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A hallmark feature of topologically ordered states of matter is thedependence of ground state degeneracy (GSD) on the topology of the manifolddetermined by the global shape of the system. Although the topology of aphysical system is practically hard to manipulate, recently it was shown thatin certain topologically ordered phases, topological defects can introduceextra topological GSD. Here the topological defects can be viewed aseffectively changing the topology of the physical system. Previous studies havebeen focusing on two spatial dimensions with point-like topological defects. Inthree dimensions, line-like topological defects can appear. They are closedloops in the bulk that can be linked and knotted, effectively leading tocomplex three dimensional manifolds in certain topologically ordered states.This paper studies the properties of such line-defects in a particular context:the lattice dislocations. We give an analytical construction, together withsupport from exact numerical calculations, for the dependence of the GSD ondislocations of certain doubled versions of the exactly solvable Kitaev's toriccode models in both two and three dimensions. We find that the GSD of the 3dmodel depends only on the total number of dislocation loops, no matter how theyare linked or knotted. The results are extended to Z_n generalizations of themodel. Additionally, we consider the phases in which the crystalline orders aredestroyed through proliferation of double dislocations. The resulting phasesare shown to host topological orders described by non-Abelian gauge theories.

Changing topology by topological defects in three-dimensional topologically ordered phases