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Non-Abelian Statistics in one dimension: topological momentum spacings and SU(2) level k fusion rules
No full textWe use a family of critical spin chain models discovered recently by one of us [M. Greiter, Map-
ping of Parent Hamiltonians, Springer, Berlin/Heidelberg 2011] to propose and elaborate that non-
Abelian, SU(2) level k = 2S anyon statistics manifests itself in one dimension through topological
selection rules for fractional shifts in the spacings of linear momenta, which yield an internal Hilbert
space of, in the thermodynamic limit degenerate states. These shifts constitute the equivalent to the
fractional shifts in the relative angular momenta of anyons in two dimensions. We derive the rules
first for Ising anyons, and then generalize them to SU(2) level k anyons. We establish a one-to-one
correspondence between the topological choices for the momentum spacings and the fusion rules of
spin 1 2 spinons in the SU(2) level k Wess–Zumino–Witten model, where the internal Hilbert space is
spanned by the manifold of allowed fusion trees in the Bratelli diagrams. Finally, we show that the
choices in the fusion trees may be interpreted as the choices between different domain walls between
the 2S + 1 possible, degenerate dimer configurations of the spin S chains at the multicritical poin
Mapping of parent hamiltonians: from abelian and non-abelian quantum hall states to exact models of critical spin chains
No full textThis 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
Microscopic formulation of the hierarchy of quantized Hall states
No full textExplicit wave functions for the hierarchy of fractionally quantized Hall states are proposed, and a method for integrating out the quasiparticle coordinates in the spherical geometry is developed. Their energies and overlaps with the exact ground states for small numbers of particles with Coulomb interactions are found to be excellent. We then generalize the adiabatic transport argument of Arovas, Schrieffer, and Wilczek to evaluate quasiparticle charges and statistics, and show that none of the proposed states is the exact ground state of any model Hamiltonian with two-body interactions only.Explicit wave functions for the hierarchy of fractionally quantized Hall states are proposed, and a method for integrating out the quasiparticle coordinates in the spherical geometry is developed. Their energies and overlaps with the exact ground states for small numbers of particles with Coulomb interactions are found to be excellent. We then generalize the adiabatic transport argument of Arovas, Schrieffer, and Wilczek to evaluate quasiparticle charges and statistics, and show that none of the proposed states is the exact ground state of any model Hamiltonian with two-body interactions only.Explicit wave functions for the hierarchy of fractionally quantized Hall states are proposed, and a method for integrating out the quasiparticle coordinates in the spherical geometry is developed. Their energies and overlaps with the exact ground states for small numbers of particles with Coulomb interactions are found to be excellent. We then generalize the adiabatic transport argument of Arovas, Schrieffer, and Wilczek to evaluate quasiparticle charges and statistics, and show that none of the proposed states is the exact ground state of any model Hamiltonian with two-body interactions only
Publisher's Note: Landau level quantization on the sphere [Phys. Rev. B <b>83</b> , 115129 (2011)]
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