Generalized non-reciprocal phase transitions in multipopulation systems

Non-reciprocal interactions are prevalent in various complex systems leading to phenomena that cannot be described by traditional equilibrium statistical physics. Although non-reciprocally interacting systems composed of two populations have been closely studied, the physics of non-reciprocal systems with a general number of populations is not well explored despite the relevance to biological systems, active matter, and driven-dissipative quantum materials. In this work, we investigate the generic features of the phases and phase transitions and emerge in $O(2)$ symmetric many-body systems with multiple non-reciprocally coupled populations, applicable to microscopic models such as networks of oscillators, flocking models, and more generally systems where each agent has a phase variable. Using symmetry and topology of the possible orbits, we systematically show that a rich variety of time-dependent phases and phase transitions arise. Examples include multipopulation chiral phases that are distinct from their two-population counterparts that emerge via a phase transition characterized by critical exceptional points, as well as limit cycle saddle-node bifurcation and Hopf bifurcation. Interestingly, we find a phase transition that dynamically restores the $\mathbb{Z}_2$ symmetry occurs via a homoclinic orbit bifurcation, where the two $\mathbb{Z}_2$ broken orbits merge at the phase transition point, providing a general route to homoclinic chaos in the order parameter dynamics for $N\geq4$ populations. Our framework provides general principles for understanding non-equilibrium heterogeneous systems and guides experimental exploration into such systems.