Higher structures in matrix product states

For a parameterized family of invertible states (short-range-entangled states) in $(1+1)$ dimensions, we discuss a generalization of the Berry phase. Using translationally-invariant, infinite matrix product states (MPSs), we introduce a gerbe structure, a higher generalization of complex line bundles, as an underlying mathematical structure describing topological properties of a parameterized family of matrix product states. We also introduce a "triple inner product" for three matrix product states, which allows us to extract a topological invariant, the Dixmier-Douady class over the parameter space.