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Non-Abelian Anyons in the Kitaev Honeycomb Model

Ahmet Tuna Bolukbasi

Published Oct 1, 2010• 1 citation
OpenAlex

Abstract

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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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.
We develop a rigorous and highly accurate technique for calculation of the Berry phase in systems with a quadratic Hamiltonian within the context of the Kitaev honeycomb lattice model. The method is based on the recently found solution of the model which uses the Jordan-Wigner-type fermionization in an exact effective spin-hardcore boson representation. We specifically simulate the braiding of two non-Abelian vortices (anyons) in a four vortex system characterized by a two-fold degenerate ground state. The result of the braiding is the non-Abelian Berry matrix which is in excellent agreement with the predictions of the effective field theory. The most precise results of our simulation are characterized by an error on the order of 10−510^{-5}10−5 or lower. We observe exponential decay of the error with the distance between vortices, studied in the range from one to nine plaquettes. We also study its correlation with the involved energy gaps and provide preliminary analysis of the relevant adiabaticity conditions. The work allows to investigate the Berry phase in other lattice models including the Yao-Kivelson model and particularly the square-octagon model. It also opens the possibility of studying the Berry phase under non-adiabatic and other effects which may constitute important sources of errors in topological quantum computation.
We develop a rigorous and highly accurate technique for the calculation of the Berry phase in systems with a quadratic Hamiltonian within the context of the Kitaev honeycomb lattice model. The method is based on the recently found solution of the model that uses the Jordan–Wigner-type fermionization in an exact effective spin-hardcore boson representation. We specifically simulate the braiding of two non-Abelian vortices (anyons) in a four-vortex system characterized by a twofold degenerate ground state. The result of the braiding is the non-Abelian Berry matrix, which is in excellent agreement with the predictions of the effective field theory. The most precise results of our simulation are characterized by an error of the order of 10−5 or lower.We observe exponential decay of the error with the distance between vortices, studied in the range of one to nine plaquettes. We also study its correlation with the involved energy gaps and provide a preliminary analysis of the relevant adiabaticity conditions. The work allows one to investigate the Berry phase in other lattice models including the Yao–Kivelson model and particularly the square–octagon model. It also opens up the possibility of studying the Berry phase under nonadiabatic and other effects that may constitute important sources of errors in topological quantum computation.
We develop a rigorous and highly accurate technique for the calculation of the Berry phase in systems with a quadratic Hamiltonian within the context of the Kitaev honeycomb lattice model. The method is based on the recently found solution of the model that uses the Jordan–Wigner-type fermionization in an exact effective spin-hardcore boson representation. We specifically simulate the braiding of two non-Abelian vortices (anyons) in a four-vortex system characterized by a twofold degenerate ground state. The result of the braiding is the non-Abelian Berry matrix, which is in excellent agreement with the predictions of the effective field theory. The most precise results of our simulation are characterized by an error of the order of 10 −5 or lower. We observe exponential decay of the error with the distance between vortices, studied in the range of one to nine plaquettes. We also study its correlation with the involved energy gaps and provide a preliminary analysis of the relevant adiabaticity conditions. The work allows one to investigate the Berry phase in other lattice models including the Yao–Kivelson model and particularly the square–octagon model. It also opens up the possibility of studying the Berry phase under non-adiabatic and other effects that may constitute important sources of errors in topological quantum computation.
In this thesis, we study the non-Abelian anyons that emerge as vortices in Ki-taev's honeycomb spin lattice model. By generalizing the solution of the model, we explicity demonstrate the non-Abelian fusion rules and the braid statistics that charaterize the anyons. This is based on showing the presence of vortices leads to zero modes in the spectrum. These can acquire finite energy due to short range vortex-vortex interactions. By studying the spectral evolution as a function of the vortex seperation, we unambigously identify the zero modes with the fusion degrees of freedom of non-Abelian anyons. To calculate the non-Abelian statistics, we show how the vortex transport can be implemented through local manipulation of the couplings. This enables us to employ the eigenstates of the model to simulate a process where a vortex winds around another. The corresponding evolution of the degenerate ground state space is given by a Berry phase, which under suitable conditions coincides with the statistics. By considering a range of finite size systems, we find a physical regime where the Berry phase gives the predicted statistics of the anoyonic vortices with high fidelity. Finally we study the full-vortex sector of the model and find that is supports a previously undiscovered topological phase. This new phase emerges from the phase with non-Abelian anyons due to their interactions. To study the transitions between the different topological phases appearing in the model, we consider Fermi surface, whose topology captures the long-range properties. Each phase is found to be characterized by a distinct number of Fermi points, with the number depending on distinct global Hamiltonian symmetries. To study how the Fermi surfaces evolve into each other at phase transitions, we consider the low-energy field theory that is described by Dirac fermions. We show that phase transition driving perturbations translate to a coupling to chiral gauge fields, that always lead to Fermi point transport. By studying this transport, we obtain analytically the extended phase space of the model and its properties.
The Kitaev honeycomb lattice system is an important model of topological materials whose phase diagram exhibits both abelian and non-abelian topological phases. The latter, a so-called Ising phase, is related to topological superconductors. Its quasiparticle excitations, which are formed by Majorana fermions attached to vortices, show non-abelian fractional statistics and are known as Ising anyons. We investigate dislocation defects in the Ising phase of the Kitaev honeycomb model. After introducing them to the system, we accordingly generalize our solution of this model to the situation with the defects. The important part of this effort is developing an appropriate Jordan-Wigner fermionization procedure. It is expected that the presence of defects manifests itself by the formation of fermionic zero-energy modes around the defect end points. We numerically confirm this expectation and further investigate properties of these modes. The computational potential of our technique is demonstrated for both diagonalization and dynamical simulations. The latter focuses on the process of fusion of the vortex zero-energy modes with the Majorana fermions attached to the defect. This process simulates fusion of non-abelian Ising anyons.
We present a solution of Kitaev's spin model on the honeycomb lattice and of related topologically ordered spin models. We employ a Jordan-Wigner-type fermionization and find that the Hamiltonian takes a BCS-type form, allowing the system to be solved by Bogoliubov transformation. Our fermionization does not employ nonphysical auxiliary degrees of freedom and the eigenstates we obtain are completely explicit in terms of the spin variables. The ground state is obtained as a BCS condensate of fermion pairs over a vacuum state which corresponds to the toric-code state with the same vorticity. We show in detail how to calculate all eigenstates and eigenvalues of the model on the torus. In particular, we find that the topological degeneracy on the torus descends directly from that of the toric-code, which now supplies four vacua for the fermions, one for each choice of periodic vs antiperiodic boundary conditions. The reduction of the degeneracy in the non-Abelian phase of the model is seen to be due to the vanishing of one of the corresponding candidate BCS ground states in that phase. This occurs in particular in the fully periodic vortex-free sector. The true ground state in this sector is exhibited and shown to be gapped away from the three partially antiperiodic ground states whenever the non-Abelian phase is gapped.