Geometric phases and quantum entanglement as building blocks for non-Abelian quasiparticle statistics

Some models describing unconventional fractional quantum Hall states predict quasiparticles that obey non-Abelian quantum statistics. The most prominent example is the Moore-Read model for the $\nu =5∕2$ state, in which the ground state is a superconductor of composite fermions, the charged excitations are vortices in that superconductor, and the non-Abelian statistics is closely linked to the degeneracy of the ground state in the presence of vortices. In this paper we develop a physical picture of the non-Abelian statistics of these vortices. Considering first the positions of the vortices as fixed, we define a set of single-particle states at and near the core of each vortex, and employ general properties of the corresponding Bogolubov–de Gennes equations to write the ground states in the Fock space defined by these single-particle states. We find all ground states to be entangled superpositions of all possible occupations of the single-particle states near the vortex cores, in which the probability for all occupations is equal, and the relative phases vary from one ground state to another. Then, we examine the evolution of the ground states as the positions of the vortices are braided. We find that as vortices move, they accumulate a geometric phase that depends on the occupations of the single-particle states near the cores of other vortices. Thus, braiding of vortices changes the relative phase between different components of a superposition, in which the occupations of these states differ, and hence transform the system from one ground state to another. These transformations, that emanate from the quantum entanglement of the occupations of single-particle states and from the dependence of the geometric phase on these occupations, are the source of the non-Abelian statistics. Finally, by exploring a “self-similar” form of the many-body wave functions of the various ground states, we show the equivalence of our picture, in which vortex braiding leads to a change in the relative phase of components in a superposition, and pictures derived previously, in which vortex braiding seemingly affects the occupations of states in the cores of the vortices.