  Recently, non-reciprocal systems have become a focus of growing interest.
Examples occur in soft and active matter, but also in engineered quantum
materials and neural (brain) networks. Here, we investigate the impact of
non-reciprocity on the collective behavior of a system of (dry) chiral active
matter. Specifically, we consider a mixture of "circle swimmers" with steric
interactions and non-reciprocal alignment couplings. Based on hydrodynamic
equations which we derive from a set of Langevin equations, we explore the
interplay of non-reciprocity, finite size, and chirality. We first consider, as
a reference, one-species systems with reciprocal couplings. Based on a linear
stability analysis and numerical simulations, we here observe three different
types of collective behavior, that is, flocking, motility-induced phase
separation, and a combination of both. Turning then to a non-reciprocal system,
we find that non-reciprocity can turn otherwise stationary instabilities into
oscillatory ones, affect the relative orientation of flocks, and, crucially,
change the general type of instability. This illustrates the drastic impact of
non-reciprocity and chirality on the emergent collective dynamics of active
matter systems, with potentially far-reaching biological implications.
