Anyonic time exchange in hierarchical Hall states: emergent Luttinger scaling and cross-correlations at finite temperature

Abstract

We address hierarchical fractional quantum Hall edges coupled by a spatially local quantum point contact within the Unified Nonequilibrium Perturbative (UNEP) framework. Without assuming a Tomonaga-Luttinger liquid (TLL) tunneling correlator, we impose a local anyonic time-exchange (ATE) constraint with phase $\barθ$, yielding a nonequilibrium fluctuation-dissipation integral equation relating DC tunneling current and noise. For thermal initial states, detailed balance closes this equation for the current. Using the Wiener-Hopf method, we find a unique physical solution of local TLL form, with scaling dimension $δ$ determined by $\barθ=πδ\pmodπ$, emerging from the solution rather than being imposed via a TLL correlator. By contrast, ATE and inversion symmetry alone fix the exchange monodromy but not the full transport dynamics. We then study the "anyon collider" beyond Laughlin states at finite temperatures. Detailed balance is broken and noise becomes super-Poissonian, but the relation between nonequilibrium and equilibrium correlators determines current and noise. The output cross-correlation and balanced generalized Fano factor depend on both $δ$ and statistical phase $θ$, identical to $\barθ$ for co-propagating modes with short-range interactions. A temperature-dependent current scale separates the zero-temperature asymptotic regime from the low-current regime, where the balanced Fano factor saturates to a function of $θ$ and $δ$ distinct from its zero-temperature value. This latter regime is particularly relevant to ``anyon collider'' experiments. These complementary regimes constrain the statistical phase and scaling dimension, subject to the symmetry $δ\mapsto1-δ$ at fixed $θ$. For independent statistical phase and scaling dimension, the thermal crossover can reverse the sign of output cross-correlations.