Topological liquid nucleation induced by vortex-vortex interactions in Kitaev's honeycomb model

Abstract

We provide a comprehensive microscopic understanding of the nucleation of topological quantum liquids, a general mechanism where interactions between non-Abelian anyons cause a transition to another topological phase, which we study in the context of Kitaev's honeycomb lattice model. For non-Abelian vortex excitations arranged on superlattices, we observe the nucleation of several distinct Abelian topological phases whose character is found to depend on microscopic parameters such as the superlattice spacing or the spin exchange couplings. By reformulating the interacting vortex superlattice in terms of an effective model of Majorana fermion zero modes, we show that the nature of the collective many-anyon state can be fully traced back to the microscopic pairwise vortex interactions. Due to RKKY-type sign oscillations in the interactions, we find that longer-range interactions beyond nearest neighbor can influence the collective state and thus need to be included for a comprehensive picture. The omnipresence of such interactions implies that corresponding results should hold for vortices forming an Abrikosov lattice in a p-wave superconductor, quasiholes forming a Wigner crystal in non-Abelian quantum Hall states or topological nanowires arranged in regular arrays.

Kitaev spin models from topological nanowire networks

“…Due to the richness of their phase diagrams [27][28][29][30][31][32], the immediate interest would be on topological phase transitions. We outlined also the conditions where wire arrays could be made to undergo more exotic transitions, such as a disorder-induced transition to a metallic state [39] or a nucleation transition due to the presence of a vortex crystal [22]. Furthermore, if local control over the array parameters can be executed with sufficient accuracy, one could even entertain the possibility of using them to test non-Abelian braiding statistics [40,41].…”

“…For sparse arrays this leads to exponential degeneracy lifting that gives a source of decoherence. For dense arrays, however, something more dramatic could happen: the Majorana modes could hybridize and form another collective topological state, very much like what can happen in Majorana mode binding vortex crystals [22]. This would require going through a phase transition, resulting in the significant degradation of the encoded information.…”

“…This has been studied in the context of the original honeycomb model [22], where different phases with Chern numbers ν = ±2, ± 4 can be realized depending on the spacing of the vortex superlattice. To verify that this same mechanism works also in the Y-K model, we plot in Fig.…”

“…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].…”

Conformal energy currents on the edge of a topological superconductor

“…As a concrete * cns08@ic.ac.uk example, we study Kitaev's honeycomb model [9]: a twodimensional spin liquid that supports topological superconducting phases with a variety of Chern numbers [10][11][12]. Due to the analytical tractability of this model, it is amenable to a wide variety of numerical studies such as finite-temperature analysis [13][14][15].…”