Bulk and edge invariants for parametrized quantum systems with symmetry

Parametrized quantum systems are a generalization of conventional topological phases of matter, where one considers families of gapped many-body systems which depend continuously on some external parameters. Familiar nontrivial examples include the spin-1/2 particle in a magnetic field and the Thouless charge pump. We construct new examples of parametrized quantum systems via the suspension isomorphism, which can be thought of as "higher" analogues of Thouless pumps. We develop bulk and boundary topological invariants which characterize such paramterized quantum systems, and establish a concrete bulk-boundary correspondence in certain cases.