We study a class of 2D spin models with the Kitaev-type couplings in mosaic structure lattices to implement topological orders. We show that they are exactly solvable by reducing them to some free Majorana fermion models with gauge symmetries. The typical case with a 4-8-8 close packing is investigated in detail to display the quantum phases with abelian and non-abelian anyons. Its topological properties characterized by Chern numbers are revealed through the edge modes of its spectrum.

Aug 6, 20070 citations

Mosaic spin models with topological order

Shiliu Yang, D. L. Zhou, C. P. Sun

arXiv:0708.0676Physical Review BNov 9, 200785 citations

arXiv:1012.5276 · New Journal of Physics

Kaleidoscope of topological phases with multiple Majorana species

G Kells, J Kailasvuori, Johannes K. Slingerland, J Vala

Published Sep 22, 2011• 35 citations

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Abstract

DOI: 10.1088/1367-2630/13/9/095014

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Kaleidoscope of topological phases with multiple Majorana species

Kells, G., Kailasvuori, J., Johannes K. Slingerland +1 more

Exactly solvable lattice models for spins and non-interacting fermions provide fascinating examples of topological phases, some of them exhibiting the localized Majorana fermions that feature in proposals for topological quantum computing. The Chern invariant nu is an important characterization of such phases. Here we look at the square-octagon variant of Kitaev's honeycomb model. It maps to spinful paired fermions and enjoys a rich phase diagram featuring distinct Abelian and non-Abelian phases with nu = 0, +/- 1, +/- 2, +/- 3 and +/- 4. The nu = +/- 1 and nu = +/- 3 phases all support localized Majorana modes and are examples of Ising and SU(2)(2) anyon theories, respectively.Exactly solvable lattice models for spins and non-interacting fermions provide fascinating examples of topological phases, some of them exhibiting the localized Majorana fermions that feature in proposals for topological quantum computing. The Chern invariant nu is an important characterization of such phases. Here we look at the square-octagon variant of Kitaev's honeycomb model. It maps to spinful paired fermions and enjoys a rich phase diagram featuring distinct Abelian and non-Abelian phases with nu = 0, +/- 1, +/- 2, +/- 3 and +/- 4. The nu = +/- 1 and nu = +/- 3 phases all support localized Majorana modes and are examples of Ising and SU(2)(2) anyon theories, respectively.

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Topological Majorana flat bands in the Kitaev model on a Bishamon-kikko lattice

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We unveil an interesting example of topological flat bands of Majorana fermions in quantum spin liquids. We study the Kitaev model on a periodically depleted honeycomb lattice, under a magnetic field within the perturbation theory. The model can be straightforwardly extended while maintaining the exact solvability, and its ground state is a quantum spin liquid as on the honeycomb lattice. As fractionalized excitations, there are unpaired localized Majorana fermions in addition to the itinerant Majorana fermions and ${\mathbb{Z}}_{2}$ fluxes. We show that, in the absence of the magnetic field, the Majorana fermions exhibit completely flat bands at zero energy, and by applying the magnetic field, these bands turn into topological flat bands with nonzero Chern number. By varying the anisotropy of the interactions and the magnitude of the magnetic field, we clarify that the system hosts a variety of topological phases that do not appear in the original honeycomb model. We emphasize that the topological flat bands underlying this rich topology come from the hybridization of the Majorana flat bands and unpaired Majorana fermions, which is unique to the flat bands of fractionalized excitations in quantum spin liquids. Our findings would stimulate the exploration of a new class of Kitaev materials exhibiting rich topology arising from topological Majorana flat bands.We unveil an interesting example of topological flat bands of Majorana fermions in quantum spin liquids. We study the Kitaev model on a periodically depleted honeycomb lattice, under a magnetic field within the perturbation theory. The model can be straightforwardly extended while maintaining the exact solvability, and its ground state is a quantum spin liquid as on the honeycomb lattice. As fractionalized excitations, there are unpaired localized Majorana fermions in addition to the itinerant Majorana fermions and ${\mathbb{Z}}_{2}$ fluxes. We show that, in the absence of the magnetic field, the Majorana fermions exhibit completely flat bands at zero energy, and by applying the magnetic field, these bands turn into topological flat bands with nonzero Chern number. By varying the anisotropy of the interactions and the magnitude of the magnetic field, we clarify that the system hosts a variety of topological phases that do not appear in the original honeycomb model. We emphasize that the topological flat bands underlying this rich topology come from the hybridization of the Majorana flat bands and unpaired Majorana fermions, which is unique to the flat bands of fractionalized excitations in quantum spin liquids. Our findings would stimulate the exploration of a new class of Kitaev materials exhibiting rich topology arising from topological Majorana flat bands.

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We study the simulation of the Kitaev spin model on quantum computers. In particular we focus on the models defined on the honeycomb, and square-octagon lattices. Using a fermionic language to describe these models reveals a region of the parameter space that is exactly solvable. We explore an ansatz that is capable of expressing the ground state in the exactly solvable region of the parameter space and extend it outside this region with good accuracy. Doing the calculation using fermions, while requiring the introduction of a non-local map from the fermionic Hilbert space to that of qubits, offers the potentially interesting application of realizing non-abelian anyons on quantum computers, and can also lead to a reduction in the number of qubits required by half.We study the simulation of the Kitaev spin model on quantum computers. In particular we focus on the models defined on the honeycomb, and square-octagon lattices. Using a fermionic language to describe these models reveals a region of the parameter space that is exactly solvable. We explore an ansatz that is capable of expressing the ground state in the exactly solvable region of the parameter space and extend it outside this region with good accuracy. Doing the calculation using fermions, while requiring the introduction of a non-local map from the fermionic Hilbert space to that of qubits, offers the potentially interesting application of realizing non-abelian anyons on quantum computers, and can also lead to a reduction in the number of qubits required by half.

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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 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.

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Shuai Yang, D. L. Zhou, C. P. Sun

Aug 6, 20070 citations

Mosaic spin models with topological order

Shiliu Yang, D. L. Zhou, C. P. Sun

We study a class of two-dimensional spin models with the Kitaev-type couplings in mosaic structure lattices to implement topological orders. We show that they are exactly solvable by reducing them to some free Majorana fermion models with gauge symmetries. The typical case with a 4-8-8 close packing is investigated in detail to display the quantum phases with Abelian and non-Abelian anyons. Its topological properties characterized by Chern numbers are revealed through the edge modes of its spectrum.We study a class of two-dimensional spin models with the Kitaev-type couplings in mosaic structure lattices to implement topological orders. We show that they are exactly solvable by reducing them to some free Majorana fermion models with gauge symmetries. The typical case with a 4-8-8 close packing is investigated in detail to display the quantum phases with Abelian and non-Abelian anyons. Its topological properties characterized by Chern numbers are revealed through the edge modes of its spectrum.

arXiv:0708.0676Physical Review BNov 9, 200785 citations

{\mathbb{Z}}_{2}

fluxes. We show that, in the absence of the magnetic field, the Majorana fermions exhibit completely flat bands at zero energy, and by applying the magnetic field, these bands turn into topological flat bands with nonzero Chern number. By varying the anisotropy of the interactions and the magnitude of the magnetic field, we clarify that the system hosts a variety of topological phases that do not appear in the original honeycomb model. We emphasize that the topological flat bands underlying this rich topology come from the hybridization of the Majorana flat bands and unpaired Majorana fermions, which is unique to the flat bands of fractionalized excitations in quantum spin liquids. Our findings would stimulate the exploration of a new class of Kitaev materials exhibiting rich topology arising from topological Majorana flat bands.