Anyonic exchange in the time domain is tied to Luttinger type scaling
We consider Fractional Quantum Hall (FQH) edges with a spatially local Quantum Point Contact (QPC). Within the Unified Nonequilibrium Perturbative (UNEP) framework, without assumptions on the underlying Hamiltonian $H_{0}$ for the edges, we search for the associated backscattering DC current and noise compatible with the anyonic time exchange (ATE) constraint with a phase $\barθ$. For that, we infer a nonequilibrium fluctuation-dissipation relation that explicitly involves $\barθ$ and yields an integral equation connecting the nonequilibrium DC current and noise. On one hand, we assume initial thermal states, so that the DC noise is Poissonian. Then the integral equation for the DC current is shown, through the Wiener-Hopf technique, to admit the unique TLL local solution. Therefore, $\barθ$ is necessarily tied to the scaling dimension $δ$, which is robust with respect to edge interactions. On the other hand, we address the "anyon collider" setup where DC noise is super-Poissonian. As the difference between nonequilibrium and equilibrium correlators is fixed, the integral equation admits a unique solution for both nonequilibrium DC backscattering current and super-Poissonian noise, whose explicit temperature dependence is thus determined.