Rigorous calculations of non-Abelian statistics in the Kitaev honeycomb model

Bolukbasi, Ahmet Tuna; Vala, Jiří

doi:10.1088/1367-2630/14/4/045007

Cited by 14 publications

(17 citation statements)

References 33 publications

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“…This process can be viewed as simulating adiabatic vortex transport, which has been employed to verify the non-Abelian statistics of the vortices [32,33] and to uncover the oscillating interactions between them [8]. Here we will employ this equivalence between coupling and gauge configurations to simulate perturbations in vortex lattices.…”

Section: A Simulating Vortex Transport By Tuning the Spin Exchange Cmentioning

confidence: 99%

Perturbed vortex lattices and the stability of nucleated topological phases

2014

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We study the stability of nucleated topological phases that can emerge when interacting non-Abelian anyons form a regular array. The studies are carried out in the context of Kitaev's honeycomb model, where we consider three distinct types of perturbations in the presence of a lattice of Majorana mode binding vortices -spatial anisotropy of the vortices, dimerization of the vortex lattice and local random disorder. While all the nucleated phases are stable with respect to weak perturbations of each kind, strong perturbations are found to result in very different behavior. Anisotropy of the vortices stabilizes the strong-pairing like phases, while dimerization can recover the underlying non-Abelian phase. Local random disorder, on the other hand, can drive all the nucleated phases into a gapless thermal metal state. We show that all these distinct behavior can be captured by an effective staggered tight-binding model for the Majorana modes. By studying the pairwise interactions between the vortices, i.e. the amplitudes for the Majorana modes to tunnel between vortex cores, the locations of phase transitions and the nature of the resulting states can be predicted. We also find that due to oscillations in the Majorana tunneling amplitude, lattices of Majorana modes may exhibit a Peierls-like instability, where a dimerized configuration is favored over a uniform lattice. As the nature of the nucleated phases depends only on the Majorana tunneling, our results are expected to apply also to other system supporting localized Majorana mode arrays, such as Abrikosov lattices in p-wave superconductors, Wigner crystals in Moore-Read fractional quantum Hall states or arrays of topological nanowires.

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Section: A Simulating Vortex Transport By Tuning the Spin Exchange Cmentioning

confidence: 99%

Perturbed vortex lattices and the stability of nucleated topological phases

2014

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“…We outlined also the conditions where wire arrays could be made to undergo more exotic transitions, such as a disorder-induced transition to a metallic state [39] or a nucleation transition due to the presence of a vortex crystal [22]. Furthermore, if local control over the array parameters can be executed with sufficient accuracy, one could even entertain the possibility of using them to test non-Abelian braiding statistics [40,41].…”

Section: Discussionmentioning

confidence: 99%

“…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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“…While the Berry phase and non-Abelian information of quasiparticles moving adiabatically around each other have been studied numerically [35][36][37][38][39][40][41], the full modular S matrix and the corresponding fusion rules for the microscopic models hosting the non-Abelian topological states have been lacking, due to the computational difficulty of directly dealing with and distinguishing different quasiparticles. Recently, there is growing interest on characterizing topological order through the quantum entanglement information [33,34,[42][43][44][45][46][47][48][49][50].…”

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confidence: 99%

Identifying Non-Abelian Topological Order through Minimal Entangled States

2014

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The topological order is equivalent to the pattern of long-range quantum entanglements, which cannot be measured by any local observable. Here we perform an exact diagonalization study to establish the non-Abelian topological order through entanglement entropy measurement. We focus on the quasiparticle statistics of the non-Abelian Moore-Read and Read-Rezayi states on the lattice boson models. We identify multiple independent minimal entangled states (MESs) in the groundstate manifold on a torus. The extracted modular S matrix from MESs faithfully demonstrates the Majorana quasiparticle or Fibonacci quasiparticle statistics, including the quasiparticle quantum dimensions and the fusion rules for such systems. These findings support that MESs manifest the eigenstates of quasiparticles for the non-Abelian topological states and encode the full information of the topological order. Introduction.-One of the most striking phenomena in the fractional quantum Hall (FQH) system is the emergent fractionalized quasiparticles obeying Abelian [1] or non-Abelian [2-4] braiding statistics. Interchange of two Abelian quasiparticles leads to a nontrivial phase acquired by their wavefunction, whereas interchange of two non-Abelian quasiparticles results in an operation of matrix to the degenerating groundstate space and the final state will depend on the order of operations being carried out. The non-Abelian quasiparticles and its braiding statistics are fundamentally important for understanding the topological order and also have potential application in topologically fault-tolerant quantum computation [5][6][7]. So far, such quasiparticles have not been definitely identified in nature. However, it is generally believed that they exist in the FQH systems at filling factor ν = 5/2 [8] and 12/5 [9], described by the Moore-Read (MR) [2,10,11] and Read-Rezayi (RR) states [4,12], respectively. The topological band model for optical lattices with bosonic particles is another promising platform to realize the non-Abelian topological states [13][14][15][16][17][18][19][20][21][22][23]. In the MR and RR states, the quasiparticles satisfy the following characteristic fusion rules that specify how the quasiparticles combine and fuse into more than one type of quasiparticles [7]:

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Rigorous calculations of non-Abelian statistics in the Kitaev honeycomb model

Cited by 14 publications

References 33 publications

Perturbed vortex lattices and the stability of nucleated topological phases

Perturbed vortex lattices and the stability of nucleated topological phases

Kitaev spin models from topological nanowire networks

Identifying Non-Abelian Topological Order through Minimal Entangled States

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