  Topological phases of quantum matter defy characterization by conventional
order parameters but can exhibit quantized electro-magnetic response and/or
protected surface states. We examine such phenomena in a model for
three-dimensional correlated complex oxides, the pyrochlore iridates. The model
realizes interacting topological insulators with and without time-reversal
symmetry, and topological Weyl semimetals. We use cellular dynamical mean field
theory, a method that incorporates quantum-many-body effects and allows us to
evaluate the magneto-electric topological response coefficient in correlated
systems. This invariant is used to unravel the presence of an interacting axion
insulator absent within a simple mean field study. We corroborate our bulk
results by studying the evolution of the topological boundary states in the
presence of interactions. Consequences for experiments and for the search for
correlated materials with symmetry-protected topological order are given.
