[Paper Review] Extracting the scaling dimension of quantum Hall quasiparticles from current correlations

Noam Schiller, Yuval Oreg+1 more|arXiv (Cornell University)|Nov 9, 2021

Quantum and electron transport phenomena88 references30 citations

TL;DR

This paper proposes a model-independent method to extract the scaling dimension Δ of fractional quantum Hall quasiparticles using thermal tunneling noise at a quantum point contact (QPC), by measuring the Fano factor's dependence on edge temperatures. The approach enables robust, universal extraction of Δ—comparable in reliability to charge measurement via shot noise—offering a critical tool for distinguishing non-Abelian candidate theories at filling ν = 5/2.

ABSTRACT

Fractional quantum Hall quasiparticles are generally characterized by two quantum numbers: electric charge $Q$ and scaling dimension $\Delta$. For the simplest states (such as the Laughlin series) the scaling dimension determines the quasiparticle's anyonic statistics (the statistical phase $ heta=2\pi\Delta$). For more complicated states (featuring counterpropagating modes or non-Abelian statistics) knowing the scaling dimension is not enough to extract the quasiparticle statistics. Nevertheless, even in those cases knowing the scaling dimension facilitates distinguishing different candidate theories for describing the quantum Hall state at a particular filling (such as PH-Pfaffian and anti-Pfaffian at $ u=5/2$). Here we propose a scheme for extracting the scaling dimension of quantum Hall quasiparticles from thermal tunneling noise produced at a quantum point contact. Our scheme makes only minimal assumptions about the edge structure and features the level of robustness, simplicity, and model independence comparable to extracting the quasiparticle charge from tunneling shot noise.

Motivation & Objective

• To develop a robust, model-independent method for measuring the scaling dimension Δ of fractional quantum Hall quasiparticles.
• To overcome the limitations of voltage-bias-dependent measurements, which suffer from non-universal tunneling amplitude effects.
• To enable discrimination between competing candidate theories for non-Abelian states (e.g., at ν = 5/2) using a measurable, universal quantity.
• To leverage temperature as a tunable control parameter in QPC experiments to access Δ via the Fano factor's thermal dependence.
• To provide a theoretically grounded, numerically tractable framework for extracting Δ from experimentally accessible noise and current data.

Proposed method

• Proposes measuring the Fano factor F = S/(2eIT) as a function of edge temperatures T1 and T2, where S is excess noise and IT is tunneling current.
• Derives an exact, universal expression for the Fano factor in the weak tunneling limit: F = (2Q/πe) × Im[ψ(2Δ + iQV/(2πkBT))], where ψ is the digamma function.
• Uses non-equilibrium Keldysh formalism and Luttinger liquid theory to compute noise and current correlations at a QPC with counterpropagating edge modes.
• Derives numerically stable integral forms for noise (S0T, STT) and current (IT) in terms of hyperbolic and trigonometric functions of temperature and voltage.
• Applies contour deformation and analytic continuation techniques to handle singularities in the complex plane, enabling numerical convergence.
• Validates the approach in limiting regimes: equal temperatures (T1 = T2 = T), dominant temperature (eV ≪ kBT), and dominant voltage (eV ≫ kBT), showing consistency with known limits.

Experimental results

Research questions

• RQ1 Can the scaling dimension Δ of FQH quasiparticles be extracted from thermal tunneling noise in a way that is robust to non-universal tunneling effects?
• RQ2 Does the Fano factor’s dependence on edge temperature T1 and T2 provide a universal signature of the quasiparticle scaling dimension Δ?
• RQ3 How does the proposed method compare in robustness and universality to existing voltage-bias-based measurements of Δ?
• RQ4 To what extent can this method distinguish between competing candidate theories for non-Abelian FQH states (e.g., PH-Pfaffian vs. anti-Pfaffian at ν = 5/2) via Δ?
• RQ5 What are the quantitative limits of the method in the regimes of low and high voltage relative to temperature?

Key findings

• The Fano factor F is universally determined by the quasiparticle charge Q and scaling dimension Δ via F = (2Q/πe) × Im[ψ(2Δ + iQV/(2πkBT))], independent of tunneling amplitude.
• In the equal-temperature limit (T1 = T2 = T), the Fano factor exhibits universal scaling with T/V, enabling direct extraction of Δ from experimental data.
• In the low-voltage, high-temperature regime (eV ≪ kBT), the Fano factor scales as F ≈ f(Δ) × (kBT1/eV), where f(Δ) is a function of Δ alone, allowing Δ to be extracted from the slope.
• In the high-voltage, low-temperature regime (eV ≫ kBT), the Fano factor approaches F ≈ Q/e + (kB T1/eV)(1 − 4Δ) + O((kBT/eV)²), showing a linear correction in 1/V proportional to (1 − 4Δ), which directly probes Δ.
• The method is robust against non-universal tunneling amplitude variations because the Fano factor F is independent of the tunneling matrix element η.
• Numerical evaluation of the derived integral expressions (e.g., B26–B27) confirms convergence and stability, enabling practical implementation in experiments with tunable edge temperatures.