String flux mechanism for fractionalization in topologically ordered phases

We construct a family of exactly solvable spin models that illustrate a 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 Zₙ quantum double models defined on specially constructed d-dimensional lattices, and possess Zₙ topological order for d≥2. The models have a unitary, internal symmetry G, where G is an arbitrary finite group. The simplest string flux model is a Z₂ toric code defined on a bilayer square lattice, where G=Z₂ is layer-exchange symmetry. In general, by varying the pattern of Zₙ flux in the ground state, any desired fractionalization class [element of H²(G,Zₙ)] can be realized for the Zₙ charge excitations. While the string flux models are not gauge theories, they map to Zₙ gauge theories in a certain limit, where they follow a magnetic route for the emergence of low-energy gauge structure. The models are analyzed by studying the action of G symmetry on Zₙ 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∈H²(G,Zₙ) give rise to the same phase. We conclude with a discussion of open issues and future directions.