Bolukbasi, Ahmet Tuna
Non-Abelian Anyons in the Kitaev Honeycomb Model.
PhD thesis, National University of Ireland Maynooth.
The non-Abelian Berry phase is an essential feature of non-Abelian anyons for
the realization of topological quantum computation. This thesis is primarily a study
about the numerical calculation of the Berry phase of non-Abelian anyons in the
Kitaev honeycomb lattice model. It is also a guide for experimental the realizations
of the actual brading process.
We give an introduction to the theory of non-Abelian anyons, briefly discussing in
what kind of systems they are realized, and their possible use in topological quantum
computation. Non-Abelian anyons are studied within the Kitaev honeycomb model
where they are realized on the plaquettes of the honeycomb lattice. The Kitaev honeycomb
model can be solved exactly by using various fermionization methods. In
this thesis, we review a solution based on Jordan-Wigner types of fermions which
transform Hamiltonian to a fermionic quadratic form. This kind of fermionization
procedure is quite general and can be applied to any trivalent spin lattice models.
Moreover, we introduce Hartree-Fock-Bogoliubov method to solve general quadratic
fermionic Hamiltonian and employ Bloch-Messiah theorem in order to write ground
state wave function explicitly. Later, we apply these methods to honeycomb model
and study the eigenstates of the model so that we can do the Berry phase calculation.
The final chapter explains the details of the numerical calculation of the non-Abelian
Berry phase. First, we show how to create and adiabatically move vortices in the honeycomb
model. A brief review of the Berry phase is given including some discussion
about a numerical approach. Later on Thouless’ representation of the ground state is
introduced to calculate the Berry phase. All these theoretical tools are applied to a
4 vortex configuration of the model to calculate the non-Abelian Berry phase of the
system on a particular path in the parameter space.
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