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Nonabelian Anyons attached to Superconducting Islands in FQH Liquids

Hisham Sati, Urs Schreiber

TL;DR

This work addresses whether nonabelian anyons can emerge in fractional quantum Hall liquids via superconducting islands. It replaces the conventional 3D Chern-Simons viewpoint with a 5D Maxwell-Chern-Simons framework and $2$-Cohomotopy flux quantization, then reduces to observable 3D topological data while preserving covariant structure. For an $n$-punctured disk hosting $n$ islands, the topological state space becomes irreps of a framed spherical braid group (quotiented by $\mathrm{rot}$), yielding nonabelian braiding around islands in addition to abelian Hopfion-type phases; parastatistics appear for small $n$. This establishes a robust topological mechanism by which superconducting islands could attach nonabelian anyons to an FQH liquid, guiding experimental efforts to realize island-induced topological qubits.

Abstract

The idea that topologically protected quantum states, such as anyons, may be attached to super/semiconductor heterostructures has received enormous attention, but experimental signatures in 1D systems remain elusive. Here we revisit theoretical underpinnings of anyons in 2D fractional quantum Hall (FQH) systems, whose signatures have been experimentally observed by independent groups. Invoking novel theorems about the Hopfion or $\mathbb{C}P^1$-model understood as flux quantization in 2-Cohomotopy, we demonstrate a robust prediction for possibly nonabelian anyonic states induced by superconducting islands.

Nonabelian Anyons attached to Superconducting Islands in FQH Liquids

TL;DR

This work addresses whether nonabelian anyons can emerge in fractional quantum Hall liquids via superconducting islands. It replaces the conventional 3D Chern-Simons viewpoint with a 5D Maxwell-Chern-Simons framework and -Cohomotopy flux quantization, then reduces to observable 3D topological data while preserving covariant structure. For an -punctured disk hosting islands, the topological state space becomes irreps of a framed spherical braid group (quotiented by ), yielding nonabelian braiding around islands in addition to abelian Hopfion-type phases; parastatistics appear for small . This establishes a robust topological mechanism by which superconducting islands could attach nonabelian anyons to an FQH liquid, guiding experimental efforts to realize island-induced topological qubits.

Abstract

The idea that topologically protected quantum states, such as anyons, may be attached to super/semiconductor heterostructures has received enormous attention, but experimental signatures in 1D systems remain elusive. Here we revisit theoretical underpinnings of anyons in 2D fractional quantum Hall (FQH) systems, whose signatures have been experimentally observed by independent groups. Invoking novel theorems about the Hopfion or -model understood as flux quantization in 2-Cohomotopy, we demonstrate a robust prediction for possibly nonabelian anyonic states induced by superconducting islands.
Paper Structure (11 sections, 1 theorem, 19 equations, 2 figures)

This paper contains 11 sections, 1 theorem, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Critique of Effective Lagrangians
  4. Proper Topological Limit
  5. Proper Flux Quantization
  6. Relation to Hopfion/CP1-Model
  7. Covariantized Observables
  8. Results
  9. Modelling Superconducting Islands
  10. Deriving the Anyon Attachment
  11. Conclusion

Key Result

Proposition 3.1

[proposition]AppearanceOfBraidGroup The topological quantum states on the $n$-punctured disk according to ObservablesOnPuncturedDisk fall into unitary irreps of the subgroup of the $\mathrm{rot}$-quotient of the framed spherical braid group with $n+1$ strands on framed braids whose total framing number is a multiple of $n+1$.

Figures (2)

  • Figure 1: A disk (hosting an FQH fluid) with $n$ islands (punctures) is homeomorphic to a sphere with $n+1$ punctures. From the view of the complement bulk, all punctures are at infinity\ref{['CompactificationOfPuncturedDisk']}, which entails with \ref{['MapsVanishingAtInfinity']} that magnetic flux does not enter here.
  • Figure 2: Spherical braids\ref{['TheSphericalBraidGroup']} are braids of worldlines of punctures on the 2-sphere (cf. \ref{['PuncturedDiskAndSphere']}). These are almost the same as ordinary Artin braids of punctures in the plane. (Shown is the element $b_1 b_2 b_1 b_2 b_1$). The only difference being that on the sphere, the Artin braid $(b_1 \cdots b_n)(b_n \cdots b_1)$ (where all punctures are fixed except one, which is circling the others) is topologically trivial.

Theorems & Definitions (1)

  • Proposition 3.1