  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.

Strongly Correlated Electrons