  Active systems exhibit spontaneous flows induced by self-propulsion of
microscopic constituents and can reach a nonequilibrium steady state without an
external drive. Constructing the analogy between the quantum anomalous Hall
insulators and active matter with spontaneous flows, we show that topologically
protected sound modes can arise in a steady-state active system in continuum
space. We point out that the net vorticity of the steady-state flow, which acts
as a counterpart of the gauge field in condensed-matter settings, must vanish
under realistic conditions for active systems. The quantum anomalous Hall
effect thus provides design principles for realizing topological metamaterials.
We propose and analyze the concrete minimal model and numerically calculate its
band structure and eigenvectors, demonstrating the emergence of nonzero bulk
topological invariants with the corresponding edge sound modes. This new type
of topological active systems can potentially expand possibilities for their
experimental realizations and may have broad applications to practical active
metamaterials. Possible realization of non-Hermitian topological phenomena in
active systems is also discussed.
