  We construct a family of exactly solvable spin models that illustrate a novel
mechanism for fractionalization in topologically ordered phases, dubbed the
string flux mechanism. The essential idea is that an anyon of a topological
phase can be endowed with fractional quantum numbers when the string attached
to it slides over a background pattern of flux in the ground state. The string
flux models that illustrate this mechanism are $Z_n$ quantum double models
defined on specially constructed $d$-dimensional lattices, and possess $Z_n$
topological order for $d \geq 2$. The models have a unitary, internal symmetry
$G$, where $G$ is an arbitrary finite group. The simplest string flux model is
a $Z_2$ toric code defined on a bilayer square lattice, where $G = Z_2$ is
layer-exchange symmetry. In general, by varying the pattern of $Z_n$ flux in
the ground state, any desired fractionalization class [element of $H^2(G,
Z_n)$] can be realized for the $Z_n$ charge excitations. While the string flux
models are not gauge theories, they map to $Z_n$ gauge theories in a certain
limit, where they follow a novel magnetic route for the emergence of low-energy
gauge structure. The models are analyzed by studying the action of $G$ symmetry
on $Z_n$ charge excitations, and by gauging the $G$ symmetry. The latter
analysis confirms that distinct fractionalization classes give rise to distinct
quantum phases, except that classes $[\omega], [\omega]^{-1} \in H^2(G, Z_n)$
give rise to the same phase. We conclude with a discussion of open issues and
future directions.
