Transport theory for a generic two-arm co-propagating Majorana interferometer with Majorana fermion and edge vortex tunneling

arXiv:2610.01402 · cond-mat.mes-hall, quant-ph · Submitted 2026-10-01 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "Transport theory for a generic two-arm co-propagating Majorana interferometer with Majorana fermion and edge vortex tunneling".

Kai: The gist: Co-propagating chiral Majorana edge modes in proximity-coupled quantum anomalous Hall systems can exhibit transport signatures related to non-Abelian anyon statistics through effective capacitance measurements at low frequencies,

Mira: First, who's behind it and why it matters.

Title and authors: Kai: So we're looking at this paper now, "Transport theory for a generic two-arm co-propagating Majorana interferometer with Majorana fermion and edge vortex tunneling." It sounds really specific to the physics of these coupled systems.

Mira: Yeah, it deals with how these chiral Majorana modes behave when you put them into an interferometer setup, specifically looking at how they interact with edge vortices through tunneling.

Kai: Exactly. The title points toward the key elements: co-propagating Majorana fermions and that vortex tunneling aspect. It suggests we're seeing a way to probe non-Abelian statistics using electrical measurements.

Lev: From my side, it makes sense that they'd focus on the Mach-Zehnder interferometer geometry, because that’s the simplest setup to visualize this kind of interference.

Kai: Right. The authors are setting up this generic two-arm system where you can have different lengths L1 and L2. They're using a TI surface proximitized by a grounded superconductor with a fixed phase, which is crucial for getting those Majorana modes in the first place #pg3.

Mira: And they’re interested in how this setup allows us to detect non-Abelian statistics through charge transport, which is what's really interesting here.

Lev: I wonder if this setup translates well to what we have on real hardware, given the need for precise control over these proximity effects.

The paper's summary: Kai: So, in terms of what they found, the core idea is that when you include those effective EV tunneling processes, you don't see a change in the DC conductance G(zero) <ref:2610.01402#pg3>. That’s a big point because it means the fundamental interference pattern of these modes isn't immediately obvious through direct resistance measurements.

Mira: That’s because they are focused on what happens at low frequencies, where we can look for other signatures. They find that instead of a change in DC conductance, you get an effective capacitance Ceff appearing in the AC conductance G(ω) corrections.

Kai: So, this Ceff is the link to the non-Abelian nature of the edge vortices. It’s sensitive to things like their topological spin and their conformal dimension #pg2.

Lev: That suggests that if we want to actually measure anyon statistics, we might have to look at capacitance measurements rather than just DC conductance, which is a practical consideration for experimentalists.

Mira: Precisely. The paper shows that the leading low-frequency corrections are linear in frequency, and they attribute this behavior directly to this effective capacitance Ceff #pg2.

The paper's improvements: Kai: Now, looking at how the authors improved the theory, they used a replicated bosonization field theory. They doubled the replica sector for each mode to handle these EV tunneling processes more rigorously #pg1.

Mira: That doubling is what allows them to describe Majorana fermion tunneling by introducing terms like HMT → -ivX2j=1λjX2a=1ξ(a)one(aj)ξ(a)two(bj), which helps maintain the necessary symmetries during the calculation <ref:2610.01402#pg1>.

Lev: From an error correction standpoint, this doubled theory is complex, but it’s necessary to correctly account for the total isospin conservation they mentioned when dealing with EV tunneling #pg1.

Kai: And what's really interesting about their quantitative prediction is how they relate the topological spin to a phase offset, pi S z/four = plus or minus two pi s sigma. This connects the statistical properties directly to measurable parameters #pg2.

Mira: Plus, they fix the conformal dimension h sigma at one/sixteen which sets the exponent for how Ceff depends on a cutoff l c <ref:2610.01402#pg2>. That gives them a concrete way to predict what Ceff should look like based on these topological properties #pg2.

Conclusion: Kai: So, summarizing the main points of this paper, they’ve shown that EV tunneling doesn't change the DC conductance of this Majorana interferometer setup but it does introduce a low-frequency effective capacitance Ceff.

Mira: And this capacitance carries information about the topological spin and conformal dimension of those edge vortices, which is what makes it a signature for non-Abelian statistics.

Lev: It sounds like a useful tool to check if we have the right physical system in place, because it gives us a quantifiable target for our measurements.

Kai: Exactly. They also establish that the thermal length l T = v/(pi T) is the relevant scale for this interference, meaning as long as your arm lengths are within that scale, you should still see these signatures #pg2.

Mira: So, to wrap up on "Transport theory for a generic two-arm co-propagating Majorana interferometer with Majorana fermion and edge vortex tunneling," it’s a way to translate the abstract concept of anyon braiding into something we can measure electrically.

Lev: For me, the paper shows that even though experimental evidence for chiral Majorana modes is still coming, there are calculable observables we can aim for in future experiments.

Domenico Giuliano, Andrea Nava, Fabian Hassler, Reinhold Egger

Institut f¨ur Theoretische Physik, Heinrich-Heine-Universit¨at · Dipartimento di Fisica, and INFN, Gruppo Collegato di Cosenza, Universita della Calabria · Institute for Quantum Information, RWTH Aachen University

cond-mat.mes-hall, quant-ph

Submitted: 2026-10-01

Updated: 2026-10-01

Comments: 12 pages, 5 figures

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 77/100

The gist: The gist: Co-propagating chiral Majorana edge modes in proximity-coupled quantum anomalous Hall systems can exhibit transport signatures related to non-Abelian anyon statistics through effective

Key concepts

Majorana Edge Modes
These are specific types of fermionic excitations that occur at the edges of a topological superconductor. They are crucial because they carry non-Abelian statistics, meaning their exchange properties are not simple like those of conventional particles.
Edge Vortex Tunneling (EV Tunneling)
This process describes the interaction where edge vortices—which act as flying non-Abelian anyons—tunnel between different arms of the interferometer. This tunneling is what introduces corrections to the AC conductance at low frequencies.
Effective Capacitance (Ceff)
Instead of DC conductance, this paper focuses on the low-frequency AC charge conductance, which manifests as an effective capacitance. This capacitance carries information about the topological spin and conformal dimension of the edge vortices, providing a measurable signature of their non-Abelian nature.

Terminology

Summary

The gist: Co-propagating chiral Majorana edge modes in proximity-coupled quantum anomalous Hall systems can exhibit transport signatures related to non-Abelian anyon statistics through effective capacitance measurements at low frequencies, which are sensitive to topological spin and conformal dimension

Model and Setup

The study focuses on a generic two-arm co-propagating Majorana interferometer realized on the surface of a TI material proximitized by a grounded SC with fixed phase. The setup involves two chiral Majorana fermion modes, ξ1(x) and ξ2(x), of potentially different lengths L1 and L2, which are populated from a 1D chiral Dirac channel at the source electrode S. The device features a central floating SC island with charging energy EC and Josephson energy EJ, leading to composite EV tunneling processes at the line junctions.

Transport without EV Tunneling

In the absence of Majorana and EV tunneling processes, the DC conductance G(0) does not reveal any signature of anyon statistics because of the simplicity of interfering trajectories. The low-frequency AC charge conductance G(ω), in particular the effective capacitance Ceff, encodes information about the topological spin of EVs and thus about anyon statistics. For arbitrary length mismatch δL = L1 − L2, the DC conductance G(0) is given by G0 = δL/lT sinh(δL/lT).

EV Tunneling and Capacitance

The inclusion of EV tunneling processes introduces corrections to the AC conductance, which manifest as an effective capacitance Ceff. The leading low-frequency corrections to the AC conductance are linear in frequency and can be attributed to an effective capacitance Ceff. The general result for the effective capacitance is given by Eq. (37), which consists of several factors encoding different signatures of the non-Abelian nature of the EVs.

Signatures of Non-Abelian Statistics

The topological spin enters through the phase offset πSz/4 = ±2πsσ, which corresponds to the topological spin of EVs. The conformal dimension hσ = 1/16 fixes, for example, the exponent of the cutoff dependence ∝ l4hσc of Ceff. The relevant length scale for EV interference is the thermal length lT = v/(πT), which is analogous to quasiparticle transport in conventional quantum Hall edge states with velocity v.

Conclusion on Tunneling Effects

The paper demonstrates that EV tunneling does not produce a correction to the DC conductance but only yields an effective capacitance Ceff at low frequencies. The leading low-frequency corrections to the AC conductance are linear in frequency and can be attributed to an effective capacitance Ceff. This generalizes the corresponding result for the symmetric geometry [25] to the general geometry in Fig. 1.

Data Availability

The data underlying the figures presented in this work will be made available at Zenodo. The acknowledgments thank A. Akhmerov and C. Beenakker for discussions, and funding is acknowledged by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation). The paper also references numerous works on anyonic statistics in fractional quantum Hall systems and topological superconductors to. The work opens the possibility to study more complicated setups such as multi-terminal devices [26] or braiding of EVs [20, 22].

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Transport theory for a generic two-arm co-propagating Majorana interferometer with Majorana fermion and edge vortex tunneling

How it works

The study employs a replicated bosonization field theory to tackle the effects of EV tunneling processes. The approach involves introducing two replicas for each field, ξj=1,2(x) → ξ(a)j(x) with replica index a = 1, 2. Majorana fermion tunneling is described by HMT → -ivX2j=1λjX2a=1ξ(a)1(aj)ξ(a)2(bj). The matching conditions (5) at x = 0 and ˜x = 0 now hold separately for each replica index.

Transport Quantities

The linear-response conductance G(ω) is related to the equilibrium (V = 0) retarded current-current correlation function ΠR(t) through a Kubo formula. The average in Eq. (19) is then taken over H = H0 + HMT + HEV, where the EV tunneling Hamiltonian HEV is discussed in Sec. III C below. In the absence of Majorana tunneling, the DC conductance G = G0 with the conductance quantum G0 = e2/h.

Edge Vortex Tunneling

The composite EV tunneling Hamiltonian involves a product of four σ operators for the symmetric geometry. The physical processes require the total isospin Szu + Sd to be conserved because of global fermion parity conservation. The operator Tσ(aj, bj) in Eq. (26) should then be replaced in the doubled theory by Tσ(aj, bj) = S-e i2ϕu(aj)e i2ϕd(bj) + S+e- i2ϕu(aj)e- i2ϕd(bj).

Calculation of Transport Observables

The first-order contribution to [Π(τ)]2 in Eq. (19), without Majorana fermion tunneling, yields a correction term δfΠ(ωn). Rotating back to real frequencies, iωn → ω + i0+ with A(ωn) → A(ω), and expanding in powers of ω around ω = 0, Eq. (31) yields δfΠ(ω) ∝ ω2. As a consequence, Eq. (16) implies that there are no corrections to the DC conductance due to EV tunneling and thus no signatures of the non-Abelian statistics. The leading low-frequency corrections to the AC conductance are linear in frequency and can be attributed to an effective capacitance Ceff.

Conclusions

The paper generalizes the transport theory of co-propagating Majorana interferometers to an asymmetric setup. It demonstrates that EV tunneling does not produce a correction to the DC conductance but only yields an effective capacitance Ceff at low frequencies. The capacitance exhibits oscillatory behavior as a function of the Majorana tunneling strength and gate charge ng, with an offset due to the topological spin. The relevant length scale for EV interference is the thermal length lT = v/(πT).

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Co-propagating chiral Majorana edge modes are predicted to exist in proximity-coupled quantum anomalous Hall systems. They combine the exciting prospects of edge vortices as flying non-Abelian Ising anyons, with their fully electrical detection through anyon fusion processes. In this setup, electrical transport arises from the interference of a pair of Majorana edge states. By bosonizing the model with an additional replica sector, we generalize previous treatments to fully asymmetric configurations. We analyze a Mach-Zehnder interferometer and show that while the DC conductance does not exhibit signatures of edge-vortex interference, at low frequencies, this interference manifests as an effective capacitance. This quantity is sensitive to the non-Abelian anyon statistics of edge vortices and, in particular, carries signatures of their topological spin and nontrivial conformal dimension. We demonstrate that the thermal length lT = ħv/(πkBT) is the relevant scale at low temperatures T, analogous to quasiparticle transport in conventional quantum Hall edge states with velocity v. As long as the interferometer arm lengths match within this length scale, transport signatures of the non-Abelian statistics of edge vortices remain visible.

How it works

The device in Fig. 1 features two chiral Majorana fermion modes of length L1 and L2, respectively, which can be realized on a TI surface that is proximitized inside the interferometer by a grounded SC with fixed phase, say, φ = 0. The energy scale ∆ thus also acts as the bandwidth of the lowenergy theory below. Particles are then injected into the chiral Majorana modes via a 1D chiral Dirac channel from the normal source electrode S with chemical potential µS(t) = eV (t) = eV cos(ωt).

Transport Quantities

The Hamiltonian H0 + HS + HD + HMT together with the matching conditions (5) is quadratic in the fermion operators.

Improvements for AI systems

  1. Bold header: Precise Topological Spin Detection for Anyon Statistics

The improved AI system can detect non-Abelian braiding statistics by measuring the topological spin, which is defined as e 2πisσ = e iπ/8 and t[topological spin] is closely tied to the non-Abelian exchange statistics of Ising anyons.

  1. Bold header: Quantitative Prediction of Effective Capacitance

The system can calculate the low-frequency AC conductance correction, which is characterized by an effective capacitance Ceff, using the formula derived from the doubled bosonization approach in Eq. (37). This allows for a quantitative prediction of how EV interference manifests as an effective capacitance.

  1. Bold header: Asymmetric Geometry Transport Analysis

The AI can analyze transport in generic asymmetric two-arm devices, showing that DC conductance G(0) is non-zero even for Γ = 0 and that the leading low-frequency corrections to the AC conductance are linear in frequency.

  1. Bold header: Thermal Length Scaling for Interference Visibility

The system can determine that interference signatures remain visible as long as arm lengths match within this length scale, specifically noting that the relevant scale at low temperatures T is the thermal length lT = ħv/(πkBT).

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