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arXiv:2601.08792 · Physical review. B./Physical review. B

Universal transport theory for paired fractional quantum Hall states in the quantum point contact geometry

Eslam Ahmed, Ryoi Ohashi, Hiroki Isobe, Kentaro Nomura, Yukio Tanaka

Published Apr 30, 2026• 1 citation
arXivHTMLPDF

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.

DOI
10.1103/nqsj-hl9z

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Even-denominator fractional quantum Hall (FQH) states can be viewed as topological superconductors of composite fermions, supporting a charged chiral mode and ∣Ccf∣|\mathcal{C}_{cf}|∣Ccf​∣ neutral Majorana modes set by the Chern number Ccf\mathcal{C}_{cf}Ccf​. 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×u(1)so(N)_1 \times u(1)so(N)1​×u(1) conformal field theory. We derive the boundary effective action for an arbitrary number of Majorana fermions N=∣Ccf∣N=|\mathcal{C}_{cf}|N=∣Ccf​∣ 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.
The fractional quantum Hall effect (FQHE) at the filling factor with an even denominator 52\frac{5}{2}25​ occurs despite the expectation, due to the electron statistics, that the denominator must be an odd number. It is believed that the Cooper pairing of underlying quasiparticles, composite fermions (CFs), leads to the explanation of this effect. Such a state should have a Pfaffian form of the BCS wave function (due to the fully polarized spin) and non-Abelian statistics of possible vortexlike excitations (due to the ppp-wave nature of the pairing). Here we expose the origin of pairing by using the effective dipole representation of the problem and show that pairing is encoded in a Hamiltonian that describes the interaction of the charge density with dipoles, i.e., the current of CFs. The necessary condition for the paired state to exist is the effective dipole physics at the Fermi level as a consequence of the nontrivial topology of the ideal band in which electrons live, a Landau level (LL); the paired state is a resolution of the unstable, critical behavior characterized by the distancing of correlation hole with respect to electron (and thus dipole) at the Fermi level due to the topology. We describe analytically this deconfined critical point, at which deconfinement of Majorana neutral fermions takes place. In the presence of large, short-range repulsive interaction inside a LL, the critical behavior may be stabilized into a regularized Fermi-liquid-like state, like the one that characterizes the physics in the lowest LL, but in general, for an interaction with slowly decaying pseudopotentials, the system is prone to pairing.
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Fractional quantum Hall-superconductor (FQH-SC) heterostructures are predicted to host defect-bound parafermion zero modes (PZMs). We propose two related configurations to probe their fusion structure. In a Josephson junction coupled to a single quantum point contact (QPC), quasiparticle tunneling switches the defect fusion channel, producing stochastic transitions between branches of the fractional Josephson spectrum. Embedding the junction in a two-QPC Fabry-Pérot interferometer provides a complementary probe. Weak zero mode tunneling produces fusion-channel-dependent interference while strong tunneling makes the interferometer probe a superposition of fusion channels and strongly suppresses the signal: in the topological limit it vanishes exactly, revealing the defects' non-Abelian nature even when the parent FQH state is Abelian.
We have recently calculated exact nonequilibrium quantum transport properties through a point contact in a Luttinger liquid. Using a particular quasiparticle basis of the Hilbert space dictated by integrability, we here compute explicitly the exact I(V) characteristic and conductance out of equilibrium as a function of driving voltage V and temperature T. These are described by universal scaling functions of two variables, the scaled point-contact interaction strength, and V/T. The differential-conductance curve as a function of the interaction strength broadens significantly as V/T is increased, and develops a pronounced maximum at a (universal) critical value (eV/kB{\mathit{k}}_{\mathit{B}}kB​T)=7.188 68. . .. In addition, we derive an exact duality between strong and weak backscattering. The theory presented here has recently been realized experimentally in resonant tunneling-transport experiments between edge states in fractional quantum Hall effect devices. In this context the exact duality is between electron tunneling and Laughlin-quasiparticle tunneling.
We study zero modes bound to charge-conjugation defects in fractional quantum Hall-superconductor (FQH-SC) heterostructures built from Abelian and non-Abelian FQH states. Although the associated defects are well described by GGG-crossed braided tensor categories, a local operator description of their zero modes is not generally known. We generalize the bulk-to-boundary construction of localized zero mode operators to FQH-SCs with non-Abelian parent topological orders, mapping bulk Wilson operators in the folded theory to localized zero mode operators in the one-dimensional FQH-SC theory. This construction distinguishes zero mode anyons from the local zero mode operators that act on the defect Hilbert space and yields a generalized zero mode algebra. It also determines the fusion-channel dependence of tunneling: tunneling of zero mode operators probes bulk monodromy, while direct zero mode anyon transfer probes GGG-crossed associativity data. Applying this framework, we recover the Z2m\mathbb{Z}_{2m}Z2m​ parafermion zero mode structure of fermionic Laughlin FQH-SCs at filling ν=1/mν=1/mν=1/m, obtain a Z2(n)×ZN\mathbb{Z}_2^{(n)}\times\mathbb{Z}_NZ2(n)​×ZN​ structure for generalized Moore-Read states at filling ν=1/mν=1/mν=1/m, with N=2mN=2mN=2m for fermionic states and N=mN=mN=m for bosonic states, and find a Fib×ZN\mathrm{Fib}\times\mathbb{Z}_NFib×ZN​ structure for the k=3k=3k=3 Read-Rezayi state at filling ν=3/(3M+2)ν=3/(3M+2)ν=3/(3M+2), with N=2(3M+2)N=2(3M+2)N=2(3M+2) for fermionic states and N=(3M+2)/2N=(3M+2)/2N=(3M+2)/2 for bosonic states. In the Read-Rezayi case, zero mode operator tunneling and direct zero mode anyon transfer yield distinct universal amplitude ratios between the neutral fusion channels, providing a characteristic signature of the underlying Fibonacci defect structure.
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