Many-Body Invariants for Topological Insulators: Multipole, Chern, and Hinge States

We propose many-body invariants for the broad classes of topological insulators including Chern insulators, chiral hinge insulators, and multipole insulators. Unlike band indices which only work for non-interacting band insulators, our invariants can detect non-trivial topology of quantum many-body wave functions hence applicable to fully interacting quantum systems. To this end, we design several unitaries whose expectation values on many-body ground states serve as the invariants. We show that the unitaries detect the coefficients of the topological field theory, which are the defining characteristics of topological insulators. This allows us to develop a new way of evaluating Chern numbers, and also the many-body invariant for chiral hinge insulator. Furthermore, we will also show that boundary observables such as the edge-localized polarizations and the corner charge can be measured purely by the many-body unitaries when endowed with appropriate background geometry.