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

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.

207 pages, 12 figures; ISBN 978-3-642-24383-7; Springer Tract of Modern Physics, Vol 244, 2011

References in corpus (16)

• Quantum spin-liquid emerging in two-dimensional correlated Dirac fermions
• Particle-hole symmetry and the Pfaffian state
• Particle-Hole Symmetry and the $\nu =5/2$ Quantum Hall State
• Anyons and the quantum Hall effect - a pedagogical review
• An exact chiral spin liquid with non-Abelian anyons
• Fractional Quantum Hall States and Jack Polynomials
• The entanglement gap and a new principle of adiabatic continuity
• Experimental studies of the fractional quantum Hall effect in the first excited Landau level
• Spin Hamiltonian for which the Chiral Spin Liquid is the Exact Ground State
• Non-local order in gapless systems: Entanglement Spectrum in Spin Chains
• The non-Abelian Interferometer
• Non-Abelian Statistics in a Quantum Antiferromagnet
• Valence bond solids for SU(n) spin chains: exact models, spinon confinement, and the Haldane gap
• Pseudopotentials for Multi-particle Interactions in the Quantum Hall Regime
• Paired composite fermion wavefunctions
• Parent Hamiltonian for the Chiral Spin Liquid

Cited by in corpus (15)

• Momentum-Space Entanglement Spectrum of Bosons and Fermions with Interactions
• Quantum spin models for the SU(n)_1 Wess-Zumino-Witten model
• Repulsive Interactions in Quantum Hall Systems as a Pairing Problem
• A family of spin-S chain representations of SU(2)_k Wess-Zumino-Witten models
• Infinite Matrix Product States for long range SU(N) spin models
• Universal entanglement spectra in critical spin chains
• A hierarchy of exactly solvable spin-1/2 chains with so(N)_1 critical points
• Parent Hamiltonian Reconstruction of Jastrow-Gutzwiller Wavefunctions
• Fibonacci anyon excitations of one-dimensional dipolar lattice bosons
• Pfaffian structures and certain solutions to BKP hierarchies II. Multiple integrals
• Resonating valence bond realization of spin-1 non-Abelian chiral spin liquid on the torus
• Pfaffian-like ground states for bosonic atoms and molecules in one-dimensional optical lattices
• Trapped-ion quantum simulation of tunable-range Heisenberg chains
• Conformal field theory and the non-abelian $SU(2{)}_k$ chiral spin liquid
• Spinon bases in supersymmetric CFTs