Universal Transport Theory for Paired Fractional Quantum Hall States in the Quantum Point Contact Geometry

TL;DR

The paper develops a unified transport theory for paired even-denominator FQH states whose edges are described by $so(N)_1\times u(1)$ CFT, with $N=|\mathcal{C}_{cf}|$ Majorana modes. It uses a non-perturbative instanton framework to establish a weak-strong duality between strong quasiparticle tunneling and weak electron tunneling, and derives the scaling dimension of the fundamental quasiparticle tunneling operator as $\Delta_{QP}=\frac{2\nu+N}{8}$. For typical fractions and small $N$, $\Delta_{QP}<1$ so the system flows to an insulating fixed point in the IR, with universal conductance scalings $G(T)\sim T^{2/\nu}$ or $G(V)\sim V^{2/\nu}$ independent of $N$, while UV behavior retains $N$-dependent corrections that fingerprint the neutral Majorana sector. The analysis provides a practical electrical diagnostic for Majorana content, linking topological order to measurable transport exponents and offering a general framework for QPCs at topological edges beyond specific candidate states.

Abstract

Even-denominator fractional quantum Hall (FQH) states can be viewed as topological superconductors of composite fermions, supporting a charged chiral mode and $|\mathcal{C}_{cf}|$ neutral Majorana modes set by the Chern number $\mathcal{C}_{cf}$. Despite ongoing efforts, distinguishing the many competing paired phases remains an open problem. In this work, we propose a unified theory of charge transport across a quantum point contact (QPC) for general paired FQH states described by an $so(N)_1 \times u(1)$ conformal field theory. We derive the boundary effective action for an arbitrary number of Majorana fermions $N=|\mathcal{C}_{cf}|$ and develop a non-perturbative instanton approximation to describe tunneling processes. We establish a weak-strong duality relating strong quasiparticle tunneling to weak electron tunneling. We calculate the scaling dimensions of the tunneling operators and demonstrate that while the weak-coupling fixed point is generally unstable, the strong-coupling fixed point is stable for physically relevant filling fractions and number of Majorana fermions. These transport exponents provide a distinct experimental fingerprint to identify the topological phases of even-denominator FQH states.

Figures (1)

• Figure 1: Schematic of a quantum point contact geometry for a paired fractional Quantum Hall state. The top ($t$) and bottom (b) edges carry a charged boson mode $\phi$ and $N$ neutral Majorana modes $\psi_1,\dots,\psi_N$. The QPC constriction is located at $x=0$. (a) Weak quasiparticle tunneling regime with coupling strength $\Gamma_{QP}$. (b) The electron tunneling regime with coupling strength $\Gamma_{ET}$.