DC Conductance of X-shaped Majorana Interferometer reveals Non-Abelian Anyon Statistics

The gist: The four-terminal, X-shaped chiral Majorana interferometer enables direct detection of non-Abelian statistics of Ising anyons via linear-response DC conductance tensor in charge transport experiments.

Theoretical Framework and Model

The study utilizes a four-terminal, X-shaped chiral Majorana interferometer with a central floating superconducting island to probe the non-Abelian statistics of Ising anyons through DC conductance measurements The device is realized on a topological insulator surface proximitized by a conventional superconductor and magnetic materials of opposite magnetization, leading to 1D chiral Majorana fermion modes at interfaces The low-energy theory is derived using chiral bosonization, which describes the four 1D chiral Majorana modes as two neutral auxiliary 1D chiral Dirac fermion modes represented by two chiral boson fields The Hamiltonian for the system is initially given by H0 = -iv/2X n,j∈[1,2]Z L0 dx ξn,j (x)∂xξn,j (x).

Majorana Tunneling and Edge Vortex Dynamics

The device incorporates two types of tunneling processes: Majorana fermion tunneling between different Majorana edges at the Josephson line junctions and composite edge vortex tunneling events The Majorana fermion tunneling Hamiltonian HMT involves terms like H(a,d)MT = λa/π ∂xϕ1(a1) + λd/π ∂xϕ2(a1), which can be gauged away by a unitary transformation Furthermore, EV tunneling processes are included via operators like Ta = σ1,1(a1)σ2,1(a1) and Td = σ2,2(a1)σ1,2(a1), which lead to terms like HEV ∝ Γ X ± e∓2πing/2∓ Sz e±i/2ϕ1(a1). The composite EV tunneling processes in the vacuum fusion channel are described by HEV = Γe-2πing e i/2 [ϕ1(a1)+ϕ2(a1)] e - i/2 [ϕ1(a2)+ϕ2(a2)] + h.c.

Renormalization Group Analysis and Anyon Statistics

The Renormalization Group (RG) analysis of the effective action Seff[w1, w2] reveals a flow toward a strong-coupling fixed point when ∆ > v/δa The RG equations for the couplings Γ and Γ˜ show that both Majorana tunneling amplitudes and phases φ± stay marginal to all orders in perturbation theory. In the regime ∆ > v/δa, the fermionic fusion channel is dynamically generated by the interplay of EV tunneling (rate ∝ Γ) and Majorana fermion tunneling (∝ φbc). The effective coupling Γ˜ in the weak-coupling regime is estimated as Γ˜eff ≃ -Γδλ/v ∆/ε˜ 1/2 ln(∆/ε˜), which serves as an estimate for the rate of composite EV tunneling processes.

DC Conductance Tensor Signatures

The DC conductance tensor Gjj′ is calculated using a Kubo formula, and it is found to be completely isotropic, G = G˜ 1 1! In the absence of EV tunneling or Majorana tunneling events, the steady-state DC currents flowing into the drain electrodes must vanish altogether. A finite low-energy conductance G˜ is expected only in the simultaneous presence of both EV tunneling (rate ∝ Γ) and Majorana fermion tunneling (amplitude ∝ δλ). The phase shift π/4 in the conductance oscillations with respect to a gate-tunable charge parameter ng displays an offset related to the anyon topological spin, providing a signature for non-Abelian braiding statistics.

Strong Coupling Regime Results

In the strong-coupling regime, where Γ → ∞ and Γ˜ → −∞, the linear-response DC conductance tensor is found to be G˜/G0 ≃ ∆/4Γ 1 1 + 2πΓδa/v. Even in this limit, the finite result for G˜ in Eq. (3.21) can provide evidence for the non-Abelian statistics of Ising anyons. This finite conductance is a remarkable consequence of the multi-terminal geometry allowing anyon fusion processes in the fermionic channel, which contrasts with two-arm interferometers.

Conclusions

Multi-terminal interferometer geometries harboring chiral Majorana modes can provide conceptually simple, and hopefully soon realizable, setups which allow experimentalists to probe the non-Abelian statistics of Ising anyons in terms of easily accessible linear-response DC transport observables. The mere observation of a finite low-energy conductance already represents a signature for the non-Abelian statistics of Ising anyons. The resulting isotropic conductance tensor is a nontrivial experimental check of the theory.

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Prepared for submission (JQuant)

A JOURNAL BY SISSA MEDIALAB JQuant DC Conductance of X-shaped Majorana Interferometer reveals Non-Abelian Anyon Statistics D. Giuliano a,b,c A. Nava a,∗ R. Egger a and F. Hassler d Institut für Theoretische Physik, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany b I.N.F.N., Gruppo collegato di Cosenza, Arcavacata di Rende, I-87036 Cosenza, Italy c Dipartimento di Fisica, Università della Calabria Arcavacata di Rende, I-87036 Cosenza, Italy d Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany contacts: domenico.giuliano@fis.unical.it a.nava@hhu.de, egger@hhu.de, hassler@physik.rwth-aachen.de ABSTRACT We propose a four-terminal, X-shaped chiral Majorana interferometer with a central floating superconducting island, enabling the direct detection of the non-Abelian statistics of Ising anyons via the linear-response DC conductance tensor in charge transport experiments We show that in such a multi-terminal Majorana interferometer, both the vacuum and the fermionic fusion channel for Ising anyons are possible This is in contrast to two-arm interferometers, where only the vacuum fusion channel is accessible and the DC conductance contribution from edge vortices always vanishes Using a low-energy effective theory derived via chiral bosonization, we find that in the X-shaped interferometer, the DC conductance tensor is completely isotropic, yielding a non-zero conductance when simultaneous edge-vortex and Majorana tunneling activates the fermionic fusion channel Apart from conductance oscillations in a gate-tunable charge parameter, which display an offset related to the anyon topological spin measuring a finite conductance can already provide direct evidence for non-Abelian statistics in this geometry arXiv:2607.24171v1 [quant-ph] 27 Jul 2026

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CONTENTS 1 Introduction The fractional exchange statistics of anyonic excitations has garnered a lot of attention after recent experimental measurements of the statistical braiding phase of Abelian anyons in fractional quantum Hall (FQH) devices [1, 2] An outstanding open challenge is to demonstrate the non-Abelian braiding statistics of the more complex non-Abelian anyons predicted to arise, e.g., at specific FQH filling factors and in topological superconductor (SC) geometries Such excitations could form the basis for topological quantum information processing schemes [3] The simplest type of non-Abelian anyon excitation is given by Ising anyons [3], which correspond to vortices harboring a zero-energy Majorana zero mode (MZM) When realized as excitations in the chiral one-dimensional (1D) edge modes of FQH samples or in other topologically nontrivial materials, such edge vortices (EVs) are intrinsically mobile (“flying”), and thus are attractive candidates for transporting quantum information [4] Moreover, as we discuss below, flying Ising anyons can allow experimentalists to probe the expected non-Abelian fusion rules and the associated braiding statistics of Ising anyons directly in DC charge transport experiments Flying Ising anyons are intimately related to their spatially localized MZMs [5], which have been discussed in recent experiments on proximitized semiconductor heterostructures and tunable quantum dot geometries and may provide practically useful Majorana qubits, see, e.g., Refs.

1. Bold Header: Non-Abelian Statistics Detection via DC Conductance Tensor Analysis. The improved AI system can perform direct detection of non-Abelian statistics by analyzing the linear-response DC conductance tensor in charge transport experiments, moving beyond previous limitations where only the vacuum fusion channel was accessible.

2. Bold Header: Isotropic Conductance Signature Recognition. The system will identify a completely isotropic conductance tensor, which is parametrized by a single conductance G˜ at low energies, as a key signature for non-Abelian statistics, providing an interesting challenge to experimentally test these predictions.

3. Bold Header: Topological Spin Measurement via Phase Shift Extraction. The AI can extract the signature of the topological spin by measuring the phase shift π/4 in Eq. (3.12) of conductance oscillations with respect to a gate-tunable charge parameter, which is connected to the topological spin of Ising anyons.

4. Bold Header: Dynamic Generation of Fermionic Fusion Channel. The system can predict that the fermionic fusion channel for Ising anyons is dynamically activated when both EV tunneling and Majorana tunneling occur, as shown in the derivation where a finite coupling Γ˜ will be dynamically activated by the simultaneous presence of EV tunneling (Γ) and Majorana fermion tunneling (δλ).