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 $l_T = \hbar v/(πk_B T)$ 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.
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.
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. Here, Ising anyons are realizable as edge vortices nucleated at Josephson line junctions defining the superconducting island, where both edge-vortex and Majorana-fermion tunneling processes can occur. 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.
In the Kitaev honeycomb model, a weak magnetic field gaps the gapless Majorana cone, driving the system into a gapped non-Abelian phase with Ising topological order. The edge supports a single chiral Majorana mode with chiral central charge c−=1/2, yielding a half-quantized thermal Hall conductance κxy/T=π2kB2/6h. We rigorously investigate whether observing this half-quantized value uniquely identifies the non-Abelian Ising anyon sector, or whether Abelian topological orders or extrinsic mechanisms can produce the same signature. By combining the edge-bulk correspondence with the classification of strictly two-dimensional bosonic topological orders, we show that a chiral central charge c−=1/2 (mod 8) is the unique fingerprint of the Ising non-Abelian anyon theory among minimal bosonic topological orders. Abelian bosonic orders necessarily have integer c−, with the minimal nonzero value being c−=1. Thus, a robustly half-quantized κxy/T uniquely distinguishes the Majorana sector from Abelian alternatives, provided the system is a genuine 2D bosonic spin system free of invertible E8 stacking or phonon contamination. ## 1. Introduction The Kitaev honeycomb model is an exactly solvable model of a quantum spin liquid, characterized by fractionalized excitations known as anyons. In the absence of a magnetic field, the model hosts a gapless Majorana cone. A weak magnetic field in the [111] direction gaps this cone and induces a non-Abelian phase with Ising topological order. This phase supports a single chiral Majorana edge mode, which contributes to a quantized thermal Hall conductance. The central question addressed in this preprint is whether the observation of a half-quantized thermal Hall conductance, corresponding to a chiral central charge c−=1/2, uniquely identifies the non-Abelian Ising anyon sector, or if Abelian topological orders can mimic this signature. ## 2. Background The thermal Hall conductance $\kappa_{x
The discovery of the fractional quantum Hall effect stimulated the investigation of anyons, particles with fractional statistics which are neither bosons nor fermions. This thesis focuses on the study of quantum Hall states which may support non-Abelian anyons. We first address the validity of assumptions used in the numerical study of such states, and then proceed with analyzing different experiments which can detect non-Abelian fractional statistics. We quantitatively analyze the two-point contact interferometer experiment, which is hoped to display clear-cut, direct evidence of non-Abelian fractional statistics. We calculate the temperature and voltage dependence of the interference experiment outcome, and the signal attenuation due to finite temperature loss of coherence. We then analyze the edge theory of a family of non-Abelian quantum Hall states in the second Landau level, and examine the tunneling between these states and a quantum dot. This tunneling problem maps onto the multi-channel Kondo problem, and will allow distinguishing between different quantum Hall states. Finally, we use the same theoretical methods for analyzing Sagnac interference in the conductance of a carbon nanotube loop, a one-dimensional system.
Majorana edge reconstruction and the ν=5/2 non-Abelian thermal Hall puzzle
Pioneering thermal transport measurements on two-dimensional electron gases in high magnetic fields have demonstrated that the quantized Hall state at filling factor ν=5/2 has a thermal Hall conductance κ quantized in half-integer multiples of κ0=π2kB2T/3h. Half-integer κ/κ0 is a signature of neutral Majorana edge modes, in turn linked to the presence of non-Abelian anyon excitations in the bulk. However, the experimentally observed value of κ corresponds to the 'PH-Pfaffian' state, in tension with numerical studies which instead favor either the Pfaffian or the AntiPfaffian. A variety of mechanisms have been invoked to explain this discrepancy, but have been either ruled out by further experiments or else involve fine-tuning. Building on density-matrix-renormalization group studies of physically realistic edges and analytic calculations of edge structure, we propose an alternative resolution of this puzzle involving an 'edge reconstruction' solely involving the neutral Majorana sector of the theory. Such a Majorana edge reconstruction can "screen'' a Pfaffian or AntiPfaffian bulk, so that transport signatures become indistinguishable from those of the PH-Pfaffian. We argue that this physically natural scenario is consistent with experiment.
Majorana zero modes (MZMs) localized in superconducting vortices are predicted to realize Ising anyon non-Abelian statistics. In finite-size vortex lattices, overlapping MZM wavefunctions generate exponentially decaying energy splittings. We investigate whether these finite-size effects merely perturb the ideal Ising braid group representation or fundamentally alter it into a distinct, dynamically dependent representation. By modeling a two-dimensional topological superconductor with an effective low-energy Hamiltonian incorporating exponentially decaying overlap terms, we numerically evaluate the time-evolution operator during adiabatic braiding. Our analysis shows that finite-size effects introduce a non-commuting dynamical phase that modifies the ideal Ising braid matrix. This results in a time-dependent braid matrix that alters expected interferometric conductance oscillations. Specifically, the conductance oscillations exhibit a phase shift proportional to the braiding time and inter-vortex distance, scaling as δϕ∝τΔe−R/ξ [to verify]. The non-Abelian nature appears degraded, with the braid matrix depending on the adiabatic trajectory and duration rather than solely on its topological class. This suggests interferometry can dynamically detect these finite-size modifications. ## 1. Introduction Majorana zero modes (MZMs) in topological superconductors are predicted to exhibit non-Abelian braiding statistics, forming the basis for topological quantum computation. In a p+ip superconductor or similar platforms, vortices trap MZMs whose braiding is described by the Ising anyon model. However, realistic systems are finite, and MZMs localized at neighboring vortices have overlapping wavefunctions, leading to exponentially small energy splittings. While often treated as a negligible perturbation, the impact of these splittings on the braid group representation during finite-time braiding remains an open question. Here, we analyze the
We naturally obtain the NOT and CNOT logic gates, which are key pieces of quantum computing algorithms, in the framework of the non-Abelian Chern–Simons–Higgs theory in two spatial dimensions. For that, we consider the anyonic quantum vortex topological excitations and show that Majorana anyons, namely, self-adjoint combinations of these vortices and anti-vortices, have in general non-Abelian statistics. The associated unitary monodromy braiding matrices become the required logic gates in the special case when the vortex spin is s = 1/4, which corresponds to the case of Ising non-Abelian anyons, found in different quantum computing systems. We explicitly construct the vortex field operators, show that they carry both magnetic flux and charge and obtain their Euclidean correlation functions by using the method of quantization of topological excitations, which is based on the order-disorder duality. These correlators are in general multivalued, the number of sheets being determined by the vortex spin. This, by its turn, is proportional to the vacuum expectation value of the Higgs field and therefore can be tuned by both the free parameters of the Higgs potential and the temperature.