Nonreciprocal phases of matter

Nonreciprocal interactions in which the influence of A on B differs from that of B on A are abundant in physical, chemical, biological, and ecological systems, and are known to give rise to oscillatory states. Yet, it remains unclear whether these states represent true phases of matter: Can they maintain long-range order in spatially extended, noisy environments in the thermodynamic limit? And what kinds of phase transitions do they exhibit? To address these questions, we introduce a minimal generalization of the Ising model with two species having opposing goals. We demonstrate that oscillatory phases are stable in three dimensions but not in two, and that nonreciprocity changes the critical exponents from those of the Ising model to those of the XY model. We further extend this framework to a nonreciprocal XY model and develop a Harris-like criterion that determines when nonreciprocity fundamentally alters universal behavior. Finally, we apply these insights to a recent model of biomolecular condensates, predicting exotic dynamical phases and suggesting experimental tests.