Aspects of Non-Abelian Many Body Physics

Author: Bradford, Kent B.

Year: 1997

Degree: Dissertation (Ph.D.)

Advisor: Preskill, John P.

Committee Members: Preskill, John P.; Politzer, Hugh David; Wise, Mark B.; Weichman, Peter B.

Option: Physics

DOI: 10.7907/rn7x-3882

Abstract

The general formulation of quantum statistical mechanics hints at interesting generalizations of the usual Bose/Fermi framework in two spatial dimensions. Anyon statistics, which is essentially a continuous interpolation between Bose and Fermi statistics, is relevant to the Fractional Quantum Hall Effect in two-dimensional (i.e., thin layer) condensed matter systems. In addition, the possibility of non-abelian statistics, in which the multi-particle wavefunction transforms as a representation of a non-abelian group under the exchange of indistinguishable particles, has been explored. Spontaneously broken non-abelian gauge theories in (2 + 1) dimensions often have stable topological defects, called non-abelian vortices, that experience non-abelian statistics. In addition, it has been suggested that degenerate quasihole multiplets in Quantum Hall systems also transform as non-abelian representations of the braid group under particle exchange. In this thesis, I explore the braiding properties of systems of two-cycle flux vortices in a residual S₃ discrete gauge group. The individual vortices are uncharged, but multi-vortex states can have Cheshire charge. The uncharged sectors all have non-vanishing bosonic subspaces, as do the non-abelian charged trivial flux sectors. A kinetic Hamiltonian for vortices on a periodic lattice is constructed. There is a modification to the translational symmetry in the periodically identified direction for non-trivial Z₂ charged sectors. The ground state energies for various three and four vortex sectors is numerically determined. Typically, the ground state is bosonic, with a gap separating it from a non-abelian subspace.

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