Schreiber Understanding FQH via Algebraic Topology

An article that we are finalizing at CQTS:

---

• Hisham Sati$\;\;\;$and$\;\;\;$ Urs Schreiber

  
  

  Understanding Fractional Quantum Hall Systems

  via the Algebraic Topology of exotic Flux Quanta

  
  

  download draft: pdf

Abstract. Fractional quantum Hall systems (FQH) are a main contender for future hardware realizing topologically protected registers (“topological qbits”) subject to topologically protected operations (“topological quantum gates”), both plausibly necessary ingredients for future quantum computers at useful scale, but remaining only partially understood.

Here we present a novel non-Lagrangian effective description of FQH systems, 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 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 (a cousin of Hypothesis H in high energy physics), the results here are rigorously derived and as such might usefully inform future laboratory searches for novel anyonic phenomena in FQH systems and hence for topological quantum hardware.

---

partly surveyed in these lecture notes:

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

based on:

• Flux Quantization

• Flux Quantization on M5-Branes

• Abelian Anyons on Flux-Quantized M5-Branes

• Anyons on M5-Probes of Seifert 3-Orbifolds

  (which in turn is based on: The Character Map in Equivariant Twistorial Cohomotopy)

Related presentations:

• Homotopy Theory for Topological Quantum Computing

• Non-Lagrangian construction of abelian CS/FQH theory

• Rethinking Topological Quantum Logic