Interacting non-Abelian anyons as Majorana fermions in the honeycomb lattice model

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 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.

Perturbed vortex lattices and the stability of nucleated topological phases

“…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.…”

“…These interactions mean that the topological degeneracy will only be exact in the limit of infinite quasiparticle separation [5]. Microscopics of this degeneracy lifting have been analyzed in several systems including Moore-Read fractional quantum Hall states [6], p-wave superconductors [7], Kitaev's honeycomb model [8], and topological nanowires [9]. As any realization of topological order will ultimately be in a finite system, the anyons are forced to be in proximity to each other and thus the interactions are rarely negligible.…”

“…By studying the system specific pairwise tunneling amplitudes [6][7][8], the collective state of the system can be predicted from the generalized Majorana tight-binding model presented here. In addition to the honeycomb model [19] and its generalizations [22,23], Majorana arrays may naturally occur as Abrikosov vortex lattices in p-wave superconductors [24] or as quasihole Wigner crystals in Moore-Read fractional quantum Hall states [25].…”

Kitaev spin models from topological nanowire networks

“…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.…”

“…For instance, when ν is odd, vortices themselves bind Majorana modes. One could thus employ these collective states to study the characteristic vortex interactions [37] that can lead to a nucleation transition when a vortex crystal forms [22,38]. Other directions could be the emergence of a disorder induced thermal metal state unique to Majorana modes [36,39], the non-Abelian statistics of the vortices [40,41], or impurity effects [33,34,42].…”