Topological Degeneracy and Vortex Manipulation in Kitaev’s Honeycomb Model

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

doi:10.1103/physrevlett.101.240404

Cited by 42 publications

(64 citation statements)

References 29 publications

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“…In the next section, we introduce the model as well as its main properties. In particular, we discuss the importance of the boundary conditions and insist on the role played by conserved quantities 15 and the constraints resulting from them. In Sec.…”

Section: For Details)mentioning

confidence: 99%

Perturbative approach to an exactly solved problem: Kitaev honeycomb model

Vidal

Schmidt

Dusuel³

2008

Phys. Rev. B

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We analyze the gapped phase of the Kitaev honeycomb model perturbatively in the isolated-dimer limit. Our analysis is based on the continuous unitary transformations method which allows one to compute the spectrum as well as matrix elements of operators between eigenstates, at high order. The starting point of our study consists in an exact mapping of the original honeycomb spin system onto a square-lattice model involving an effective spin and a hardcore boson. We then derive the low-energy effective Hamiltonian up to order 10 which is found to describe an interacting-anyon system, contrary to the order 4 result which predicts a free theory. These results give the groundstate energy in any vortex sector and thus also the vortex gap, which is relevant for experiments. Furthermore, we show that the elementary excitations are emerging free fermions composed of a hardcore boson with an attached spin-and phase-operator string. We also focus on observables and compute, in particular, the spin-spin correlation functions. We show that they admit a multiplaquette expansion that we derive up to order 6. Finally, we study the creation and manipulation of anyons with local operators, show that they also create fermions, and discuss the relevance of our findings for experiments in optical lattices.

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Section: For Details)mentioning

confidence: 99%

Perturbative approach to an exactly solved problem: Kitaev honeycomb model

Vidal

Schmidt

Dusuel³

2008

Phys. Rev. B

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“…We first study the quantum Ising chain in a transverse magnetic field, perhaps one of the most celebrated systems in condensed matter for offering a tractable solution and rich physics, one with plentiful studies even in the context of quenching [7,8,[28][29][30][31]. The second system, the Kitaev honeycomb model, is also special in its analytically soluble structure [32][33][34][35][36][37][38][39] and has also received significant attention in the context of quenching [9]. The transverse Ising model maps to a p-wave superconducting chain [40], while the honeycomb lattice model maps to a p + ip superconductor coupled to a Z 2 gauge field [32,33,35,36], the latter thus being a natural two-dimensional extension of the former.…”

Section: Introductionmentioning

confidence: 99%

Topological blocking in quantum quench dynamics

et al. 2014

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We study the nonequilibrium dynamics of quenching through a quantum critical point in topological systems, focusing on one of their defining features: ground-state degeneracies and associated topological sectors. We present the notion of "topological blocking," experienced by the dynamics due to a mismatch in degeneracies between two phases, and we argue that the dynamic evolution of the quench depends strongly on the topological sector being probed. We demonstrate this interplay between quench and topology in models stemming from two extensively studied systems, the transverse Ising chain and the Kitaev honeycomb model. Through nonlocal maps of each of these systems, we effectively study spinless fermionic p-wave paired topological superconductors. Confining the systems to ring and toroidal geometries, respectively, enables us to cleanly address degeneracies, subtle issues of fermion occupation and parity, and mismatches between topological sectors. We show that various features of the quench, which are related to Kibble-Zurek physics, are sensitive to the topological sector being probed, in particular, the overlap between the time-evolved initial ground state and an appropriate low-energy state of the final Hamiltonian. While most of our study is confined to translationally invariant systems, where momentum is a convenient quantum number, we briefly consider the effect of disorder and illustrate how this can influence the quench in a qualitatively different way depending on the topological sector considered.

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“…Their properties, depending on the Chern number ν, are encoded in the low-energy part of the energy spectrum of the corresponding vortex sector. 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%

“…One example however is in understanding the robustness of the system to virtual processes which at an intermediate stage involve the excitation of fermions or vortices. This strength is exploited in the weak J limit and allows the low-energy sector of the full spin model to be perturbatively mapped to a toric code Hamiltonian [4,[55][56][57]. Note that only in the 0-fermion sector can the vortex eigenvalues of the full Kitaev spin model be exactly identified with the eigenvalues of the toric code excitations; see for example [57].…”

Section: Appendix C: Fermions Vortices and Spinsmentioning

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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Topological Degeneracy and Vortex Manipulation in Kitaev’s Honeycomb Model

Cited by 42 publications

References 29 publications

Perturbative approach to an exactly solved problem: Kitaev honeycomb model

Perturbative approach to an exactly solved problem: Kitaev honeycomb model

Topological blocking in quantum quench dynamics

Kitaev spin models from topological nanowire networks

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