  Coherently driven atomic gases inside optical cavities hold great promise for
generating rich dynamics and exotic states of matter. It was shown recently
that an exotic topological superradiant state exists in a two-component
degenerate Fermi gas coupled to a cavity, where local order parameters coexist
with global topological invariants. In this work, we characterize in detail
various properties of this exotic state, focusing on the feedback interactions
between the atoms and the cavity field. In particular, we demonstrate that
cavity-induced interband coupling plays a crucial role in inducing the
topological phase transition between the conventional and topological
superradiant states. We analyze the interesting signatures in the cavity field
left by the closing and reopening of the atomic bulk gap across the topological
phase boundary and discuss the robustness of the topological superradiant state
by investigating the steady-state phase diagram under various conditions.
Furthermore, we consider the interaction effect and discuss the interplay
between the pairing order in atomic ensembles and the superradiance of the
cavity mode. Our work provides many valuable insights into the unique
cavity--atom hybrid system under study and is helpful for future experimental
exploration of the topological superradiant state.
