Abstract:
Topological quantum information processing relies on adiabatic braiding of nonabelian quasiparticles. Performing the braiding operations in finite time introduces transitions out of the ground-state manifold and deviations from the nonabelian Berry phase. We show that these errors can be eliminated by suitably designed counterdiabatic correction terms in the Hamiltonian. We implement the resulting shortcuts to adiabaticity for simple protocols of nonabelian braiding and show that the error suppression can be substantial even for approximate realizations of the counterdiabatic terms.

Abstract:
We study an anyon model in a toric honeycomb lattice. The ground states and the low-lying excitations coincide with those of Kitaev toric code model and then the excitations obey mutual semionic statistics. This model is helpful to understand the toric code of anyons in a more symmetric way. On the other hand, there is a direct relation between this toric honeycomb model and a boundary coupled Ising chain array in a square lattice via Jordan-Wigner transformation. We discuss the equivalence between these two models in the low-lying sector and realize these anyon excitations in a conventional fermion system.

Abstract:
We apply the SU(2) slave fermion formalism to the Kitaev honeycomb lattice model. We show that both the Toric Code phase (the A phase) and the gapless phase of this model (the B phase) can be identified with p-wave superconducting phases of the slave fermions, with nodal lines which, respectively, do not or do intersect the Fermi surface. The non-Abelian Ising anyon phase is a $p+ip$ superconducting phase which occurs when the B phase is subjected to a gap-opening magnetic field. We also discuss the transitions between these phases in this language.

Abstract:
We study exactly both the ground-state fidelity susceptibility and bond-bond correlation function in the Kitaev honeycomb model. Our results show that the fidelity susceptibility can be used to identify the topological phase transition from a gapped A phase with Abelian anyon excitations to a gapless B phase with non-Abelian anyon excitations. We also find that the bond-bond correlation function decays exponentially in the gapped phase, but algebraically in the gapless phase. For the former case, the correlation length is found to be $1/\xi=2\sinh^{-1}[\sqrt{2J_z -1}/(1-J_z)]$, which diverges around the critical point $J_z=(1/2)^+$.

Abstract:
We study the reduced fidelity and reduced fidelity susceptibility in the Kitaev honeycomb model. It is shown that the reduced fidelity susceptibility of two nearest site manifest itself a peak at the quantum phase transition point, although the one-site reduced fidelity susceptibility vanishes. Our results directly reveal that the reduced fidelity susceptibility can be used to characterize the quantum phase transition in the Kitaev honeycomb model, and thus suggest that the reduced fidelity susceptibility is an accurate marker of the topological phase transition when it is properly chosen, despite of its local nature.

Abstract:
We develop a rigorous and highly accurate technique for calculation of the Berry phase in systems with a quadratic Hamiltonian within the context of the Kitaev honeycomb lattice model. The method is based on the recently found solution of the model which uses the Jordan-Wigner-type fermionization in an exact effective spin-hardcore boson representation. We specifically simulate the braiding of two non-Abelian vortices (anyons) in a four vortex system characterized by a two-fold degenerate ground state. The result of the braiding is the non-Abelian Berry matrix which is in excellent agreement with the predictions of the effective field theory. The most precise results of our simulation are characterized by an error on the order of $10^{-5}$ or lower. We observe exponential decay of the error with the distance between vortices, studied in the range from one to nine plaquettes. We also study its correlation with the involved energy gaps and provide preliminary analysis of the relevant adiabaticity conditions. The work allows to investigate the Berry phase in other lattice models including the Yao-Kivelson model and particularly the square-octagon model. It also opens the possibility of studying the Berry phase under non-adiabatic and other effects which may constitute important sources of errors in topological quantum computation.

Abstract:
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 ground-state 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 multi-plaquette 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.

Abstract:
We demonstrate that the anyon statistics and three-loop statistics of various 2d and 3d topological phases can be derived using semiclassical nonlinear Sigma model field theories with a topological $\Theta$-term. In our formalism, the braiding statistics has a natural geometric meaning: The braiding process of anyons or loops leads to a nontrivial field configuration in the space-time, which will contribute a braiding phase factor due to the $\Theta$-term.

Abstract:
We study the effect of coupling magnetic impurities to the honeycomb lattice spin-1/2 Kitaev model in its spin liquid phase. We show that a spin-S impurity coupled to the Kitaev model is associated with an unusual Kondo effect with an intermediate coupling unstable fixed point K_c J/S separating topologically distinct sectors of the Kitaev model. We also show that the massless spinons in the spin liquid mediate an interaction of the form S_{i\alpha}^{2}S_{j\beta}^{2}/R_{ij}^{3} between distant impurities unlike the usual dipolar RKKY interaction S_{i\alpha}S_{j\alpha}/R_{ij}^{3} noted in various 2D impurity problems with a pseudogapped density of states of the spin bath. Furthermore, this long-range interaction is possible only if the impurities (a) couple to more than one neighboring spin on the host lattice and (b) the impurity spin is not a spin-1/2.$

Abstract:
Wave functions describing quasiholes and electrons in nonabelian quantum Hall states are well known to correspond to conformal blocks of certain coset conformal field theories. In this paper we explicitly analyse the algebraic structure underlying the braiding properties of these conformal blocks. We treat the electrons and the quasihole excitations as localised particles carrying charges related to a quantum group that is determined explicitly for the cases of interest. The quantum group description naturally allows one to analyse the braid group representations carried by the multi-particle wave functions. As an application, we construct the nonabelian braid group representations which govern the exchange of quasiholes in the fractional quantum Hall effect states that have been proposed by N. Read and E. Rezayi, recovering the results of C. Nayak and F. Wilczek for the Pfaffian state as a special case.