Symmetry breaking in active non-reciprocal systems

In many biological and synthetic active matter systems, self-propelled constituents interact through couplings that violate action-reaction symmetry -- a property known as non-reciprocity. Both activity and non-reciprocity independently drive such systems far from thermodynamic equilibrium. In this thesis, we investigate how these two features interplay and how non-reciprocal orientational couplings give rise to distinct forms of symmetry breaking in the collective behavior of active binary mixtures. To this end, we consider a paradigmatic model of a polar active binary mixture with steric repulsion between all particles, intraspecies alignment, and, crucially, non-reciprocal orientational couplings between different species. Starting from a microscopic description in terms of Langevin equations, we derive a mean-field continuum model and perform analytical linear stability analyses. In the reciprocal limit, the model exhibits collective behaviors such as flocking (parallel motion), antiflocking (antiparallel single-species flocks), and clustering via motility-induced phase separation. In the non-reciprocal regime, by combining continuum field theory with particle-based simulations, we identify two qualitatively distinct dynamical regimes. In the weakly coupled regime, i.e., below the parity-time (PT) symmetry-breaking threshold, non-reciprocal opposing alignment goals of different species lead to asymmetric clustering akin to partial demixing. Clusters of predominantly one species emerge. Thus, non-reciprocal orientational couplings can induce asymmetric scalar density dynamics. Both continuum and particle-level descriptions capture the asymmetric clustering phenomenon, but dynamical features such as chase-and-run behavior only become apparent in particle simulations. In the strongly coupled regime, above the PT-symmetry-breaking threshold, transitions to PT-broken, time-dependent states are marked by so-called exceptional points (EPs) in the field-theoretical description. At the particle level, non-reciprocity induces spontaneous rotations -- yet without full, homogeneous synchronization of all particles. Instead, we observe a variety of behaviors, ranging from synchronized demixed clusters to chimera-like states. The spontaneous chirality of particles increases with non-reciprocity and peaks at coupling strengths corresponding to field-theoretical EPs. Having identified these distinct dynamical regimes, a natural next question concerns how far the system is driven from equilibrium and how this relates to the onset of symmetry breaking. To quantify time-reversal symmetry breaking and the system's distance from equilibrium across the various collective states, we compute the informatic entropy production rate. At the particle level, it increases with non-reciprocity and exhibits pronounced peaks at coupling strengths associated with EPs. These peaks mirror the susceptibility of the polarization order parameter. A complementary field-theoretical analysis confirms this correspondence: in the long-wavelength limit, the entropy production rate scales with the polarization susceptibilities. Together, our analytical and numerical results reveal the diverse effects of non-reciprocity across scales. We show that orientational couplings alone can strongly influence scalar density dynamics, and that EPs are not merely field-theoretical constructs but leave distinct, observable signatures at the particle level. This scale-bridging analysis lays a foundation for exploring whether biological or artificial systems might harness the strongly non-equilibrium dynamics near EPs to achieve functional behaviors.