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

Abstract

A hallmark feature of topologically ordered states of matter is the dependence of ground state degeneracy (GSD) on the topology of the manifold determined by the global shape of the system. Although the topology of a physical system is practically hard to manipulate, recently it was shown that in certain topologically ordered phases, topological defects can introduce extra topological GSD. Here the topological defects can be viewed as effectively changing the topology of the physical system. Previous studies have been focusing on two spatial dimensions with point-like topological defects. In three dimensions, line-like topological defects can appear. They are closed loops in the bulk that can be linked and knotted, effectively leading to complex 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 with support from exact numerical calculations, for the dependence of the GSD on dislocations of certain doubled versions of the exactly solvable Kitaev's toric code models in both two and three dimensions. We find that the GSD of the 3d model depends only on the total number of dislocation loops, no matter how they are linked or knotted. The results are extended to Z_n generalizations of the model. Additionally, we consider the phases in which the crystalline orders are destroyed through proliferation of double dislocations. The resulting phases are shown to host topological orders described by non-Abelian gauge theories.