Effect of inter-edge interaction in a quantum Hall collider

Technical Summary: Effect of Inter-Edge Interaction in a Quantum Hall Collider

Problem Statement
Fractional quantum Hall (FQH) colliders offer a time-domain approach to measuring anyon exchange phases via braiding, distinct from the spatial braiding used in interferometers. While theoretical predictions for Laughlin states (e.g., $\nu = 1/3$) align with experimental observations, a significant discrepancy exists for hierarchical states such as $\nu = 2/5$. Theoretical models predict a braiding phase of $3\pi/5$ accompanied by a negative tunneling current. However, experiments report a negative generalized Fano factor, which implies a positive tunneling current. Previous attempts to resolve this, such as accounting for the finite width of anyonic wavepackets, provided qualitative explanations but left the mechanism incomplete. This paper investigates the specific role of short-range inter-edge interactions in a multi-mode FQH collider to explain the observed sign reversal in the tunneling current and the resulting negative Fano factor.

Methodology
The authors model a $\nu = 2/5$ FQH collider, which supports two co-propagating edge modes with conductances $1/3(e^2/h)$ and $1/15(e^2/h)$. The system is described using bosonization techniques, where edge excitations are represented by bosonic fields $\phi_i(x,t)$.

• Hamiltonian: The edge Hamiltonian includes bare velocities $v_i$ and a short-range inter-edge interaction parameter $\tilde{u}$. This interaction couples the two edge modes.
• Diagonalization: The interacting Hamiltonian is diagonalized into two eigenmodes ($\tilde{\phi}_1, \tilde{\phi}_2$) with renormalized velocities $\tilde{v}_1$ and $\tilde{v}_2$. The original anyon operators are transformed into this new basis, revealing that an injected anyon fractionalizes into components propagating along these distinct eigenmodes.
• Tunneling Formalism: The tunneling current at the central quantum point contact (QPC) is calculated using the Keldysh technique to the lowest order in tunneling coupling. The calculation accounts for the time-domain braiding of injected anyons with quasi-particle-quasi-hole (QP-QH) pairs created at the collider.
• Poissonian Averaging: The study transitions from single anyon injection to a dilute, stochastic stream of anyons (Poissonian distribution). The authors analyze the time-dependent tunneling current by averaging over the number of incoming anyons, distinguishing between "correlated" and "uncorrelated" contributions based on the time delay $\Delta\tau$ between the arrival of the fast and slow eigenmodes.

Key Contributions and Results

1. Fractionalization and Decomposition: The presence of inter-edge interaction causes a dilute anyon beam to decompose into two components:

   • A correlated component, where both eigenmodes arrive within the relevant time window, preserving the standard interaction-invariant braiding phase.
   • An uncorrelated component, arising when the time delay $\Delta\tau$ between eigenmodes exceeds the decay time of the integrand (long-junction limit). In this regime, the fast and slow modes arrive at the collider QPC at times that are statistically independent regarding the tunneling events.

2. Time-Domain Braiding Dynamics:

   • In the absence of interaction ($\tilde{u}=0$), the tunneling current is negative, consistent with the theoretical phase $3\pi/5$.
   • When interaction is switched on, the injected anyon fractionalizes. The fast eigenmode arrives first, contributing a partial braiding phase that results in a positive current. The slow mode arrives later, contributing a negative current.
   • The total current is determined by the competition between the time delay $\Delta\tau$ and the decay time $T_{decay}$ of the integrand.

3. Resolution of the Sign Discrepancy:

   • In the long-junction limit ($\Delta\tau \gg T_{decay}$), the uncorrelated component dominates the integral. The effective braiding phase becomes a sum of interaction-dependent phases ($\sum \sin(2\pi \Lambda^j_{\alpha\gamma})$) rather than the invariant mutual phase.
   • This interaction-dependent phase yields a positive tunneling current, which reverses the sign predicted by non-interacting theories.
   • Consequently, the generalized Fano factor becomes negative (approaching $P \simeq -1$ for strong interactions), matching experimental observations at $\nu = 2/5$.

4. Finite Width Effects: The authors further incorporate the finite spatial width of the anyonic wavepackets. While this shifts the Fano factor values, the qualitative result remains robust: the interplay between strong inter-edge interactions and finite wavepacket width explains the experimentally observed negative Fano factor.

Significance
The paper provides a robust theoretical interpretation of the anomalous negative Fano factor observed in $\nu = 2/5$ FQH colliders. It demonstrates that inter-edge interactions are not merely a perturbation but fundamentally alter the time-domain braiding process by inducing fractionalization into correlated and uncorrelated components. The dominance of the uncorrelated component in the long-junction limit reverses the tunneling current sign, resolving the conflict between previous theoretical predictions and experimental data. The framework established here is presented as extendable to other multi-mode collider configurations, offering new insights into the interplay between inter-edge interactions and fractional statistics in quantum Hall systems.