Schreiber FQH Anyons

An article that we are finalizing at CQTS:

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• Fractional Quantum Hall Anyons via

  the Algebraic Topology of Surplus Flux Quanta

  
  

  manuscript:

  • pdf (v3, substantial revision of fine-print)

  • arXiv:2505.22144 (v2)

Abstract. Fractional quantum Hall systems (FQH), with their experimentally observed anyonic topological order, are a desirable future hardware platform for intrinsically stable quantum registers (“topological qbits”) and operations (“topological quantum gates”). But theoretical questions remain, notably concerning the desired realization of nonabelian anyon statistics.

Here we analyze a novel non-Lagrangian effective description of FQH anyons based on global quantization of surplus flux in extraordinary non-abelian cohomology theories, specifically in unstable Cohomotopy. Topological quantum order manifests as local systems of gapped Hilbert spaces over the covariantized quantized flux moduli spaces.

Assuming that FQH surplus flux is effectively quantized in 2-Cohomotopy theory (a cousin of Hypothesis H in high-energy physics), we rigorously compute covariant flux monodromy and classify its unitary irreducible representations. This reproduces fine details such as of modular topological order on the torus; but it also predicts potential new routes to nonabelian statistics in FQH systems which may inform experimental searches for topological quantum hardware.

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Exposition in:

• Rethinking FQH Anyons (talk)

• Rethinking Topological Quantum (lightning talk)

Based on:

• Flux Quantization

• Flux Quantization on M5-Branes

• Cohomotopy, Framed Links, and Abelian Anyons

• Anyons on M5-Probes of Seifert 3-Orbifolds

  (which in turn is based on: The Character Map in Twisted Equivariant Nonabelian Cohomology)

• Engineering of Anyons on M5-Probes via Flux-Quantization

  (lecture notes)

Followup articles:

• Nonabelian Anyons attached to Superconducting Islands in FQH Liquids

• Identifying Anyonic Topological Order in Fractional Quantum Anomalous Hall Systems

• Higher-dimensional Anyons via Higher Cohomotopy

• Non-Perturbative SDiff Covariance of FQH Excitations

Related presentations:

• Homotopy Theory for Topological Quantum Computing

• Rethinking FQH Anyons – talk

• Non-Lagrangian construction of abelian CS/FQH theory

• The $1 Million Puzzle: Hadrons and Quantum Computers