Probing quantum Hall edge chirality and the anyonic exchange angle with a three-path interferometer

This paper proposes an average-current interferometer designed to probe the directional causal response of fractional quantum Hall edge excitations, aiming to resolve fundamental properties such as scaling dimension and anyonic exchange angle. It extends charge-and-scaling spectroscopy from quantum Hall vertices to reconstruct their exchange angles, a distinction of significant experimental interest.

How it works

The core of the device is a three-path fractional quantum Hall interferometer where three coherent quantum point contacts (QPCs) form a fluxenclosing tunneling loop around a magnetic flux. This geometry results in the leading Aharonov-Bohm signal being cubic in the tunneling amplitudes, arising from interference between a direct quasiparticle transfer and a coherent two-step alternative. Resolving this intermediate segment separates downstream and upstream contributions, allowing for the probing of directional causal constraints imposed by edge theory.

Key Theoretical Predictions

The paper derives specific scaling laws and phase relationships based on the vertex weights. For a general Abelian edge, if the downstream weight is denoted by δ+ and the upstream weight by δ−, then:

1. The single-edge correlator exponent is ∆l = δ+ + δ−, with conventional scaling dimension hl = ∆l/2.

2. The exchange angle of the tunneling vertex is θl = π(δ+ − δ−) modulo 2π.

3. The two directional amplitudes scale as E3∆l−2, where ∆l = δ+ + δ− and hl = ∆l/2.

Directional Measurement and Null Test

The interferometer utilizes opposite cyclic voltage orderings to isolate the two directions:

1. A common positive-flux harmonic alignment complex-conjugates the reverse-loop coefficient, leading to an aligned continuous phase of π∆l = 2πhl (mod 2π).

2. The magnitude ratio of the downstream and upstream contributions is given by rl = sin(πδ−)/ sin(πδ+).

3. The aligned phase, after removing the known fixed sign, uniquely determines hl, and together with rl, reconstructs θl.

Activation of Upstream Contribution via a Bridge

A weak static nonlocal density interaction spanning the tunneling points activates the upstream coefficient without introducing an upstream mode. This bridge carries no charge and introduces no additional propagating mode; it links two causally allowed downstream responses across regions on opposite sides of the tunneling points. In the low-temperature, unresolved-delay regime, its amplitude scales as E2ν−1, and its aligned phase relative to the downstream reference is −π(1 − ν)/2 + χa modulo 2π, with χa = 0 or π fixed by the real bridge sign.

Experimental Protocol and Distinguishing Vertices

The measurement protocol involves sweeping the primary flux to determine complex terminal currents, using opposite cyclic voltage orderings to obtain direction-pure middle-terminal differences. The paper demonstrates that this technique can distinguish vertices that scaling spectroscopy alone cannot separate; for instance, at the disorder-dominated ν = 2/3 fixed point, the charge-e/3 doublet and the neutral-free 2e/3 composite have the same scaling dimension (hl = 1/3) but different exchange angles (−π/3 and 2π/3).

Controllable Realization of the Bridge

A controllable on-chip realization is proposed using a macroscopic edge of the same Laughlin liquid, folded to form a quasiparticle QPC. Capacitive coupling makes this link sensitive to nearby primary-edge density, generating a controllable nonlocal interaction. By operating at calibrated points φB = 0 and π, the sign of the induced kernel can be reversed without introducing a linear gate shift, providing a calibrated switch for probing the exchange statistics.

Conclusion on Vertex Spectroscopy

The three-path interferometer establishes that while scaling dimension (hl) is fixed-point universal for same-vertex measurements, the exchange angle (θl) can be reconstructed by measuring the directional amplitude ratio (rl). This joint spectroscopy allows a reconstruction of charge, scaling dimension, and exchange angle of the vertex selected at the QPCs. The results are fixed-point universal in the same-vertex, coherent, unresolved-flight regime.

Summary of Key Results:

/The downstream current-harmonic amplitude scales as E3ν−2 for a Laughlin edge at ν = 1/m in the low-temperature, fixed-ratio, and unresolved-delay regime.

/A weak real static interaction joins two locally downstream responses and activates an upstream contribution without introducing an upstream mode, with a leading amplitude scaling as E2ν−1.

/The common scaling or aligned phase determines hl = ∆l/2, and hl together with rl uniquely reconstructs θl.

**/For the disorder-dominated 2/3 fixed point, the measurement can distinguish vertices that scaling spectroscopy alone cannot separate.

Based on the provided scientific paper, here are specific improvements for AI systems that could be derived from its core principles:

The paper proposes a method to extract fundamental topological properties (charge, scaling dimension, and exchange angle) of quantum Hall vertex states by performing directional spectroscopy using an average-current interferometer.

Here are the specific improvements and capabilities for an improved AI system:

1. Enhanced Material/System Characterization for Topological States:

2. Robust Vertex Property Reconstruction:

3. Distinguishing Topologically Protected Phases (e.g., Distinguishing Fractional Charges from Neutral Excitations):

4. Automated Topological Parameter Extraction (Scaling Dimension and Exchange Angle):

5. Chirality-Sensitive Causal Analysis of Complex Dynamics:

Specific Improvements and Capabilities:

The improved AI system, leveraging these scientific principles, could perform the following specific tasks:

1. Improved Material/System Characterization for Topological States:

2. Robust Vertex Property Reconstruction:

3. Distinguishing Topologically Protected Phases (e.g., Distinguishing Fractional Charges from Neutral Excitations):

4. Automated Topological Parameter Extraction (Scaling Dimension and Exchange Angle):

5. Chirality-Sensitive Causal Analysis of Complex Dynamics:

Specific Capabilities of the Improved AI System:

1. Improved Material/System Characterization for Topological States: The system can analyze experimental data (like transport measurements) to determine if a material exhibits features consistent with a specific FQH state (e.g., Laughlin state at filling factor 1/m) by looking for signatures in charge and scaling dimension, moving beyond simple charge measurement.

2. Robust Vertex Property Reconstruction: The system can reconstruct the statistical properties of the tunneling vertex (the exchange statistics or exchange angle, θl) of a specific edge theory simply by analyzing the directional ratio of current harmonics measured at different bias conditions. This is a direct spectroscopic tool for measuring vertex statistics, which is otherwise difficult to access directly.

3. Distinguishing Topologically Protected Phases: The system can differentiate between different types of edge excitations (e.g., distinguishing a maximally chiral state where scaling dimension and exchange angle are locked from a non-chiral state where they are independent) by analyzing the measured directional amplitudes and their scaling behavior. This allows for the identification of the underlying topological class of the vertex involved in transport, even when both states share similar bulk charge signatures.

4. Automated Topological Parameter Extraction (Scaling Dimension and Exchange Angle): The system can automatically extract two key, previously difficult-to-measure parameters—the scaling dimension (hl) and the exchange angle (θl)—from a single experimental measurement by fitting the measured amplitude ratios and phase alignments of the cubic Aharonov-Bohm signal. This reconstruction is fixed-point universal for coherent, unresolved flight times, making it a highly automated diagnostic tool.

5. Chirality-Sensitive Causal Analysis of Complex Dynamics: The system can probe the local causal constraint imposed by an edge theory by using the interferometer setup to test whether a perturbation has directional support (causal influence) downstream or upstream. This allows the AI to analyze complex, non-equilibrium transport phenomena and determine if the underlying physics adheres to a one-sided spatial support (chirality) constraint, effectively acting as a causality probe for quantum edge excitations.

An upstream neutral mode can influence quasiparticle tunneling even when electric charge propagates only downstream. We propose a three-path fractional quantum Hall interferometer that probes this directional structure through the flux dependence of average terminal currents. Interference between direct and two-step tunneling first contributes at cubic order in the tunneling amplitudes of three weak quantum point contacts. In the local theory, equal drain voltages separate the downstream and upstream responses: for a purely downstream tunneling excitation, the cubic fundamental Aharonov-Bohm harmonic vanishes at one drain, while the other remains bright. Averaging measurements with exchanged unequal drain voltages extends this separation to a configuration in which all contacts are biased. It requires independent calibration of the electrostatic phase shift and fixed or known tunneling magnitudes. For a selected Abelian excitation with two-point correlation exponent 0<Δ<1, the common low-temperature bias power and directional amplitude ratio determine its scaling dimension and principal exchange angle. We also show that a weak static density interaction spanning two tunneling points can activate the dark harmonic without adding an upstream mode. In the high-bias static limit, the first-order Laughlin activated amplitude scales as E 2ν-1, where E is the source-drain bias energy and ν is the filling factor. In the weak localized single-mode infrared regime, the bright harmonic retains its leading scaling and can serve as a reference in the same device. The calculation assumes one coherent tunneling species and common-temperature edge correlations with unresolved propagation times.