Mapping of Parent Hamiltonians: from Abelian and non-Abelian Quantum Hall States to Exact Models of Critical Spin Chains

TL;DR

This paper presents an exact critical spin chain model with arbitrary spin S, extending the Haldane--Shastry model, and explores the transition from abelian to non-abelian spinon statistics through Hall state mappings.

Contribution

It introduces a new exactly solvable spin chain model with non-abelian statistics derived from quantum Hall state mappings, generalizing the Haldane--Shastry model.

Findings

The model includes the Haldane--Shastry as a special case.

Spinons obey non-abelian statistics in the general model.

Topological fractional momentum spacings characterize the non-abelian behavior.

Abstract

This monograph introduces an exact model for a critical spin chain with arbitrary spin S, which includes the Haldane--Shastry model as the special case S=1/2. While spinons in the Haldane--Shastry model obey abelian half-fermi statistics, the spinons in the general model introduced here obey non-abelian statistics. This manifests itself through topological choices for the fractional momentum spacings. The general model is derived by mapping exact models of quantized Hall states onto spin chains. The book begins with pedagogical review of all the relevant models including the non-abelian statistics in the Pfaffian Hall state, and is understandable to every student with a graduate course in quantum mechanics.