Nonlocal order parameters for states with topological electromagnetic response

Chern insulators are states of matter characterized by their quantized Hall conductance but also by their singular response to monopole configurations of an external electromagnetic field. In this paper, we exploit this response to provide a classification for these states. We demonstrate that for each Chern-insulator state, including the trivial state, the response defines an associated operator that can be interpreted as the insertion of local charges and monopoles at two points separated in space. This operator decays algebraically in the monopole separation distance only for its associated state but exponentially in all other states. Crucially, the operators do not depend on any microscopic properties, and therefore constitute a general set of nonlocal order parameters. We support this claim with numerical evaluation of the order parameters in a simple lattice model, and find excellent agreement. Our construction is well suited for generalization to other states with topological electromagnetic response, and we use the states with a quantized magnetoelectric effect in three dimensions as an example. Aside from providing insights into topological states of matter, our construction can also be exploited to efficiently diagnose such states numerically.