730 research outputs found
Interacting non-Abelian anyons as Majorana fermions in the honeycomb lattice model
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.Comment: 22 pages, 12 figures, published versio
Non-Abelian statistics as a Berry phase in exactly solvable models
We demonstrate how to directly study non-Abelian statistics for a wide class
of exactly solvable many-body quantum systems. By employing exact eigenstates
to simulate the adiabatic transport of a model's quasiparticles, the resulting
Berry phase provides a direct demonstration of their non-Abelian statistics. We
apply this technique to Kitaev's honeycomb lattice model and explicitly
demonstrate the existence of non-Abelian Ising anyons confirming the previous
conjectures. Finally, we present the manipulations needed to transport and
detect the statistics of these quasiparticles in the laboratory. Various
physically realistic system sizes are considered and exact predictions for such
experiments are provided.Comment: 10 pages, 3 figures. To appear in New Journal of Physic
Topological Degeneracy and Vortex Manipulation in Kitaev's Honeycomb Model
The classification of loop symmetries in Kitaev's honeycomb lattice model provides a natural framework to study the Abelian topological degeneracy. We derive a perturbative low-energy effective Hamiltonian that is valid to all orders of the expansion and for all possible toroidal configurations. Using this form we demonstrate at what order the system's topological degeneracy is lifted by finite size effects and note that in the thermodynamic limit it is robust to all orders. Further, we demonstrate that the loop symmetries themselves correspond to the creation, propagation, and annihilation of fermions. We note that these fermions, made from pairs of vortices, can be moved with no additional energy cost
Rigorous Calculations of Non-Abelian Statistics in the Kitaev Honeycomb Model
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 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.Comment: 27 pages, 9 figures, 3 appendice
A Short Introduction to Topological Quantum Computation
This review presents an entry-level introduction to topological quantum computation -- quantum computing with anyons. We introduce anyons at the system-independent level of anyon models and discuss the key concepts of protected fusion spaces and statistical quantum evolutions for encoding and processing quantum information. Both the encoding and the processing are inherently resilient against errors due to their topological nature, thus promising to overcome one of the main obstacles for the realisation of quantum computers. We outline the general steps of topological quantum computation, as well as discuss various challenges faced it. We also review the literature on condensed matter systems where anyons can emerge. Finally, the appearance of anyons and employing them for quantum computation is demonstrated in the context of a simple microscopic model -- the topological superconducting nanowire -- that describes the low-energy physics of several experimentally relevant settings. This model supports localised Majorana zero modes that are the simplest and the experimentally most tractable types of anyons that are needed to perform topological quantum computation
Diagnosing Topological Edge States via Entanglement Monogamy
Topological phases of matter possess intricate correlation patterns typically probed by entanglement entropies or entanglement spectra. In this Letter, we propose an alternative approach to assessing topologically induced edge states in free and interacting fermionic systems. We do so by focussing on the fermionic covariance matrix. This matrix is often tractable either analytically or numerically, and it precisely captures the relevant correlations of the system. By invoking the concept of monogamy of entanglement, we show that highly entangled states supported across a system bipartition are largely disentangled from the rest of the system, thus, usually appearing as gapless edge states. We then define an entanglement qualifier that identifies the presence of topological edge states based purely on correlations present in the ground states. We demonstrate the versatility of this qualifier by applying it to various free and interacting fermionic topological systems
Scaling Properties of Random Walks on Small-World Networks
Using both numerical simulations and scaling arguments, we study the behavior
of a random walker on a one-dimensional small-world network. For the properties
we study, we find that the random walk obeys a characteristic scaling form.
These properties include the average number of distinct sites visited by the
random walker, the mean-square displacement of the walker, and the distribution
of first-return times. The scaling form has three characteristic time regimes.
At short times, the walker does not see the small-world shortcuts and
effectively probes an ordinary Euclidean network in -dimensions. At
intermediate times, the properties of the walker shows scaling behavior
characteristic of an infinite small-world network. Finally, at long times, the
finite size of the network becomes important, and many of the properties of the
walker saturate. We propose general analytical forms for the scaling properties
in all three regimes, and show that these analytical forms are consistent with
our numerical simulations.Comment: 7 pages, 8 figures, two-column format. Submitted to PR
Exact Chiral Spin Liquids and Mean-Field Perturbations of Gamma Matrix Models on the Ruby Lattice
We theoretically study an exactly solvable Gamma matrix generalization of the
Kitaev spin model on the ruby lattice, which is a honeycomb lattice with
"expanded" vertices and links. We find this model displays an exceptionally
rich phase diagram that includes: (i) gapless phases with stable spin fermi
surfaces, (ii) gapless phases with low-energy Dirac cones and quadratic band
touching points, and (iii) gapped phases with finite Chern numbers possessing
the values {\pm}4,{\pm}3,{\pm}2 and {\pm}1. The model is then generalized to
include Ising-like interactions that break the exact solvability of the model
in a controlled manner. When these terms are dominant, they lead to a trivial
Ising ordered phase which is shown to be adiabatically connected to a large
coupling limit of the exactly solvable phase. In the limit when these
interactions are weak, we treat them within mean-field theory and present the
resulting phase diagrams. We discuss the nature of the transitions between
various phases. Our results highlight the richness of possible ground states in
closely related magnetic systems.Comment: 9 pages, 9 figure
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