The Symmetry Perspective (Birkhäuser, 2002).
Perturbation Theory for Linear Operators 2nd edn (Springer, 1984). Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits. Exceptional points in optics and photonics. Sound isolation and giant linear nonreciprocity in a compact acoustic circulator. Spatially uniform traveling cellular patterns at a driven interface. Parity-breaking transitions of modulated patterns in hydrodynamic systems. Bifurcations in distributed kinetic systems with aperiodic instability. Kuramoto model of coupled oscillators with positive and negative coupling parameters: an example of conformist and contrarian oscillators. Temporal association in asymmetric neural networks. Kuramoto model for excitation-inhibition-based oscillations. Are steadily moving crystals unstable? Phys. Statistical mechanics where Newton’s third law is broken. Spatiotemporal order and emergent edge currents in active spinner materials. Group formation and cohesion of active particles with visual perception–dependent motility. How many dissenters does it take to disorder a flock? New J. Hierarchical group dynamics in pigeon flocks. Pairing, waltzing and scattering of chemotactic active colloids. Synchronization and collective dynamics in a carpet of microfluidic rotors. Our work lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point-enforced pattern formation and hysteresis. Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. We describe the emergence of these phases using insights from bifurcation theory 22, 23 and non-Hermitian quantum mechanics 24, 25. The resulting phase transitions are controlled by spectral singularities called exceptional points 21. We illustrate this mechanism with simple robotic demonstrations. Here we show that non-reciprocity leads to time-dependent phases in which spontaneously broken continuous symmetries are dynamically restored.
Although wave propagation in non-reciprocal media has recently been closely studied 1, 16, 17, 18, 19, 20, less is known about the consequences of non-reciprocity on the collective behaviour of many-body systems. Non-reciprocity occurs, for instance, in active matter 1, 2, 3, 4, 5, 6, non-equilibrium systems 7, 8, 9, networks of neurons 10, 11, social groups with conformist and contrarian members 12, directional interface growth phenomena 13, 14, 15 and metamaterials 16, 17, 18, 19, 20. Out of equilibrium, a lack of reciprocity is the rule rather than the exception.
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