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.

💡 Research Summary

This paper presents a systematic investigation into the rich landscape of dynamical phases and phase transitions that emerge in systems composed of three or more populations interacting via non-reciprocal couplings. Moving beyond the well-studied two-population case, the authors develop a general framework based on symmetry and topology to classify non-equilibrium phases in O(2)-symmetric models, which encompass systems like coupled oscillators, flocking models, and non-reciprocal XY models.

The core of the analysis relies on the Ott-Antonsen equations (or their generalizations) as a concrete mean-field model. In the ordered phase where each population is synchronized, the slow dynamics are governed by the evolution of phase variables. A key insight is leveraging the system’s O(2) ~ SO(2) ⋊ Z2 symmetry. The spontaneous breaking of the SO(2) symmetry leads to ordering, while non-reciprocity can further spontaneously break the remaining Z2 (phase-conjugation) symmetry, leading to chiral “chase” dynamics.

To classify possible time-dependent phases, particularly limit cycles, the authors analyze the topology of orbits on the N-dimensional torus (T^N), the phase space for the N phases. They introduce winding numbers to characterize how an orbit wraps around this torus. A crucial technical step involves reducing the dynamics to equations solely for the phase differences φ_ab = φ_a - φ_b, which are decoupled from the dynamics of the total phase Φ due to the SO(2) symmetry. This “reduced system” often exhibits clear periodic orbits even when the full dynamics appear quasiperiodic, allowing for a robust topological classification.

The main findings are:

1. Novel Chiral Phases: For three populations, distinct chiral phases are identified. Beyond the “2-chiral phase” where only two populations chase each other, a new “3-chiral phase” exists where all three populations sequentially chase one another (A→B→C→A…). These phases are topologically distinct, meaning a phase transition must occur to switch between them.
2. Diverse Transition Mechanisms: The transitions between these dynamical phases occur via various bifurcations, including those characterized by critical exceptional points (where eigenvectors coalesce at the transition due to non-Hermitian dynamics), saddle-node bifurcations of limit cycles, and Hopf bifurcations.
3. Dynamical Z2 Symmetry Restoration: A particularly novel discovery is a phase transition that dynamically restores the Z2 symmetry. In this phase, the Z2 symmetry is instantaneously broken, but applying the Z2 operation followed by time evolution for half the limit cycle period returns the system to the same state. This transition occurs via a symmetric homoclinic orbit bifurcation, where two Z2-broken orbits merge into a single, dynamically Z2-restored orbit.
4. Route to Higher-Order Dynamics and Chaos: The framework is extended to systems with four or more populations. The homoclinic bifurcation associated with Z2 restoration is identified as a generic route leading to homoclinic chaos in the order parameter dynamics for N≥4, connecting the study of non-reciprocal phases to complex and chaotic dynamics.

In conclusion, this work establishes a general symmetry- and topology-based principle for understanding the complex dynamical phases in heterogeneous, non-equilibrium systems with multiple interacting populations. It provides a theoretical roadmap for exploring such phenomena in diverse contexts including biological networks, active matter, and driven-dissipative quantum materials.