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Topological degeneracy and vortex dynamics in the Kitaev honeycomb model

Graham Kells, Ahmet Tuna Bolukbasi, Ville Lahtinen, Johannes K. Slingerland, Jiannis K. Pachos, Jiří Vala

Published Apr 17, 2008• 1 citation
OpenAlex

Abstract

We investigate the loop symmetries of Kitaev's honeycomb lattice model. These provide a natural framework to study the abelian topological phase. We show that in the thermodynamic limit, the abelian phase on a torus is topologically degenerate to all orders of the Brillioun-Wigner expansion. We then demonstrate that these symmetries correspond to the evolution of fermions in closed loops. Importantly, we demonstrate that these fermions are made from energetically confined pairs of vortices and that their transport happens with no additional energy cost. This has important implications for the gapped and gapless phases.

Related papers

Topological Degeneracy and Vortex Manipulation in Kitaev’s Honeycomb Model

Graham Kells, Ahmet Tuna Bolukbasi, Ville Lahtinen +3 more

The classification of loop symmetries in Kitaev's honeycomb lattice model provides a natural framework to study the Abelian topological degeneracy. We derive a perturbative low-energy effective Hamiltonian that is valid to all orders of the expansion and for all possible toroidal configurations. Using this form we demonstrate at what order the system's topological degeneracy is lifted by finite size effects and note that in the thermodynamic limit it is robust to all orders. Further, we demonstrate that the loop symmetries themselves correspond to the creation, propagation, and annihilation of fermions. We note that these fermions, made from pairs of vortices, can be moved with no additional energy cost.

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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.

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1 Interacting non-Abelian anyons as Majorana fermions in the honeycomb lattice model

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We analyze low energy spectral properties of small toroidal configurations of the Kitaev honeycomb spin model in the Abelian topological phase. We begin with a brief classification of honeycomb lattices on a torus. Then, using the Brillouin–Wigner perturbation theory, we explain the low order finite size effects that can occur in these systems and show how they affect their ground state topological degeneracy. Finally, we demonstrate the accuracy of the perturbative method by means of exact diagonalization, and use the insights into the finite size effects to reconstruct the topological degeneracy in a small example system.

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Lattice Defects in the Kitaev Honeycomb Model

John Brennan, Jiří Vala

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.

The Journal of Physical Chemistry A•Feb 17, 2016•13 citations
We investigate the loop symmetries of Kitaev's honeycomb lattice model. These provide a natural framework to study the abelian topological phase. We show that in the thermodynamic limit, the abelian phase on a torus is topologically degenerate to all orders of the Brillioun-Wigner expansion. We then demonstrate that these symmetries correspond to the evolution of fermions in closed loops. Importantly, we demonstrate that these fermions are made from energetically confined pairs of vortices and that their transport happens with no additional energy cost. This has important implications for the gapped and gapless phases.
The classification of loop symmetries in Kitaev's honeycomb lattice model provides a natural framework to study the Abelian topological degeneracy. We derive a perturbative low-energy effective Hamiltonian that is valid to all orders of the expansion and for all possible toroidal configurations. Using this form we demonstrate at what order the system's topological degeneracy is lifted by finite size effects and note that in the thermodynamic limit it is robust to all orders. Further, we demonstrate that the loop symmetries themselves correspond to the creation, propagation, and annihilation of fermions. We note that these fermions, made from pairs of vortices, can be moved with no additional energy cost.
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.
We study the collective states of interacting non-Abelian anyons that emerge in Kitaev's honeycomb lattice model. Vortex–vortex interactions are shown to lead to the lifting of topological degeneracy and the energy is found to exhibit oscillations that are consistent with Majorana fermions being localized at vortex cores. We show how to construct states corresponding to the fusion channel degrees of freedom and obtain the energy gaps characterizing the stability of the topological low-energy spectrum. To study the collective behavior of many vortices, we introduce an effective lattice model of Majorana fermions. We find the necessary conditions for the model to approximate the spectrum of the honeycomb lattice model, and show that bi-partite interactions are responsible for the lifting of degeneracy also in many-vortex systems.
We study the collective states of interacting non-Abelian anyons that emerge in Kitaev's honeycomb lattice model. Vortex–vortex interactions are shown to lead to the lifting of topological degeneracy and the energy is found to exhibit oscillations that are consistent with Majorana fermions being localized at vortex cores. We show how to construct states corresponding to the fusion channel degrees of freedom and obtain the energy gaps characterizing the stability of the topological low-energy spectrum. To study the collective behavior of many vortices, we introduce an effective lattice model of Majorana fermions. We find the necessary conditions for the model to approximate the spectrum of the honeycomb lattice model, and show that bi-partite interactions are responsible for the lifting of degeneracy also in many-vortex systems.
We analyze low energy spectral properties of small toroidal configurations of the Kitaev honeycomb spin model in the Abelian topological phase. We begin with a brief classification of honeycomb lattices on a torus. Then, using the Brillouin–Wigner perturbation theory, we explain the low order finite size effects that can occur in these systems and show how they affect their ground state topological degeneracy. Finally, we demonstrate the accuracy of the perturbative method by means of exact diagonalization, and use the insights into the finite size effects to reconstruct the topological degeneracy in a small example system.
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.