Universal SPT invariants using swap operators*

We will describe many-body topological invariants/order parameters for SPT phases that involve acting with symmetry operators and a swap operator. To do so, we use the universal notion of distinguishing between different phases of matter: phases are equivalence classes defined in the thermodynamic limit by (i) adiabatic continuity, and (ii) adding trivial degrees of freedom. In the process, we will also formulate gauge-invariant versions of these order parameters. Since our arguments for the universality of such many-body invariants do not depend on the spatial dimension we are working in, this discussion can help serve as a basis for generalizations to higher spatial dimensions. We will also show how to picture the invariant by putting the system on a spacetime manifold and by thinking in terms of TQFT partition functions.