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Non-Lagrangian Construction of Anyons via Flux Quantization in Cohomotopy

Hisham Sati, Urs Schreiber

TL;DR

The paper addresses the challenge of rigorously understanding anyonic topological order beyond traditional Lagrangian Chern-Simons descriptions by proposing a flux quantization framework grounded in $2$-cohomotopy. It argues that surplus FQH flux is best described by a non-Lagrangian, exotic quantization with classifying space $\mathbb{C}P^1$, yielding the correct braiding, topological order, and edge phenomena, while remaining compatible with abelian CS predictions in many regimes. It further predicts novel defect anyons arising from flux-expelling superconducting islands and extends the framework to fractional Chern insulators, where Berry curvature plays the role of flux and monodromy resides in reciprocal momentum space $\widehat{\mathbb{T}}^2$. Overall, the approach offers a unifying, non-Lagrangian description that aligns with known FQH behavior and opens new experimental pathways for detecting non-abelian defect anyons and momentum-space anyons in FQAH systems.

Abstract

We provide a brief invitation to the novel understanding of anyonic topological order in fractional quantum (anomalous) Hall systems, via "extraordinary" quantization of effective magnetic flux in Cohomotopy -- following our presentation at ISQS29.

Non-Lagrangian Construction of Anyons via Flux Quantization in Cohomotopy

TL;DR

The paper addresses the challenge of rigorously understanding anyonic topological order beyond traditional Lagrangian Chern-Simons descriptions by proposing a flux quantization framework grounded in -cohomotopy. It argues that surplus FQH flux is best described by a non-Lagrangian, exotic quantization with classifying space , yielding the correct braiding, topological order, and edge phenomena, while remaining compatible with abelian CS predictions in many regimes. It further predicts novel defect anyons arising from flux-expelling superconducting islands and extends the framework to fractional Chern insulators, where Berry curvature plays the role of flux and monodromy resides in reciprocal momentum space . Overall, the approach offers a unifying, non-Lagrangian description that aligns with known FQH behavior and opens new experimental pathways for detecting non-abelian defect anyons and momentum-space anyons in FQAH systems.

Abstract

We provide a brief invitation to the novel understanding of anyonic topological order in fractional quantum (anomalous) Hall systems, via "extraordinary" quantization of effective magnetic flux in Cohomotopy -- following our presentation at ISQS29.

Paper Structure

This paper contains 12 sections, 10 equations, 4 figures.

Table of Contents

  1. Motivation and Introduction
  2. Topological quantum and Anyons.
  3. Solitonic Anyons in FQH Systems
  4. Fractional Quantum Hall Anyons.
  5. Flux quantization -- Ordinary and exotic.
  6. Relation to Chern-Simons theory.
  7. Anyon worldlines and their braiding.
  8. General covariance and Topological order
  9. Defect Anyons in FQH Systems
  10. Superconducting Islands in FQH Systems.
  11. Solitonic Anyons in FQAH Systems
  12. Conclusion and Outlook

Figures (4)

  • Figure 1: Topological quantum gates by adiabatic braiding of anyon worldlines.
  • Figure 2: Anyons in FQH systems are (quasi-hole vortices associated with) surplus flux magnetic flux quanta (relative to a given rational filling fraction of $K$ flux quanta per electron) through an electron gas occupying an effectively 2-dimensional semiconducting surface $\Sigma^2$. This suggests SS25-FQH that FQH anyons are to be understood in terms of an exotic effective flux quantization lawSS25-Flux.
  • Figure 3: Some (blackboard-)framed Wilson loop/links and their total crossing number/writhe.
  • Figure 4: Where FQH anyons are solitonic quanta of concentrations of magnetic flux (cf. Fig. \ref{['Flux']}), super-conducting island in the semi-conductor substrate will tend to expel magnetic flux. Hypothesis h implies/predicts that, if adiabatically movable, such islands behave like defect anyons.