The Intermingling of Symmetry and Parametrization in Matrix Product States

In the study of quantum spin systems, it is by now well-known that interesting phases of quantum matter can arise from gapped ground states when the system is invariant under a symmetry group G or when the system varies continuously with a parameter in a topological space X. In these cases, phases are characterized by indices taking values in group cohomology of G or the cohomology of X, respectively. The situation where one has both a symmetry and a parametrization is much less studied but can lead to interesting phases even when both the aforementioned indices are trivial. In this talk, I will discuss work in progress on a simple construction for general index for symmetry protected parametrized systems of matrix product states and will show some illustrative examples.