (19) we employ the W-H technique. We start by decomposing the function as $F(x) = F_+(x) + F_-(x)$, where $F_\pm(x)$ are analytic in the upper (lower) half-plane. In this representation, the equation takes the standard W-H form,

$$F_{-}(x) + K(x)F_{+}(x) = H(x), \quad (20)$$

with

$$K(x) = \frac{\coth(x) - i \cot \theta}{\coth(x) + i \cot \theta}, \quad H(x) = \frac{1}{x[\coth(x) + i \cot \theta]} \quad (21)$$

The next step consists in the multiplicative factorization of the kernel, $K(x) = K_+(x)K_-(x)$, followed by dividing Eq. (20) by $K_-(x)$. We then perform an additive decomposition of the inhomogeneous term, $H(x)/K_-(x) = R_+(x) + R_-(x)$. Using Liouville’s theorem, we obtain

$$F_{+}(x)K_{+}(x) - R_{+}(x) = R_{-}(x) - \frac{F_{-}(x)}{K_{-}(x)} = E(x), \quad (22)$$

where $E(x)$ is an entire analytic function. If $E(x)$ remains bounded at infinity, Liouville’s theorem implies that it must be constant. Moreover, if $E(x) \to \infty \to 0$, then $E(x) = 0$ for all $x$. As it is shown in (3), this is indeed the case. Consequently, the general W-H solution reads

$$K(x) = \frac{R_{+}(x)}{K_{+}(x)} + R_{-}(x)K_{-}(x). \quad (23)$$

To proceed, we express the kernel in terms of Gamma functions, allowing an explicit factorization:

$$K_{+}(x) = \frac{\Gamma\left[1 - \frac{ix}{\pi} - \frac{\theta}{\pi}\right]}{\Gamma\left[-\frac{ix}{\pi} + \frac{\theta}{\pi}\right]}, \quad K_{-}(x) = \frac{\Gamma\left[\frac{ix}{\pi} + \frac{\theta}{\pi}\right]}{\Gamma\left[1 + \frac{ix}{\pi} - \frac{\theta}{\pi}\right]}, \quad (24)$$

with the asymptotic behavior $|K_\pm(x)| \sim x^{\pm(1-2\theta/\pi)}$. We now turn to the inhomogeneous part of the equation, defined as

$$R(x) = \frac{\sinh x \sin \theta}{\pi x} \left| \Gamma \left[ 1 - \frac{\theta}{\pi} + \frac{ix}{\pi} \right] \right|^2. \quad (25)$$

To perform the additive decomposition $R(x) = R_+(x) + R_-(x)$, we note that if $R(x)$ has $n(m)$ simple poles $p_n(q_m)$ in the upper (lower) half-plane with residues $a_n(b_m)$, then

$$R_{+}(x) = \frac{1}{2\pi i} \int_{\Gamma_{-}} \frac{R(t) dt}{t - x} = \sum_{n=0}^{+\infty} \frac{a_n}{p_n - x}, \quad (26)$$

$$R_{-}(x) = -\frac{1}{2\pi i} \int_{\Gamma_{+}} \frac{R(t) dt}{t - x} = -\sum_{m=0}^{+\infty} \frac{b_m}{q_m - x}. \quad (27)$$

Since $\cot \theta$ is defined mod $\pi$, we adopt from now the associated phase that is restricted to $[0, \pi]$, keeping however the same notation for simplicity.

Here $x$ lies in the analytic strip of $R(x)$, $-i\theta < x < i\theta$, and $R(x) \sim \sin\theta \pi^{2\theta/\pi-1}|x|^{-2\theta/\pi}$ as $|x| \to \infty$. The contours $\Gamma_\pm$ enclose the poles in the upper and lower half-planes, respectively. Carrying out the contour integration in Eq. (26), we find

$$R_{+}(x) = -\frac{i}{2x} \tan \theta \left[ \frac{\Gamma\left[1 - \frac{\theta}{\pi}\right]}{\Gamma\left[\frac{\theta}{\pi}\right]} - \frac{\Gamma\left[1 - \frac{\theta}{\pi} - \frac{ix}{\pi}\right]}{\Gamma\left[\frac{\theta}{\pi} - \frac{ix}{\pi}\right]} \right], \quad (28)$$

$$R_{-}(x) = \frac{i}{2x} \tan \theta \left[ \frac{\Gamma\left[1 - \frac{\theta}{\pi}\right]}{\Gamma\left[\frac{\theta}{\pi}\right]} - \frac{\Gamma\left[1 - \frac{\theta}{\pi} + \frac{ix}{\pi}\right]}{\Gamma\left[\frac{\theta}{\pi} + \frac{ix}{\pi}\right]} \right]. \quad (29)$$

It is straightforward to verify that the leading terms in $R_\pm(x)$ cancel the pole at $x = 0$ on the real axis. Finally, from Eq. (22) we obtain

$$F_{+}(x) = -\frac{i}{2x} \tan \theta \left[ \frac{\Gamma\left[1 - \frac{\theta}{\pi}\right]}{\Gamma\left[\frac{\theta}{\pi}\right]} \frac{\Gamma\left[\frac{\theta}{\pi} - \frac{ix}{\pi}\right]}{\Gamma\left[1 - \frac{ix}{\pi} - \frac{\theta}{\pi}\right]} - 1 \right], \quad (30)$$
with the property $F_-(x) = F_+(x)$. The resulting expression for the full solution reads

$$F(x) = F_+(x) + F_-(x) = \frac{\sinh x \left| \Gamma\left[\frac{\theta}{\pi} + \frac{ix}{\pi}\right] \right|^2}{x \Gamma\left[\frac{\theta}{\pi}\right]^2}, \quad (31)$$

which indeed satisfies $F(0) = 1$. By rewriting this solution in terms of tunneling current we are coming to

$$\begin{aligned} I_{\text{tun}}(\omega_{\text{dc}}) &= \omega_{\text{dc}} G_{\text{tun}}(0) F(\omega_{\text{dc}}/2\omega_{\text{th}}) \\ &= 2\omega_{\text{th}} G_{\text{tun}}(0) \sinh\left(\frac{\omega_{\text{dc}}}{2\omega_{\text{th}}}\right) \frac{\left|\Gamma\left[\frac{\theta}{\pi} + \frac{i\omega_{\text{dc}}}{2\pi\omega_{\text{th}}}\right]\right|^2}{\Gamma\left[\frac{\theta}{\pi}\right]^2}. \end{aligned} \quad (32)$$

This solution reproduces the standard TLL result for the tunneling current, with a scaling dimension $\delta$ corresponding to the angle $\theta$ modulo $\pi$, chosen such that $0 < \delta < 1$. We have also verified that the second Kramers–Krönig relation in Eq. (15) is satisfied. However, there remains an indeterminacy in $G_{\text{tun}}(0)$ due to the homogeneity of the Kramers–Krönig integral equation with respect to this quantity. As explained in End Matter , the nonequilibrium current in the regime $\omega_{\text{dc}} \gg \omega_{\text{th}}$ depends solely on $\omega_{\text{dc}}$, which fixes the power-law behavior $G_{\text{tun}}(0) \propto \omega_{\text{th}}^{2\delta-2}$, with a prefactor determined by the microscopic details of the model and the energy cutoff. This prefactor is proportional to the reflection coefficient at the QPC, hence to $|\xi|^2$ in Eq. (2). We emphasize, however, that this TLL solution is itself independent of the specific form of the operator A: it follows entirely from the two simultaneous constraints—the exchange relation in Eq. (3) and the detailed-balance condition in Eq. (16).

**Discussion and conclusion** Let us summarize our analysis. We imposed the braiding link in the time domain as an additional constraint on a fully general model underlying UNEPT. This led us to derive the braiding fluctuation–dissipation theorem (FDT) introduced in Ref. [41], and consequently an integral equation relating DC noise and current. Then we enforced the detailed-balance condition on the two correlators, without assuming any inversion symmetry, so that DC noise becomes Poissonian, and we get a braiding integral equation for a unique observable,

the DC current which serves as the distinctive signature of the underlying model.

We solved this equation using the W-H method and, remarkably, obtained a unique solution: the TLL prediction, thus a scaling behavior. Even though inversion symmetry was not imposed, the resulting solution restores it. The scaling parameter is defined as $\delta = \theta/\pi - n$, where n is such that $0 < \delta < 1$, indicating that the braiding phase in time is determined by the scaling dimension and thus inherits its nonuniversality.

In fact, contrary to the common claim that $\theta$ is topologically protected, our results show that the braiding link in the time domain is fully determined by the edge dynamics that is, by the precise edge Hamiltonian, which may be affected by nonuniversal interaction effects. Thus the direct inference of the temporal braiding from its spatial counterpart can be justified only for a chiral TLL (see End Matter ). Therefore, in the presence of nonuniversal effects, the spatial braiding phase, being genuinely topologically protected, cannot systematically be captured through the temporal braiding phase, which merely reflects the nonuniversal scaling dimension.

The identification between $\delta$ and $\theta$ was the hypothesis underlying the anyon collider analysis, within the specific TLL model [15]. Within this context, the detailed-balance condition does not hold, and the DC noise becomes super-Poissonian [38, 39], but we can check that the time-exchange FDT is valid. However, since the equilibrium components of the correlators are constrained by detailed balance, it is not straightforward to construct alternative models that simultaneously satisfy the time-exchange constraint. This limitation defines one of the key perspectives of the present approach. More general nonequilibrium situations may offer additional flexibility in the form of the retarded response function, and thus in the possible structures of the two fundamental building blocks: the DC current and the DC noise.

The possibility of linking the time braiding statistical phase to a measurable DC current provides a new perspective on fractional excitations in strongly correlated systems, bridging theoretical constructs and experimentally accessible observables.

## END MATTER

**Braiding link in the chiral TLL** We start by a simple fractional filling factor (Laughlin states for which one has two oppositely chiral edges with bosonic fields labeled by $\zeta = u, d$ (see Fig. 1), obeying

$$[\Phi_{\zeta}(x), \Phi_{\zeta'}(x')] = \mp i\pi\delta_{\zeta,\zeta'} \text{sign}(x-x'), \quad (33)$$

(where the upper (down) sign is related to $u(d)$ field) and defining quasiparticle fields as

$$\Psi_{\zeta}(x) = e^{\mp ikx + i\sqrt{\nu}\Phi_{\zeta}(x)}/\sqrt{2\pi a}. \quad (34)$$

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, introducing the frequency cutoff $\omega_c$. Using the Baker–Campbell–Hausdorff formula, one obtains the exchange relation for the quasiparticle fields:

$$\Psi_{\zeta}(x) \Psi_{\zeta}^{\dagger}(x') = e^{\pm i\pi\nu \text{sign}(x-x')} \Psi_{\zeta}^{\dagger}(x') \Psi_{\zeta}(x),$$
(which reveals an exchange fractional phase in the space domain given by $\theta = \pi\nu$.

If the Hamiltonian $\mathcal{H}_0$ in Eq. (1) is a chiral TLL

$$ \mathcal{H}_0 = \frac{v}{4\pi} \sum_{\zeta=u,d} \int dx \, (\partial_x \phi_{\zeta}(x))^2, \qquad (35) $$

one has, in the interaction representation, $\Phi_{\zeta,i}(x,t) = e^{\frac{i}{\hbar}\mathcal{H}_{0t}}\Phi_{\zeta}(x)e^{-\frac{i}{\hbar}\mathcal{H}_{0t}} = \Phi_{\zeta}(x - \zeta vt)$. Choosing spatial locality by setting $x = x'$, one obtains the braiding relation in the time domain:

$$ \Psi_{\zeta,i}^{\dagger}(x,t) \Psi_{\zeta,i}(x,t') = e^{-i\theta \operatorname{sign}(t-t')} \times \Psi_{\zeta,i}(x,t') \Psi_{\zeta,i}^{\dagger}(x,t), \qquad (36) $$

this leads to a similar relation for A with a doubled phase: $A_i^\dagger(t)A_i(t') = e^{-2i\theta \operatorname{sign}(t-t')}A_i(t')A_i^\dagger(t)$. Notice that this link holds for operators and therefore does not depend on the initial nonequilibrium stationary distribution $\rho_{\text{neq}}$. We then average this equality with respect to $\rho_{\text{neq}}$, which gives the exchange link in Eq. (3).

**Detailed balance equation** Let's consider the correlator

$$ \begin{aligned} X_{\leftarrow}(t) = \langle A_{i}^{\dagger}(t) A_{i}(0) \rangle &= \operatorname{Tr}(e^{-\beta \mathcal{H}_{0}} e^{\frac{i}{\hbar} \mathcal{H}_{0t}} A_{i}^{\dagger}(0) e^{-\frac{i}{\hbar} \mathcal{H}_{0t}} A_{i}(0)) \\ &= \operatorname{Tr}(e^{-\beta \mathcal{H}_{0}} A_{i}(0) e^{\frac{i}{\hbar} \mathcal{H}_{0}(t+i\hbar\beta)} A_{i}^{\dagger}(0) e^{-\frac{i}{\hbar} \mathcal{H}_{0}(t+i\hbar\beta)}) = \langle A_{i}(0) A_{i}^{\dagger}(t+i\hbar\beta) \rangle = X_{\leftarrow}(t+i\hbar\beta). \end{aligned} \qquad (37) $$

Taking the Fourier transformation of the expression above

$$ X_{\leftarrow}(\omega) = \int dt \, e^{i\omega t} X_{\leftarrow}(t + i\hbar\beta) = e^{\beta\hbar\omega} X_{\leftarrow}(\omega), \quad \beta = 1/\hbar\omega_{th}. \qquad (38) $$

**Spectral decomposition** By using Eq. (4) for the thermal states one can obtain

$$ \begin{aligned} \hbar^2 X_{\leftarrow}(\omega) &= 2\pi \sum_{m,n} |\langle n | A | m \rangle|^2 e^{-\beta \hbar \omega_m} \delta(\omega - (\omega_n - \omega_m)), \\ \hbar^2 X_{\leftarrow}(\omega) &= 2\pi \sum_{m,n} |\langle m | A | n \rangle|^2 e^{-\beta \hbar \omega_m} \delta(\omega + (\omega_n - \omega_m)), \end{aligned} \qquad (39) $$

from which the detailed balance condition immediately follows. For the zero temperature limit $T \to 0$, the average is taken over the ground state $m = 0$. In this case, $X_{\leftarrow}(\omega)$ obviously vanishes for $\omega > 0$ due to the delta function, and the expression for the tunneling current thus takes the form

$$ I_{\text{tun}}(\omega_{\text{dc}})|_{T \to 0} = e^* X_{\leftarrow}(\omega = \omega_{\text{dc}}) = \frac{2\pi e^*}{\hbar^2} \sum_n |\langle n | A | 0 \rangle|^2 \delta(\omega_{\text{dc}} - (\omega_n - \omega_0)). \qquad (40) $$

On the one hand this function depends only on $\omega_{\text{dc}}$. On the other hand the expression Eq. (32) in the limit $\omega_{\text{dc}} \gg \omega_{th}$ yields

$$ I_{\text{tun}}(\omega_{\text{dc}}) \sim G_{\text{tun}}(0) (\omega_{\text{dc}}/\omega_{\text{th}})^{2\delta-1}. \qquad (41) $$

The finiteness of this result at $T \to 0$ therefore requires that $G_{\text{tun}}(0)$ exhibits a power-law temperature dependence.

**UNEPT conditions.** 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 $\rho_{\text{neq}}$ commutes with $\mathcal{H}_0$, i.e., $[\rho_{\text{neq}}, \mathcal{H}_0] = 0$, thus it is stationary, being diagonal with respect to many-body eigenstates of $\mathcal{H}_0$;

* (iii) One has: $\langle A_i(t) A_i(0) \rangle = 0$, where $A_i(t) = e^{\frac{i}{\hbar}\mathcal{H}_{0s}} A e^{-\frac{i}{\hbar}\mathcal{H}_{0s}}$ in the interaction picture.

**Analysis of asymptotics.** We now analyze the asymptotic behavior of the solution and justify the vanishing of $E(x)$ in Eq. (22). Let us assume that $E(x)$ may, in principle, be a polynomial of the form $E(x) \sim x^{\alpha}$ with an integer exponent $\alpha$. Although the asymptotic form of $F(x)$ is not known a priori, the auxiliary functions exhibit well-defined limits at $x \to \infty$, namely $|K_{\pm}(x)| \sim x^{\pm(1-2\theta/\pi)}$. Accordingly, the functions $R_{\pm}(x)$ behave as follows:
• For $\theta < \pi/2$, the second term dominates, yielding

$$R_+(x) \sim \frac{i \tan \theta}{2} \pi^{\frac{2\theta}{\pi}-1} x^{-\frac{2\theta}{\pi}}, \quad R_-(x) \sim -\frac{i \tan \theta}{2} \pi^{\frac{2\theta}{\pi}-1} x^{-\frac{2\theta}{\pi}}. \qquad (42)$$

• For $\theta > \pi/2$, the second term decays faster than $x^{-1}$, giving

$$R_+(x) \sim -\frac{i \tan \theta}{2} \frac{\Gamma(1 - \frac{\theta}{\pi})}{\Gamma(\frac{\theta}{\pi})} x^{-1}, \quad R_-(x) \sim \frac{i \tan \theta}{2} \frac{\Gamma(1 - \frac{\theta}{\pi})}{\Gamma(\frac{\theta}{\pi})} x^{-1}. \qquad (43)$$

Hence, for large $x$, the $\pm$ components of the solution behave as

$$F_+(x) = \frac{E(x) + R_+(x)}{K_+(x)} \approx \frac{E(x)}{K_+(x)} \sim x^{\alpha-1+\frac{2\theta}{\pi}}, \quad F_-(x) = K_-(R_-(x) - E(x)) \sim x^{\alpha-1+\frac{2\theta}{\pi}}, \qquad (44)$$

with $\alpha \in \mathbb{N}$. The convergence of the principal-value integral in Eq. (14) requires $\alpha < 0$. Consequently, Liouville's theorem enforces $E(x) \to 0$, confirming that no additional entire contribution can arise. This result establishes the uniqueness and self-consistency of the W-H solution given in Eq. (31).

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