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String flux mechanism for fractionalization in topologically ordered phases

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 $[ω], [ω]^{-1} \in H^2(G, Z_n)$ give rise to the same phase. We conclude with a discussion of open issues and future directions.