  We present the concept of a feedback-based topological acoustic metamaterial
as a tool for realizing autonomous and active guiding of sound beams along
arbitrary curved paths in free two-dimensional space. The metamaterial building
blocks are acoustic transducers, embedded in a slab waveguide. The transducers
generate a desired dispersion profile in closed-loop by processing real-time
pressure field measurements through preprogrammed controllers. In particular,
the metamaterial can be programmed to exhibit analogies of quantum topological
wave phenomena, which enables unconventional and exceptionally robust sound
beam guiding. As an example, we realize the quantum valley Hall effect by
creating, using a collocated pressure feedback, an alternating acoustic
impedance pattern across the waveguide. The pattern is traversed by artificial
trajectories of different shapes, which are reconfigurable in real-time. Due to
topological protection, the sound waves between the plates remain localized on
the trajectories, and do not back-scatter by the sharp corners or imperfections
in the design. The feedback-based design can be used to realize arbitrary
physical interactions in the metamaterial, including non-local, nonlinear,
time-dependent, or non-reciprocal couplings, paving the way to new
unconventional acoustic wave guiding on the same reprogrammable platform. We
then present a non-collocated control algorithm, which mimics another quantum
effect, rendering the sound beams uni-directional.
