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

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.

List of contents

Introduction and summary.- Three models and a ground state.- From a Laughlin state to the Haldane-Shastry model.- From a bosonic Pfaffian state to an S = 1 spin chain.- Generalization to arbitrary spin S.- Conclusions and unresolved issues.- Spherical coordinates.- Fourier sums for one-dimensional lattices.- Angular momentum algebra.- Tensor decompositions of spin operators.

About the author

PHD Martin Greiter
Karlsruhe Institute of Technology, KIT, Germany
[e-mail]

Summary

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.

Product details

• Authors: Martin Greiter
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 01.12.2011
• EAN: 9783642243837
• ISBN: 978-3-642-24383-7
• No. of pages: 194
• Weight: 454 g
• Illustrations: XIV, 194 p.
• Series: Springer Tracts in Modern Physics; Springer Tracts in Modern Physics; Springer Tracts in Modern Physics. Ergebnisse der exakten Naturwissenschaften
• Subjects: Natural sciences, medicine, IT, technology > Physics, astronomy > Theoretical physics; B, Mathematische Physik, Kybernetik und Systemtheorie, Physics, Complex systems, Solid State Physics, Theoretical, Mathematical and Computational Physics, Quantum Physics, Physics and Astronomy, Electronic Devices, Spectrum analysis, spectrochemistry, mass spectrometry, Mathematical physics, Dynamical systems, Mathematical Methods in Physics, Dynamics & statics, Statistical physics, Statistical Physics and Dynamical Systems