Fractional Quantum Hall Anyons via the Algebraic Topology of Exotic Flux Quanta

TL;DR

This paper introduces a new algebraic topological framework for understanding fractional quantum Hall anyons, potentially guiding experimental searches and advancing topological quantum computing.

Contribution

It presents a non-Lagrangian effective description of FQH anyons based on global flux quantization in non-abelian cohomology theories, linking topological flux to quantum observables.

Findings

Provides a rigorous algebraic topological model for FQH anyons.

Suggests flux quantization in 2-Cohomotopy theory for FQH systems.

Offers insights that could inform laboratory experiments on anyonic phenomena.

Abstract

Fractional quantum Hall systems (FQH), due to their experimentally observed anyonic topological order, are a main contender for future hardware-implementation of error-protected quantum registers ("topological qbits") subject to error-protected quantum operations ("topological quantum gates"), both plausibly necessary for future quantum computing at useful scale, but both remaining insufficiently understood. Here we present a novel non-Lagrangian effective description of FQH anyons, based on previously elusive proper global quantization of effective topological flux in extraordinary non-abelian cohomology theories. This directly translates the system's quantum -observables, -states, -symmetries, and -measurement channels into purely algebro-topological analysis of local systems of Hilbert spaces over the quantized flux moduli spaces. Under the hypothesis -- for which we provide a fair bit of evidence -- that the appropriate effective flux quantization of FQH systems is in 2-Cohomotopy theory (a cousin of Hypothesis H in high-energy physics), the results here are rigorously derived and as such might usefully inform laboratory searches for novel anyonic phenomena in FQH systems and hence for topological quantum hardware.