Majorana Zero-Mode Parity, the Hexagon Equation, and Chiral Edge Modes in 2D Topological Superconductors

In the paper's words

In a two-dimensional topological superconductor hosting vortices with Majorana zero modes (MZMs), the exchange of vortices yields a projective representation of the braid group. We investigate whether the full anyonic exchange matrix is uniquely determined by the MZM parity operators and whether consistency with the hexagon equation of the Ising modular tensor category necessitates chiral edge modes. We show that the MZM algebra γ_iγ_j + γ_jγ_i = 2δ_ij fixes the non-Abelian structure of the exchange operator up to an overall Abelian phase.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

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How it is done

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What would falsify it

The paper's text states no explicit falsification criterion; the limits below are what it says about where the claim stops.

Where the claim stops

1. The vortices in this system do not carry $\theta_\sigma = e^{i\pi/8}$; rather, the theory is a product $\text{Ising} \times \overline{\text{Ising}}$, and the individual vortices of each layer retain their chiral spins only if the layers are decoupled.

   from 5. Discussion

2. An important open question concerns extrinsic defects.

   from 5. Discussion

3. In a non-chiral system, vortices are extrinsic defects (gauge fluxes) rather than intrinsic anyons, since a superconductor does not have intrinsic topological order in the usual sense (it has a condensate that breaks a global symmetry).

   from 5. Discussion

Where to read

1. Abstract149 words · 1 min
2. 1. Introduction203 words · 1 min
3. 2. Background354 words · 2 min
4. 3. Analysis536 words · 2 min
5. 4. Results132 words · 1 min
6. 5. Discussion392 words · 2 min
7. 6. Conclusion126 words · 1 min