Abstract
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 topological degeneracy and the energy is found 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 the necessary conditions for the model to approximate the spectrum of the honeycomb lattice model, and show that bi-partite interactions are responsible for the lifting of degeneracy also in many-vortex systems.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. Non-Abelian anyons are quasiparticles, i.e. collective states of elementary particles behaving like a single particle, that can exist in two-dimensional systems. Their exotic statistical properties make them attractive to error-resilient quantum computing. To employ them for this purpose, it is crucial to understand: (1) under what conditions they emerge, and (2), how their properties are reflected in the experimentally accessible microscopic degrees of freedom.
Main results. Kitaev's honeycomb lattice model is an experimentally realistic spin lattice model that supports vortices that can behave as non-Abelian anyons. We show that the vortices exhibit characteristic short-range oscillating interactions and study how to control them should the vortices be employed for topological quantum computation. The oscillating interactions also imply that systems with many vortices can exhibit complicated collective behavior. We show that when its parameters are fixed in a manner consistent with the underlying vortex pattern, this behavior can be modelled using an effective lattice model of Majorana fermions.
Wider implications. When the honeycomb lattice model is realized in the laboratory, most likely as polar molecules trapped in an optical lattice, our study translates directly into a scheme to manipulate the anyonic vortices and to detect them through their microscopic signatures. The effective Majorana model can also be used to model the collective behavior of non-Abelian anyons in other topologically ordered systems.