Spectrum of the non-abelian phase in Kitaev’s honeycomb lattice model

Lahtinen, Ville; Kells, Graham; Carollo, Angelo; Stitt, Timothy; Vala, Jiří; Pachos, Jiannis K.

doi:10.1016/j.aop.2007.12.009

Cited by 61 publications

(104 citation statements)

References 35 publications

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“…For a more rigorous treatment, see the original work [10] and the subsequent developments [22], [33][34][35][36][37][38]. We show how to relate the manipulation of the vortices to the manipulation of the model's physical parameters.…”

Section: The Honeycomb Lattice Modelmentioning

confidence: 95%

“…The second term is an effective magnetic field of magnitude K , which explicitly breaks time-reversal invariance. The sum in this term runs over all next to nearestneighbor triplets, as described in [22].…”

Section: The Honeycomb Lattice Modelmentioning

confidence: 99%

“…However, these states can 3 become locally distinguishable due to tunneling processes that lead to the degeneracy being lifted [21]. This has been studied in Kitaev's honeycomb lattice model [22], in the Moore-Read state [23] and in p-wave superconductors [24][25][26]. It has also been discovered recently that in many-anyon systems the tunneling processes can not only lift the degeneracy but also even drive transitions between topological phases [27][28][29].…”

Section: Introductionmentioning

confidence: 99%

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

Lahtinen

2011

New J. Phys.

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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 the topological degeneracy and the energy is discovered 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 necessary conditions for it to approximate the spectrum of the honeycomb lattice model and show that bi-partite interactions are responsible for the degeneracy lifting also in many vortex systems.

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Section: The Honeycomb Lattice Modelmentioning

confidence: 95%

Section: The Honeycomb Lattice Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Lahtinen

2011

New J. Phys.

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“…In the ν = 0 phases the vortex properties can be obtained analytically [27,55,56], but in the other phases this has to be done numerically by simulating vortex transport [40]. This has been explicitly studied in the |ν| = 1 phase of the original honeycomb model, where both the topological degeneracy [37,52] and the braid statistics [40,41] associated with the Majorana binding vortices have been verified.…”

Section: Vortices In Kitaev Spin Modelsmentioning

confidence: 99%

Kitaev spin models from topological nanowire networks

2014

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We show that networks of superconducting topological nanowires can realize the physics of exactly solvable Kitaev spin models on trivalent lattices. This connection arises from the low-energy theory of both systems being described by a tight-binding model of Majorana modes. In Kitaev spin models the Majorana description provides a convenient representation to solve the model, whereas in an array of Josephson junctions of topological nanowires it arises from localized physical Majorana modes tunneling between the wire ends. We explicitly show that an array of junctions of three wires-a setup relevant to topological quantum computing with nanowires-can realize the Yao-Kivelson model, a variant of Kitaev spin models on a decorated honeycomb lattice. Employing properties of the latter, we show that the network can be constructed to give rise to two-dimensional collective topological states characterized by Chern numbers ν = 0, ±1, and ±2, and that defects in the array can be associated with vortex-like quasiparticle excitations. In addition we show that the collective states are stable in the presence of disorder and superconducting phase fluctuations. When the network is operated as a quantum information processor, the connection to Kitaev spin models implies that decoherence mechanisms can in general be understood in terms of proliferation of the vortex-like quasiparticles.

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“…Although Kitaev model has a very special spin coupling, its very attractive properties caused a bunch of recent studies [8,9,10,11,12,13,14,15,16,17,18,19,20]. It is convenient to understand Kitaev model if one can map this model to a familiar model.…”

Section: Introductionmentioning

confidence: 99%

Gauge symmetry in Kitaev-type spin models and index theorems on odd manifolds

2008

Nuclear Physics B

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We construct an exactly soluble spin-1 2 model on a honeycomb lattice, which is a generalization of Kitaev model. The topological phases of the system are analyzed by study of the ground state sector of this model, the vortex-free states. Basically, there are two phases, A phase and B phase. The behaviors of both A and B phases may be studied by mapping the ground state sector into a general p-wave paired states of spinless fermions with tunable pairing parameters on a square lattice. In this p-wave paired state theory, the A phase is shown to be the strong paired phase, an insulating phase. The B phase may be either gapped or gapless determined by the generalized inversion symmetry is broken or not. The gapped B is the weak pairing phase described by either the Moore-Read Pfaffian state of the spinless fermions or anti-Pfaffian state of holes depending on the sign of the next nearest neighbor hopping amplitude. A phase transition between Pfaffian and anti-Pfaffian states are found in the gapped B phase. Furthermore, we show that there is a hidden SU(2) gauge symmetry in our model. In the gapped B phase, the ground state has a non-trivial topological number, the spectral first Chern number or the chiral central charge, which reflects the chiral anomaly of the edge state. We proved that the topological number is identified to the reduced eta-invariant and this anomaly may be cancelled by a bulk Wess-Zumino term of SO(3) group through an index theorem in 2+1 dimensions.

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Spectrum of the non-abelian phase in Kitaev’s honeycomb lattice model

Cited by 61 publications

References 35 publications

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

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

Kitaev spin models from topological nanowire networks

Gauge symmetry in Kitaev-type spin models and index theorems on odd manifolds

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