  Collective guidance of out-of-equilibrium systems without using external
fields is a challenge of paramount importance in active matter, ranging from
bacterial colonies to swarms of self-propelled particles. Designing strategies
to guide active matter and exploiting enhanced diffusion associated to its
motion will provide insights for application from sensing, drug delivery to
water remediation. However, achieving directed motion without breaking detailed
balance, for example by asymmetric topographical patterning, is challenging.
Here we engineer a two-dimensional periodic topographical design with detailed
balance in its unit cell where we observe spontaneous particle edge guidance
and corner accumulation of self-propelled particles. This emergent behaviour is
guaranteed by a second-order non-Hermitian skin effect, a topologically robust
non-equilibrium phenomenon, that we use to dynamically break detailed balance.
Our stochastic circuit model predicts, without fitting parameters, how guidance
and accumulation can be controlled and enhanced by design: a device guides
particles more efficiently if the topological invariant characterizing it is
non-zero. Our work establishes a fruitful bridge between active and topological
matter, and our design principles offer a blueprint to design devices that
display spontaneous, robust and predictable guided motion and accumulation,
guaranteed by out-of-equilibrium topology.
