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# Non-reciprocal phase transitions

## Abstract

Out of equilibrium, a lack of reciprocity is the rule rather than the exception. Non-reciprocity occurs, for instance, in active matter1,2,3,4,5,6, non-equilibrium systems7,8,9, networks of neurons10,11, social groups with conformist and contrarian members12, directional interface growth phenomena13,14,15 and metamaterials16,17,18,19,20. Although wave propagation in non-reciprocal media has recently been closely studied1,16,17,18,19,20, less is known about the consequences of non-reciprocity on the collective behaviour of many-body systems. Here we show that non-reciprocity leads to time-dependent phases in which spontaneously broken continuous symmetries are dynamically restored. We illustrate this mechanism with simple robotic demonstrations. The resulting phase transitions are controlled by spectral singularities called exceptional points21. We describe the emergence of these phases using insights from bifurcation theory22,23 and non-Hermitian quantum mechanics24,25. Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point-enforced pattern formation and hysteresis. Our work lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle.

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## Data availability

No data were generated during the course of this study.

## Code availability

The computer code used in this study is available on Zenodo at https://doi.org/10.5281/zenodo.4605984 under the 2-clause BSD licence.

## References

1. Shankar, S., Souslov, A., Bowick, M. J., Marchetti, M. C. & Vitelli, V. Topological active matter. Preprint at https://arxiv.org/abs/2010.00364 (2020).

2. Uchida, N. & Golestanian, R. Synchronization and collective dynamics in a carpet of microfluidic rotors. Phys. Rev. Lett. 104, 178103 (2010).

3. Saha, S., Ramaswamy, S. & Golestanian, R. Pairing, waltzing and scattering of chemotactic active colloids. New J. Phys. 21, 063006 (2019).

4. Nagy, M., Ákos, Z., Biro, D. & Vicsek, T. Hierarchical group dynamics in pigeon flocks. Nature 464, 890–893 (2010).

5. Yllanes, D., Leoni, M. & Marchetti, M. C. How many dissenters does it take to disorder a flock? New J. Phys. 19, 103026 (2017).

6. Lavergne, F. A., Wendehenne, H., Bäuerle, T. & Bechinger, C. Group formation and cohesion of active particles with visual perception–dependent motility. Science 364, 70–74 (2019).

7. van Zuiden, B. C., Paulose, J., Irvine, W. T. M., Bartolo, D. & Vitelli, V. Spatiotemporal order and emergent edge currents in active spinner materials. Proc. Natl Acad. Sci. USA 113, 12919 (2016).

8. Ivlev, A. V. et al. Statistical mechanics where Newton’s third law is broken. Phys. Rev. X 5, 011035 (2015).

9. Lahiri, R. & Ramaswamy, S. Are steadily moving crystals unstable? Phys. Rev. Lett. 79, 1150–1153 (1997).

10. Montbrió, E. & Pazó, D. Kuramoto model for excitation-inhibition-based oscillations. Phys. Rev. Lett. 120, 244101 (2018).

11. Sompolinsky, H. & Kanter, I. Temporal association in asymmetric neural networks. Phys. Rev. Lett. 57, 2861–2864 (1986).

12. Hong, H. & Strogatz, S. H. Kuramoto model of coupled oscillators with positive and negative coupling parameters: an example of conformist and contrarian oscillators. Phys. Rev. Lett. 106, 054102 (2011).

13. Malomed, B. & Tribelsky, M. Bifurcations in distributed kinetic systems with aperiodic instability. Physica D 14, 67–87 (1984).

14. Coullet, P., Goldstein, R. E. & Gunaratne, G. H. Parity-breaking transitions of modulated patterns in hydrodynamic systems. Phys. Rev. Lett. 63, 1954–1957 (1989).

15. Pan, L. & de Bruyn, J. R. Spatially uniform traveling cellular patterns at a driven interface. Phys. Rev. E 49, 483–493 (1994).

16. Fleury, R., Sounas, D. L., Sieck, C. F., Haberman, M. R. & Alù, A. Sound isolation and giant linear nonreciprocity in a compact acoustic circulator. Science 343, 516–519 (2014).

17. Brandenbourger, M., Locsin, X., Lerner, E. & Coulais, C. Non-reciprocal robotic metamaterials. Nat. Commun. 10, 4608 (2019).

18. Miri, M.-A. & Alù, A. Exceptional points in optics and photonics. Science 363, eaar7709 (2019).

19. Scheibner, C. et al. Odd elasticity. Nat. Phys. 16, 475–480 (2020).

20. Helbig, T. et al. Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits. Nat. Phys. 16, 747–750 (2020).

21. Kato, T. Perturbation Theory for Linear Operators 2nd edn (Springer, 1984).

22. Golubitsky, M. & Stewart, I. The Symmetry Perspective (Birkhäuser, 2002).

23. Kuznetsov, Y. A. Elements of Applied Bifurcation Theory (Springer, 2004).

24. Hatano, N. & Nelson, D. R. Localization transitions in non-Hermitian quantum mechanics. Phys. Rev. Lett. 77, 570–573 (1996).

25. Bender, C. M. & Boettcher, S. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998).

26. Bricard, A., Caussin, J.-B., Desreumaux, N., Dauchot, O. & Bartolo, D. Emergence of macroscopic directed motion in populations of motile colloids. Nature 503, 95–98 (2013).

27. Palacci, J., Sacanna, S., Steinberg, A. P., Pine, D. J. & Chaikin, P. M. Living crystals of light-activated colloidal surfers. Science 339, 936–940 (2013).

28. Sieberer, L. M., Huber, S. D., Altman, E. & Diehl, S. Dynamical critical phenomena in driven-dissipative systems. Phys. Rev. Lett. 110, 195301 (2013).

29. Metelmann, A. & Clerk, A. A. Nonreciprocal photon transmission and amplification via reservoir engineering. Phys. Rev. X 5, 021025 (2015).

30. Hanai, R., Edelman, A., Ohashi, Y. & Littlewood, P. B. Non-Hermitian phase transition from a polariton Bose–Einstein condensate to a photon laser. Phys. Rev. Lett. 122, 185301 (2019).

31. Hanai, R. & Littlewood, P. B. Critical fluctuations at a many-body exceptional point. Phys. Rev. Res. 2, 033018 (2020).

32. Hohenberg, P. C. & Halperin, B. I. Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977).

33. Acebrón, J. A., Bonilla, L. L., Vicente, C. J. P., Ritort, F. & Spigler, R. The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005).

34. Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I. & Shochet, O. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229 (1995).

35. Toner, J. & Tu, Y. Long-range order in a two-dimensional dynamical XY model: how birds fly together. Phys. Rev. Lett. 75, 4326–4329 (1995).

36. Sakaguchi, H. & Kuramoto, Y. A soluble active rotater model showing phase transitions via mutual entertainment. Prog. Theor. Phys. 76, 576–581 (1986).

37. Daido, H. Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions. Phys. Rev. Lett. 68, 1073–1076 (1992).

38. Das, J., Rao, M. & Ramaswamy, S. Driven Heisenberg magnets: nonequilibrium criticality, spatiotemporal chaos and control. Europhys. Lett. 60, 418–424 (2002).

39. Bonilla, L. L. & Trenado, C. Contrarian compulsions produce exotic time-dependent flocking of active particles. Phys. Rev. E 99, 012612 (2019).

40. Cross, M. C. & Hohenberg, P. C. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993).

41. Rabaud, M., Michalland, S. & Couder, Y. Dynamical regimes of directional viscous fingering: spatiotemporal chaos and wave propagation. Phys. Rev. Lett. 64, 184–187 (1990).

42. Oswald, P., Bechhoefer, J. & Libchaber, A. Instabilities of a moving nematic–isotropic interface. Phys. Rev. Lett. 58, 2318–2321 (1987).

43. Faivre, G., de Cheveigne, S., Guthmann, C. & Kurowski, P. Solitary tilt waves in thin lamellar eutectics. Europhys. Lett. 9, 779–784 (1989).

44. Brunet, P., Flesselles, J.-M. & Limat, L. Parity breaking in a one-dimensional pattern: a quantitative study with controlled wavelength. Europhys. Lett. 56, 221–227 (2001).

45. Hassan, A. U., Hodaei, H., Miri, M.-A., Khajavikhan, M. & Christodoulides, D. N. Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators. Phys. Rev. A 92, 063807 (2015).

46. Nixon, M., Ronen, E., Friesem, A. A. & Davidson, N. Observing geometric frustration with thousands of coupled lasers. Phys. Rev. Lett. 110, 184102 (2013).

47. Parto, M., Hayenga, W., Marandi, A., Christodoulides, D. N. & Khajavikhan, M. Realizing spin Hamiltonians in nanoscale active photonic lattices. Nat. Mater. 19, 725–731 (2020).

48. Ramos, A., Fernández-Alcázar, L., Kottos, T. & Shapiro, B. Optical phase transitions in photonic networks: a spin-system formulation. Phys. Rev. X 10, 031024 (2020).

49. Ashida, Y., Furukawa, S. & Ueda, M. Parity-time-symmetric quantum critical phenomena. Nat. Commun. 8, 15791 (2017).

50. Strack, P. & Vitelli, V. Soft quantum vibrations of a PT-symmetric nonlinear ion chain. Phys. Rev. A 88, 053408 (2013).

51. Biancalani, T., Jafarpour, F. & Goldenfeld, N. Giant amplification of noise in fluctuation-induced pattern formation. Phys. Rev. Lett. 118, 018101 (2017).

52. Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. Hydrodynamic stability without eigenvalues. Science 261, 578–584 (1993).

53. Winfree, A. T. The Geometry of Biological Time (Springer, 2001).

54. Khemani, V., Moessner, R. & Sondhi, S. L. A brief history of time crystals. Preprint at https://arxiv.org/abs/1910.10745 (2019).

55. You, Z., Baskaran, A. & Marchetti, M. C. Nonreciprocity as a generic route to traveling states. Preprint at https://arxiv.org/abs/2005.07684 (2020).

56. Saha, S., Agudo-Canalejo, J. & Golestanian, R. Scalar active mixtures: the nonreciprocal Cahn–Hilliard model. Preprint at https://arxiv.org/abs/2005.07101 (2020).

57. Landau, L. & Khalatnikov, I. On the anomalous absorption of sound near a second-order phase transition point. Dokl. Akad. Nauk SSSR 96, 469–472 (1954).

58. Cugliandolo, L. F. & Kurchan, J. Weak ergodicity breaking in mean-field spin-glass models. Philos. Mag. B 71, 501–514 (1995).

59. Keim, N. C., Paulsen, J. D., Zeravcic, Z., Sastry, S. & Nagel, S. R. Memory formation in matter. Rev. Mod. Phys. 91, 035002 (2019).

60. Van Hove, L. Time-dependent correlations between spins and neutron scattering in ferromagnetic crystals. Phys. Rev. 95, 1374–1384 (1954).

61. Hohenberg, P. & Krekhov, A. An introduction to the Ginzburg–Landau theory of phase transitions and nonequilibrium patterns. Phys. Rep. 572, 1–42 (2015).

62. Wilson, K. The renormalization group and the epsilon expansion. Phys. Rep. 12, 75–199 (1974).

63. Laguës, M. & Lesne, A. Invariances d’Échelle: des Changements d’États à la Turbulence (Belin, 2003).

64. Muñoz, M. A. Criticality and dynamical scaling in living systems. Rev. Mod. Phys. 90, 031001 (2018).

65. Sornette, D. Critical Phenomena in Natural Sciences (Springer, 2000).

66. van Saarloos, W. The complex Ginzburg–Landau equation for beginners. Spatio-temporal Patterns in Nonequilibrium Complex Systems Vol. XXI (eds Cladis, P. E. & Palffy-Muhoray, P.) (Addison-Wesley, 1994).

67. Aranson, I. S. & Kramer, L. The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys. 74, 99–143 (2002).

68. Golubitsky, M. & Schaeffer, D. G. Singularities and Groups in Bifurcation Theory Vol. I (Springer, 1985).

69. Golubitsky, M., Stewart, I. & Schaeffer, D. G. Singularities and Groups in Bifurcation Theory Vol. II (Springer, 1988).

70. Crawford, J. D. & Knobloch, E. Symmetry and symmetry-breaking bifurcations in fluid dynamics. Annu. Rev. Fluid Mech. 23, 341–387 (1991).

71. Chossat, P. & Lauterbach, R. Methods in Equivariant Bifurcations and Dynamical Systems (World Scientific, 2000).

72. Haken, H. (ed.) Synergetics (Springer, 1977).

73. Henkel, M., Hinrichsen, H. & Lübeck, S. Non-equilibrium Phase Transitions Vol. 1 (Springer, 2008).

74. Henkel, M. & Pleimling, M. Non-equilibrium Phase Transitions Vol. 2 (Springer, 2010).

75. Livi, R. & Politi, P. Nonequilibrium Statistical Physics: A Modern Perspective (Cambridge Univ. Press, 2017).

76. Aron, C. & Chamon, C. Landau theory for non-equilibrium steady states, SciPost Phys. 8, 074 (2020).

77. Marchetti, M. C. et al. Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143 (2013).

78. Trefethen, L. N. & Embree, M. Spectra and Pseudospectra (Princeton Univ. Press, 2005).

79. Böberg, L. & Brosa, U. Onset of turbulence in a pipe. Z. Naturforsch. A 43, 697–726 (1988).

80. Farrell, B. F. & Ioannou, P. J. Variance maintained by stochastic forcing of non-normal dynamical systems associated with linearly stable shear flows. Phys. Rev. Lett. 72, 1188–1191 (1994).

81. Dauchot, O. & Manneville, P. Local versus global concepts in hydrodynamic stability theory. J. Phys. II 7, 371–389 (1997).

82. Grossmann, S. The onset of shear flow turbulence. Rev. Mod. Phys. 72, 603–618 (2000).

83. Chomaz, J.-M. Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392 (2005).

84. Wu, J.-Z., Ma, H.-Y. & Zhou, M.-D. Vorticity And Vortex Dynamics (Springer, 2006).

85. Schmid, P. J. Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162 (2007).

86. Kerswell, R. Nonlinear nonmodal stability theory. Annu. Rev. Fluid Mech. 50, 319–345 (2018).

87. Chajwa, R., Menon, N., Ramaswamy, S. & Govindarajan, R. Waves, algebraic growth, and clumping in sedimenting disk arrays. Phys. Rev. X 10, 041016 (2020).

88. Murphy, B. K. & Miller, K. D. Balanced amplification: a new mechanism of selective amplification of neural activity patterns. Neuron 61, 635–648 (2009); correction 89, 235 (2016).

89. Hennequin, G., Vogels, T. P. & Gerstner, W. Non-normal amplification in random balanced neuronal networks. Phys. Rev. E 86, 011909 (2012).

90. Amir, A., Hatano, N. & Nelson, D. R. Non-Hermitian localization in biological networks. Phys. Rev. E 93, 042310 (2016).

91. Asllani, M. & Carletti, T. Topological resilience in non-normal networked systems. Phys. Rev. E 97, 042302 (2018).

92. Asllani, M., Lambiotte, R. & Carletti, T. Structure and dynamical behavior of non-normal networks. Sci. Adv. 4, eaau9403 (2018).

93. Baggio, G., Rutten, V., Hennequin, G. & Zampieri, S. Efficient communication over complex dynamical networks: the role of matrix non-normality. Sci. Adv. 6, eaba2282 (2020).

94. Nicolaou, Z. G., Nishikawa, T., Nicholson, S. B., Green, J. R. & Motter, A. E. Non-normality and non-monotonic dynamics in complex reaction networks. Phys. Rev. Res. 2, 043059 (2020).

95. Neubert, M. G. & Caswell, H. Alternatives to resilience for measuring the responses of ecological systems to perturbations. Ecology 78, 653–665 (1997).

96. Nelson, D. R. & Shnerb, N. M. Non-Hermitian localization and population biology. Phys. Rev. E 58, 1383 (1998).

97. Neubert, M. G., Klanjscek, T. & Caswell, H. Reactivity and transient dynamics of predator–prey and food web models. Ecol. Modell. 179, 29 (2004).

98. Townley, S., Carslake, D., Kellie-smith, O., Mccarthy, D. & Hodgson, D. Predicting transient amplification in perturbed ecological systems. J. Appl. Ecol. 44, 1243 (2007).

99. Ridolfi, L., Camporeale, C., D’Odorico, P. & Laio, F. Transient growth induces unexpected deterministic spatial patterns in the Turing process. Europhys. Lett. 95, 18003 (2011).

100. Feng, L., El-Ganainy, R. & Ge, L. Non-Hermitian photonics based on parity–time symmetry. Nat. Photon. 11, 752 (2017).

101. Makris, K., Ge, L. & Türeci, H. Anomalous transient amplification of waves in non-normal photonic media. Phys. Rev. X 4, 041044 (2014).

102. Ashida, Y., Gong, Z. & Ueda, M. Non-Hermitian physics. Preprint at https://arxiv.org/abs/2006.01837 (2020).

103. Tripathi, V., Galda, A., Barman, H. & Vinokur, V. M. Parity–time symmetry-breaking mechanism of dynamic Mott transitions in dissipative systems. Phys. Rev. B 94, 041104 (2016).

104. Bernier, N. R., Torre, E. G. D. & Demler, E. Unstable avoided crossing in coupled spinor condensates. Phys. Rev. Lett. 113, 065303 (2014).

105. Aharonyan, M. & Torre, E. G. D. Many-body exceptional points in colliding condensates. Mol. Phys. 117, 1971 (2019).

106. Mostafazadeh, A. Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian. J. Math. Phys. 43, 205 (2002).

107. Mostafazadeh, A. Pseudo-Hermiticity versus PT-symmetry. II: A complete characterization of non-Hermitian Hamiltonians with a real spectrum. J. Math. Phys. 43, 2814 (2002).

108. Mostafazadeh, A. Pseudo-Hermiticity versus PT-symmetry. III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries. J. Math. Phys. 43, 3944 (2002).

109. Bender, C. M., Berry, M. V. & Mandilara, A. Generalized PT symmetry and real spectra. J. Phys. Math. Gen. 35, L467 (2002).

110. Bender, C. M. & Mannheim, P. D. PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues. Phys. Lett. A 374, 1616–1620 (2010).

111. Mostafazadeh, A. Physics of spectral singularities. In Trends in Mathematics (eds Kielanowski, P. et al.) 145–165 (Springer, 2015).

112. Bender, C. M. Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947 (2007).

113. Weigert, S. PT-symmetry and its spontaneous breakdown explained by anti-linearity. J. Opt. B 5, S416 (2003).

114. Wigner, E. P. Normal form of antiunitary operators. J. Math. Phys. 1, 409 (1960).

115. Konotop, V. V., Yang, J. & Zezyulin, D. A. Nonlinear waves in PT-symmetric systems. Rev. Mod. Phys. 88, 035002 (2016).

116. van Kampen, N. G. Stochastic Processes in Physics and Chemistry Vol. 1 (Elsevier, 2007).

117. Risken, H. The Fokker–Planck Equation (Springer, 1989).

118. Gardiner, C. W. Handbook of Stochastic Methods (Springer, 2004).

119. Lan, G., Sartori, P., Neumann, S., Sourjik, V. & Tu, Y. The energy–speed–accuracy trade-off in sensory adaptation. Nat. Phys. 8, 422 (2012).

120. Weiss, J. B. Coordinate invariance in stochastic dynamical systems. Tellus A55, 208–218 (2003).

121. Newton, I. Philosophiæ Naturalis Principia Mathematica (1687).

122. Ermak, D. L. & McCammon, J. A. Brownian dynamics with hydrodynamic interactions. J. Chem. Phys. 69, 1352 (1978).

123. Di Leonardo, R. et al. Hydrodynamic interactions in two dimensions. Phys. Rev. E 78, 031406 (2008).

124. Lahiri, R., Barma, M. & Ramaswamy, S. Strong phase separation in a model of sedimenting lattices. Phys. Rev. E 61, 1648 (2000).

125. Kryuchkov, N. P., Ivlev, A. V. & Yurchenko, S. O. Dissipative phase transitions in systems with nonreciprocal effective interactions. Soft Matter 14, 9720 (2018).

126. Soto, R. & Golestanian, R. Self-assembly of catalytically active colloidal molecules: tailoring activity through surface chemistry. Phys. Rev. Lett. 112, 068301 (2014).

127. Agudo-Canalejo, J. & Golestanian, R. Active phase separation in mixtures of chemically interacting particles. Phys. Rev. Lett. 123, 018101 (2019).

128. Dholakia, K. & Zemánek, P. Gripped by light: optical binding. Rev. Mod. Phys. 82, 1767–1791 (2010).

129. Yifat, Y. D. et al. Reactive optical matter: light-induced motility in electrodynamically asymmetric nanoscale scatterers. Light Sci. Appl. 7, 105 (2018).

130. Peterson, C. W., Parker, J., Rice, S. A. & Scherer, N. F. Controlling the dynamics and optical binding of nanoparticle homodimers with transverse phase gradients. Nano Lett. 19, 897–903 (2019).

131. Krakauer, J. W., Ghazanfar, A. A., Gomez-Marin, A., MacIver, M. A. & Poeppel, D. Neuroscience needs behavior: correcting a reductionist bias. Neuron 93, 480 (2017).

132. Parisi, G. Asymmetric neural networks and the process of learning. J. Phys. Math. Gen. 19, L675 (1986).

133. Derrida, B., Gardner, E. & Zippelius, A. An exactly solvable asymmetric neural network model. Europhys. Lett. 4, 167 (1987).

134. Dayan, P. & Abbott, L. Theoretical Neuroscience: Computational and Mathematical Modelling of Neural Systems (MIT Press, 2001).

135. Hong, H. & Strogatz, S. H. Conformists and contrarians in a Kuramoto model with identical natural frequencies. Phys. Rev. E 84, 046202 (2011).

136. Pluchino, A., Latora, V. & Rapisarda, A. Changing opinions in a changing world: a new perspective in sociophysics. Int. J. Mod. Phys. C 16, 515 (2005).

137. Morin, A., Caussin, J.-B., Eloy, C. & Bartolo, D. Collective motion with anticipation: flocking, spinning, and swarming. Phys. Rev. E 91, 012134 (2015).

138. Ginelli, F. et al. Intermittent collective dynamics emerge from conflicting imperatives in sheep herds. Proc. Natl Acad. Sci. USA 112, 12729 (2015).

139. Dadhichi, L. P., Kethapelli, J., Chajwa, R., Ramaswamy, S. & Maitra, A. Nonmutual torques and the unimportance of motility for long-range order in two-dimensional flocks. Phys. Rev. E 101, 052601 (2020).

140. Barberis, L. & Peruani, F. Large-scale patterns in a minimal cognitive flocking model: incidental leaders, nematic patterns, and aggregates. Phys. Rev. Lett. 117, 248001 (2016).

141. Gupta, R. K., Kant, R., Soni, H., Sood, A. K. & Ramaswamy, S. Active nonreciprocal attraction between motile particles in an elastic medium. Preprint at https://arxiv.org/abs/2007.04860 (2020).

142. Maitra, A., Lenz, M. & Voituriez, R. Chiral active hexatics: giant number fluctuations, waves and destruction of order. Phys. Rev. Lett. 125, 238005 (2020).

143. Durve, M., Saha, A. & Sayeed, A. Active particle condensation by non-reciprocal and time-delayed interactions. Eur. Phys. J. E 41, 49 (2018).

144. Costanzo, A. Milling-induction and milling-destruction in a Vicsek-like binary-mixture model. Europhys. Lett. 125, 20008 (2019).

145. Lin, Z. et al. Unidirectional invisibility induced by PT-symmetric periodic structures. Phys. Rev. Lett. 106, 213901 (2011).

146. Coulais, C., Sounas, D. & Alù, A. Static non-reciprocity in mechanical metamaterials. Nature 542, 461 (2017).

147. Ghatak, A., Brandenbourger, M., van Wezel, J. & Coulais, C. Observation of non-Hermitian topology and its bulk–edge correspondence in an active mechanical metamaterial. Proc. Natl Acad. Sci. USA 117, 29561 (2020).

148. Rosa, M. I. N. & Ruzzene, M. Dynamics and topology of non-Hermitian elastic lattices with non-local feedback control interactions. New J. Phys. 22, 053004 (2020).

149. Chen, Y., Li, X., Scheibner, C., Vitelli, V. & Huang, G. Self-sensing metamaterials with odd micropolarity. Preprint at https://arxiv.org/abs/2009.07329 (2020).

150. Das, J., Rao, M. & Ramaswamy, S. Nonequilibrium steady states of the isotropic classical magnet. Preprint at https://arxiv.org/abs/cond-mat/0404071 (2004).

151. Tasaki, H. Hohenberg–Mermin–Wagner-type theorems for equilibrium models of flocking. Phys. Rev. Lett. 125, 220601 (2020).

152. Fodor, É. et al. How far from equilibrium is active matter? Phys. Rev. Lett. 117, 038103 (2016).

153. Loos, S. A. M. & Klapp, S. H. L. Thermodynamic implications of non-reciprocity. Preprint at https://arxiv.org/abs/2008.00894 (2020).

154. Loos, S. A. M., Hermann, S. M. & Klapp, S. H. L. Non-reciprocal hidden degrees of freedom: a unifying perspective on memory, feedback, and activity. Preprint at https://arxiv.org/abs/1910.08372 (2019).

155. Malzard, S., Poli, C. & Schomerus, H. Topologically protected defect states in open photonic systems with non-Hermitian charge conjugation and parity–time symmetry. Phys. Rev. Lett. 115, 200402 (2015).

156. Lee, C. H. & Thomale, R. Anatomy of skin modes and topology in non-Hermitian systems. Phys. Rev. B 99, 201103 (2019).

157. Lee, C. H., Li, L., Thomale, R. & Gong, J. Unraveling non-Hermitian pumping: emergent spectral singularities and anomalous responses. Phys. Rev. B 102, 085151 (2020).

158. Okuma, N., Kawabata, K., Shiozaki, K. & Sato, M. Topological origin of non-Hermitian skin effects. Phys. Rev. Lett. 124, 086801 (2020).

159. Hofmann, T., Helbig, T., Lee, C. H., Greiter, M. & Thomale, R. Chiral voltage propagation and calibration in a topolectrical Chern circuit. Phys. Rev. Lett. 122, 247702 (2019).

160. Lee, C. H., Li, L. & Gong, J. Hybrid higher-order skin-topological modes in nonreciprocal systems. Phys. Rev. Lett. 123, 016805 (2019).

161. Zhang, K., Yang, Z. & Fang, C. Correspondence between winding numbers and skin modes in non-Hermitian systems. Phys. Rev. Lett. 125, 126402 (2020).

162. Achenbach, J. D. Reciprocity in Elastodynamics (Cambridge Univ. Press, 2004).

163. Nassar, H. et al. Nonreciprocity in acoustic and elastic materials. Nat. Rev. Mater. (2020).

164. Potton, R. J. Reciprocity in optics. Rep. Prog. Phys. 67, 717 (2004).

165. Estep, N. A., Sounas, D. L., Soric, J. & Alù, A. Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops. Nat. Phys. 10, 923 (2014).

166. Caloz, C. et al. Electromagnetic nonreciprocity. Phys. Rev. Appl. 10, 047001 (2018).

167. Masoud, H. & Stone, H. A. The reciprocal theorem in fluid dynamics and transport phenomena. J. Fluid Mech. 879, P1 (2019).

168. Scheibner, C., Irvine, W. T. M. & Vitelli, V. Non-Hermitian band topology and skin modes in active elastic media. Phys. Rev. Lett. 125, 118001 (2020).

169. Zhou, D. & Zhang, J. Non-Hermitian topological metamaterials with odd elasticity. Phys. Rev. Res. 2, 023173 (2020).

170. Groot, S. R. D. & Mazur, P. Non-Equilibrium Thermodynamics (Dover Publications, 1962).

171. Maltman, K. & Laidlaw, W. G. Onsager symmetry and the diagonalizability of the hydrodynamic matrix. J. Math. Phys. 16, 1561 (1975).

172. Avron, J. E. Odd viscosity. J. Stat. Phys. 92, 543–557 (1998).

173. Banerjee, D., Souslov, A., Abanov, A. G. & Vitelli, V. Odd viscosity in chiral active fluids. Nat. Commun. 8, 1573 (2017).

174. Souslov, A., Dasbiswas, K., Fruchart, M., Vaikuntanathan, S. & Vitelli, V. Topological waves in fluids with odd viscosity. Phys. Rev. Lett. 122, 128001 (2019).

175. Soni, V. et al. The odd free surface flows of a colloidal chiral fluid. Nat. Phys. 15, 1188–1194 (2019).

176. Han, M. et al. Statistical mechanics of a chiral active fluid. Preprint at https://arxiv.org/abs/2002.07679 (2020).

177. Arnold, V. I. Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, 1988).

178. Bogdanov, R. I. Bifurcations of a limit cycle for a family of vector fields on the plane. Selecta Math. Sov. 1, 373 (1981); translated from Trudy Sem. Petrovsk. 2, 23–35 (1976).

179. Bogdanov, R. I. Versal deformations of a singularity of a vector field on the plane in the case of zero eigenvalues. Selecta Math. Sov. 1, 389 (1981); translated from Trudy Sem. Petrovsk. 2, 37–65 (1976).

180. Takens, F. in Global Analysis of Dynamical Systems (eds Broer, H. W. et al.) 1–63 (IOP, 2001); reprinted from Commun. Math. Inst. Rijksuniv. Utrecht 2, 1–111 (1974).

181. Nambu, Y. Quasi-particles and gauge invariance in the theory of superconductivity. Phys. Rev. 117, 648 (1960).

182. Goldstone, J. Field theories with superconductor solutions. Nuovo Cim. 19, 154–164 (1961).

183. Goldstone, J., Salam, A. & Weinberg, S. Broken symmetries. Phys. Rev. 127, 965 (1962).

184. Hidaka, Y. Counting rule for Nambu–Goldstone modes in nonrelativistic Systems. Phys. Rev. Lett. 110, 091601 (2013).

185. Watanabe, H. Counting rules of Nambu–Goldstone modes. Annu. Rev. Condens. Matter Phys. 11, 169 (2020).

186. Watanabe, H. & Murayama, H. Unified description of Nambu–Goldstone bosons without Lorentz invariance. Phys. Rev. Lett. 108, 251602 (2012).

187. Nielsen, H. & Chadha, S. On how to count Goldstone bosons. Nucl. Phys. B 105, 445 (1976).

188. Leroy, L. On spontaneous symmetry breakdown in dynamical systems. J. Phys. Math. Gen. 25, L987 (1992).

189. Minami, Y. & Hidaka, Y. Spontaneous symmetry breaking and Nambu–Goldstone modes in dissipative systems. Phys. Rev. E 97, 012130 (2018).

190. Hongo, M., Kim, S., Noumi, T. & Ota, A. Effective Lagrangian for Nambu–Goldstone modes in nonequilibrium open systems. Preprint at https://arxiv.org/abs/1907.08609 (2019).

191. Von Neumann, J. & Wigner, E. P. Über das Verhalten von Eigenwerten bei adiabatischen Prozessen Physik. Zeit. 30, 467 (1929); translated in Symmetry in the Solid State (eds Knox, R. S. & Gold, A.) (Benjamin, New York, 1964).

192. Arnold, V. I. Modes and quasimodes. Funct. Anal. Appl. 6, 94 (1972); translated from Funktsional. Anal. i Prilozhen. 6, 12–20 (1972).

193. Arnold, V. I. Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect. Selecta Mathematica 1, 1–19 (1995).

194. Seyranian, A. P., Kirillov, O. N. & Mailybaev, A. A. Coupling of eigenvalues of complex matrices at diabolic and exceptional points. J. Phys. Math. Gen. 38, 1723 (2005).

195. Julien, K. A. Strong spatial interactions with 1:1 resonance: a three-layer convection problem. Nonlinearity 7, 1655 (1994).

196. Renardy, Y. Y., Renardy, M. & Fujimura, K. Takens–Bogdanov bifurcation on the hexagonal lattice for double-layer convection. Physica D 129, 171 (1999).

197. Guckenheimer, J. A codimension two bifurcation with circular symmetry. In Multiparameter Bifurcation Theory (eds Golubitsky, M. & Guckenheimer, J. M.) 175–184 (AMS, 1986).

198. Dangelmayr, G. & Knobloch, E. The Takens–Bogdanov bifurcation with O(2) symmetry. Phil. Trans. R. Soc. Lond. A 322, 243–279 (1987).

199. Krupa, M. Bifurcations of relative equilibria. SIAM J. Math. Anal. 21, 1453 (1990).

200. Field, M. J. Equivariant dynamical systems. Trans. Am. Math. Soc. 259, 185 (1980).

201. Toner, J. & Tu, Y. Flocks, herds, and schools: a quantitative theory of flocking. Phys. Rev. E 58, 4828 (1998).

202. Geyer, D., Morin, A. & Bartolo, D. Sounds and hydrodynamics of polar active fluids. Nat. Mater. 17, 789 (2018).

203. Bain, N. & Bartolo, D. Dynamic response and hydrodynamics of polarized crowds. Science 363, 46 (2019).

204. Dean, D. S. Langevin equation for the density of a system of interacting Langevin processes. J. Phys. Math. Gen. 29, L613 (1996).

205. Bertin, E., Droz, M. & Grégoire, G. Boltzmann and hydrodynamic description for self-propelled particles. Phys. Rev. E 74, 022101 (2006).

206. Bertin, E., Droz, M. & Grégoire, G. Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis. J. Phys. A Math. Theor. 42, 445001 (2009).

207. Farrell, F. D. C., Marchetti, M. C., Marenduzzo, D. & Tailleur, J. Pattern formation in self-propelled particles with density-dependent motility. Phys. Rev. Lett. 108, 248101 (2012).

208. Chaté, H. & Mahault, B. Dry, aligning, dilute, active matter: a synthetic and self-contained overview. Preprint at https://arxiv.org/abs/1906.05542 (2019).

209. Peshkov, A., Bertin, E., Ginelli, F. & Chaté, H. Boltzmann–Ginzburg–Landau approach for continuous descriptions of generic Vicsek-like models. Eur. Phys. J. Spec. Top. 223, 1315 (2014).

210. Ihle, T. Kinetic theory of flocking: derivation of hydrodynamic equations. Phys. Rev. E 83, 030901 (2011).

211. Mahault, B., Ginelli, F. & Chaté, H. Quantitative assessment of the Toner and Tu theory of polar flocks. Phys. Rev. Lett. 123, 218001 (2019).

212. Oza, A. U. & Dunkel, J. Antipolar ordering of topological defects in active liquid crystals. New J. Phys. 18, 093006 (2016).

213. Suzuki, R., Weber, C. A., Frey, E. & Bausch, A. R. Polar pattern formation in driven filament systems requires non-binary particle collisions. Nat. Phys. 11, 839 (2015).

214. Nishiguchi, D., Nagai, K. H., Chaté, H. & Sano, M. Long-range nematic order and anomalous fluctuations in suspensions of swimming filamentous bacteria. Phys. Rev. E 95, 020601 (2017).

215. Tsai, J.-C., Ye, F., Rodriguez, J., Gollub, J. P. & Lubensky, T. C. A chiral granular gas. Phys. Rev. Lett. 94, 214301 (2005).

216. Liebchen, B. & Levis, D. Collective behavior of chiral active matter: pattern formation and enhanced flocking. Phys. Rev. Lett. 119, 058002 (2017).

217. O’Keeffe, K. P., Hong, H. & Strogatz, S. H. Oscillators that sync and swarm. Nat. Commun. 8, 1504 (2017).

218. Levis, D., Pagonabarraga, I. & Liebchen, B. Activity induced synchronization: mutual flocking and chiral self-sorting. Phys. Rev. Res. 1, 023026 (2019).

219. Burns, K. J., Vasil, G. M., Oishi, J. S., Lecoanet, D. & Brown, B. P. Dedalus: a flexible framework for numerical simulations with spectral methods. Phys. Rev. Res. 2, 023068 (2020).

220. Caussin, J.-B. et al. Emergent spatial structures in flocking models: a dynamical system insight. Phys. Rev. Lett. 112, 148102 (2014).

221. Mishra, S., Baskaran, A. & Marchetti, M. C. Fluctuations and pattern formation in self-propelled particles. Phys. Rev. E 81, 061916 (2010).

222. Grégoire, G. & Chaté, H. Onset of collective and cohesive motion. Phys. Rev. Lett. 92, 025702 (2004).

223. Aditi Simha, R. & Ramaswamy, S. Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles. Phys. Rev. Lett. 89, 058101 (2002).

224. Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence (Springer, 1984).

225. Daido, H. Population dynamics of randomly interacting self-oscillators. I: Tractable models without frustration. Prog. Theor. Phys. 77, 622 (1987).

226. Omata, S., Yamaguchi, Y. & Shimizu, H. Entrainment among coupled limit cycle oscillators with frustration. Physica D 31, 397 (1988).

227. Martens, E. A. et al. Exact results for the Kuramoto model with a bimodal frequency distribution. Phys. Rev. E 79, 026204 (2009).

228. Bonilla, L., Vicente, C. P. & Spigler, R. Time-periodic phases in populations of nonlinearly coupled oscillators with bimodal frequency distributions. Physica D 113, 79 (1998).

229. Hong, H. & Strogatz, S. H. Mean-field behavior in coupled oscillators with attractive and repulsive interactions. Phys. Rev. E 85, 056210 (2012).

230. Ott, E. & Antonsen, T. M. Low dimensional behavior of large systems of globally coupled oscillators. Chaos 18, 037113 (2008).

231. Abrams, D. M., Mirollo, R., Strogatz, S. H. & Wiley, D. A. Solvable model for chimera states of coupled oscillators. Phys. Rev. Lett. 101, 084103 (2008).

232. Pikovsky, A. & Rosenblum, M. Partially integrable dynamics of hierarchical populations of coupled oscillators. Phys. Rev. Lett. 101, 264103 (2008).

233. Martens, E. A., Bick, C. & Panaggio, M. J. Chimera states in two populations with heterogeneous phase-lag. Chaos 26, 094819 (2016).

234. Choe, C.-U., Ri, J.-S. & Kim, R.-S. Incoherent chimera and glassy states in coupled oscillators with frustrated interactions. Phys. Rev. E 94, 032205 (2016).

235. Gallego, R., Montbrió, E. & Pazó, D. Synchronization scenarios in the Winfree model of coupled oscillators. Phys. Rev. E 96, 042208 (2017).

236. Ott, E. & Antonsen, T. M. Long time evolution of phase oscillator systems. Chaos 19, 023117 (2009).

237. Watanabe, S. & Strogatz, S. H. Integrability of a globally coupled oscillator array. Phys. Rev. Lett. 70, 2391 (1993).

238. Watanabe, S. & Strogatz, S. H. Constants of motion for superconducting Josephson arrays. Physica D 74, 197 (1994).

239. Marvel, S. A., Mirollo, R. E. & Strogatz, S. H. Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action. Chaos 19, 043104 (2009).

240. Pikovsky, A. & Rosenblum, M. Dynamics of heterogeneous oscillator ensembles in terms of collective variables. Physica D 240, 872 (2011).

241. Tyulkina, I. V., Goldobin, D. S., Klimenko, L. S. & Pikovsky, A. Dynamics of noisy oscillator populations beyond the Ott–Antonsen ansatz. Phys. Rev. Lett. 120, 264101 (2018).

242. Montbrió, E., Pazó, D. & Roxin, A. Macroscopic description for networks of spiking neurons. Phys. Rev. X 5, 021028 (2015).

243. Bick, C., Goodfellow, M., Laing, C. R. & Martens, E. A. Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review. J. Math. Neurosci. 10, 9 (2020).

244. Pazó, D. & Montbrió, E. Existence of hysteresis in the Kuramoto model with bimodal frequency distributions. Phys. Rev. E 80, 046215 (2009).

245. Pietras, B., Deschle, N. & Daffertshofer, A. First-order phase transitions in the Kuramoto model with compact bimodal frequency distributions. Phys. Rev. E 98, 062219 (2018).

246. Doppler, J. et al. Dynamically encircling an exceptional point for asymmetric mode switching. Nature 537, 76 (2016).

247. Dembowski, C. et al. Encircling an exceptional point. Phys. Rev. E 69, 056216 (2004).

248. Milburn, T. J. et al. General description of quasiadiabatic dynamical phenomena near exceptional points. Phys. Rev. A 92, 052124 (2015).

249. Mailybaev, A. A., Kirillov, O. N. & Seyranian, A. P. Geometric phase around exceptional points. Phys. Rev. A 72, 014104 (2005).

250. Galda A. & Vinokur, V. M. Parity–time symmetry breaking in magnetic systems. Phys. Rev. B 94, 020408(R) (2016); erratum 100, 209902 (2019).

251. Galda, A. & Vinokur, V. M. Exceptional points in classical spin dynamics. Sci. Rep. 9, 17484 (2019).

252. Kepesidis, K. V. et al. PT-symmetry breaking in the steady state of microscopic gain–loss systems. New J. Phys. 18, 095003 (2016).

253. Graefe, E.-M., Korsch, H. J. & Niederle, A. E. Quantum–classical correspondence for a non-Hermitian Bose–Hubbard dimer. Phys. Rev. A 82, 013629 (2010).

254. Cartarius, H., Main, J. & Wunner, G. Discovery of exceptional points in the Bose–Einstein condensation of gases with attractive 1/r interaction. Phys. Rev. A 77, 013618 (2008).

255. Gutöhrlein, R., Main, J., Cartarius, H. & Wunner, G. Bifurcations and exceptional points in dipolar Bose–Einstein condensates. J. Phys. A 46, 305001 (2013).

256. Hoyle, R. Pattern Formation (Cambridge Univ. Press, 2006).

257. Cross, M. & Greenside, H. Pattern Formation and Dynamics in Nonequilibrium Systems (Cambridge Univ. Press, 2009).

258. Meron, E. Nonlinear Physics of Ecosystems (CRC Press, 2015).

259. Swift, J. & Hohenberg, P. C. Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319 (1977).

260. Coullet, P. & Fauve, S. Propagative phase dynamics for systems with galilean invariance. Phys. Rev. Lett. 55, 2857 (1985).

261. Brachet, M. E., Coullet, P. & Fauve, S. Propagative phase dynamics in temporally intermittent systems. Europhys. Lett. 4, 1017 (1987).

262. Douady, S., Fauve, S. & Thual, O. Oscillatory phase modulation of parametrically forced surface waves. Europhys. Lett. 10, 309 (1989).

263. Coullet, P. & Iooss, G. Instabilities of one-dimensional cellular patterns. Phys. Rev. Lett. 64, 866 (1990).

264. Fauve, S., Douady, S. & Thual, O. Drift instabilities of cellular patterns. J. Phys. II 1, 311 (1991).

265. Knobloch, E., Hettel, J. & Dangelmayr, G. Parity-breaking bifurcation in inhomogeneous systems. Phys. Rev. Lett. 74, 4839 (1995).

266. Armbruster, D., Guckenheimer, J. & Holmes, P. Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry. Physica D 29, 257 (1988).

267. Proctor, M. R. E. & Jones, C. A. The interaction of two spatially resonant patterns in thermal convection. Part 1. Exact 1:2 resonance. J. Fluid Mech. 188, 301 (1988).

268. Dangelmayr, G., Hettel, J. & Knobloch, E. Parity-breaking bifurcation in inhomogeneous systems. Nonlinearity 10, 1093 (1997).

269. Simon, A. J., Bechhoefer, J. & Libchaber, A. Solitary modes and the Eckhaus instability in directional solidification. Phys. Rev. Lett. 61, 2574 (1988).

270. Flesselles, J.-M., Simon, A. & Libchaber, A. Dynamics of one-dimensional interfaces: an experimentalist’s view. Adv. Phys. 40, 1 (1991).

271. Melo, F. & Oswald, P. Destabilization of a faceted smectic-A–smectic-B interface. Phys. Rev. Lett. 64, 1381 (1990).

272. Faivre, G. & Mergy, J. Dynamical wavelength selection by tilt domains in thin-film lamellar eutectic growth. Phys. Rev. A 46, 963 (1992).

273. Kassner, K. & Misbah, C. Parity breaking in eutectic growth. Phys. Rev. Lett. 65, 1458–1461 (1990).

274. Ginibre, M., Akamatsu, S. & Faivre, G. Experimental determination of the stability diagram of a lamellar eutectic growth front. Phys. Rev. E 56, 780–796 (1997).

275. Cummins, H. Z., Fourtune, L. & Rabaud, M. Successive bifurcations in directional viscous fingering. Phys. Rev. E 47, 1727–1738 (1993).

276. Bellon, L., Fourtune, L., Minassian, V. T. & Rabaud, M. Wave-number selection and parity-breaking bifurcation in directional viscous fingering. Phys. Rev. E 58, 565–574 (1998).

277. Counillon, C. et al. Global drift of a circular array of liquid columns. Europhys. Lett. 40, 37 (1997).

278. Knobloch, E. & Proctor, M. R. E. Nonlinear periodic convection in double-diffusive systems. J. Fluid Mech. 108, 291–316 (1981).

279. Cross, M. C. & Kim, K. Linear instability and the codimension-2 region in binary fluid convection between rigid impermeable boundaries. Phys. Rev. A 37, 3909–3920 (1988).

280. Cross, M. C. Traveling and standing waves in binary-fluid convection in finite geometries. Phys. Rev. Lett. 57, 2935–2938 (1986).

281. Coullet, P. H. & Spiegel, E. A. Amplitude equations for systems with competing instabilities. SIAM J. Appl. Math. 43, 776–821 (1983).

282. Cross, M. C. Structure of nonlinear traveling-wave states in finite geometries. Phys. Rev. A 38, 3593–3600 (1988).

283. Brand, H. R., Hohenberg, P. C. & Steinberg, V. Amplitude equation near a polycritical point for the convective instability of a binary fluid mixture in a porous medium. Phys. Rev. A 27, 591–593 (1983).

284. Brand, H. R., Hohenberg, P. C. & Steinberg, V. Codimension-2 bifurcations for convection in binary fluid mixtures. Phys. Rev. A 30, 2548–2561 (1984).

285. Guckenheimer, J. Multiple bifurcation problems of codimension two. SIAM J. Math. Anal. 15, 1–49 (1984).

286. Moses, E. & Steinberg, V. Flow patterns and nonlinear behavior of traveling waves in a convective binary fluid. Phys. Rev. A 34, 693–696 (1986); erratum 35, 1444–1445 (1987).

287. Walden, R. W., Kolodner, P., Passner, A. & Surko, C. M. Traveling waves and chaos in convection in binary fluid mixtures. Phys. Rev. Lett. 55, 496–499 (1985).

288. Coullet, P., Fauve, S. & Tirapegui, E. Large scale instability of nonlinear standing waves. J. Physique Lett. 46, 787–791 (1985).

289. Bensimon, D., Pumir, A. & Shraiman, B. Nonlinear theory of traveling wave convection in binary mixtures. J. Phys. France 50, 3089–3108 (1989).

290. Knobloch, E. & Moore, D. R. Minimal model of binary fluid convection. Phys. Rev. A 42, 4693–4709 (1990).

291. Schöpf, W. & Zimmermann, W. Convection in binary fluids: amplitude equations, codimension-2 bifurcation, and thermal fluctuations. Phys. Rev. E 47, 1739–1764 (1993).

292. Bressloff, P. C., Cowan, J. D., Golubitsky, M., Thomas, P. J. & Wiener, M. C. Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex. Phil. Trans. R. Soc. Lond. B 356, 299–330 (2001).

293. Bressloff, P. C., Cowan, J. D., Golubitsky, M., Thomas, P. J. & Wiener, M. C. What geometric visual hallucinations tell us about the visual cortex. Neural Comput. 14, 473–491 (2002).

294. Cho, M. W. & Kim, S. Understanding visual map formation through vortex dynamics of spin Hamiltonian models. Phys. Rev. Lett. 92, 018101 (2004).

295. Schnabel, M., Kaschube, M. & Wolf, F. Pinwheel stability, pattern selection and the geometry of visual space. Preprint at https://arxiv.org/abs/0801.3832 (2008).

296. Butler, T. C. et al. Evolutionary constraints on visual cortex architecture from the dynamics of hallucinations. Proc. Natl Acad. Sci. USA 109, 606–609 (2012).

297. Curtu, R. & Ermentrout, B. Pattern formation in a network of excitatory and inhibitory cells with adaptation. SIAM J. Appl. Dyn. Syst. 3, 191–231 (2004).

298. Adini, Y., Sagi, D. & Tsodyks, M. Excitatory–inhibitory network in the visual cortex: psychophysical evidence. Proc. Natl Acad. Sci. USA 94, 10426–10431 (1997).

299. Hensch, T. K. & Fagiolini, M. in Progress in Brain Research (eds van Pelt, J. et al.) 115–124 (Elsevier, 2005).

300. Chossat, P. & Iooss, G. The Couette–Taylor Problem (Springer, 1994).

301. Riecke, H. & Paap, H.-G. Parity-breaking and Hopf bifurcations in axisymmetric Taylor vortex flow. Phys. Rev. A 45, 8605–8610 (1992).

302. Tennakoon, S. G. K., Andereck, C. D., Hegseth, J. J. & Riecke, H. Temporal modulation of traveling waves in the flow between rotating cylinders with broken azimuthal symmetry. Phys. Rev. E 54, 5053–5065 (1996).

303. Mutabazi, I. & Andereck, C. D. Mode resonance and wavelength-halving instability in the Taylor–Dean system. Phys. Rev. E 51, 4380–4390 (1995).

304. Bot, P., Cadot, O. & Mutabazi, I. Secondary instability mode of a roll pattern and transition to spatiotemporal chaos in the Taylor–Dean system. Phys. Rev. E 58, 3089–3097 (1998).

305. Wiener, R. J. & McAlister, D. F. Parity breaking and solitary waves in axisymmetric Taylor vortex flow. Phys. Rev. Lett. 69, 2915–2918 (1992).

306. Andereck, C. D., Liu, S. S. & Swinney, H. L. Flow regimes in a circular Couette system with independently rotating cylinders. J. Fluid Mech. 164, 155–183 (1986).

307. Altmeyer, S. & Hoffmann, C. Secondary bifurcation of mixed-cross-spirals connecting travelling wave solutions. New J. Phys. 12, 113035 (2010).

308. Pinter, A., Lücke, M. & Hoffmann, C. Competition between traveling fluid waves of left and right spiral vortices and their different amplitude combinations. Phys. Rev. Lett. 96, 044506 (2006).

309. Hong, H. Periodic synchronization and chimera in conformist and contrarian oscillators. Phys. Rev. E 89, 062924 (2014).

310. Kemeth, F. P., Haugland, S. W., Schmidt, L., Kevrekidis, I. G. & Krischer, K. A classification scheme for chimera states. Chaos 26, 094815 (2016).

311. Golubitsky, M. & Stewart, I. Hopf bifurcation in the presence of symmetry. Arch. Ration. Mech. Anal. 87, 107–165 (1985).

312. Shapere, A. & Wilczek, F. Classical time crystals. Phys. Rev. Lett. 109, 160402 (2012).

313. Wilczek, F. Quantum time crystals. Phys. Rev. Lett. 109, 160401 (2012).

314. Yao, N. Y. & Nayak, C. Time crystals in periodically driven systems. Phys. Today 71, 40 (2018).

315. Prigogine, I. & Lefever, R. Symmetry-breaking instabilities in dissipative systems. II. J. Chem. Phys. 48, 1695–1700 (1968).

316. Giergiel, K., Miroszewski, A. & Sacha, K. Time crystal platform: from quasicrystal structures in time to systems with exotic interactions. Phys. Rev. Lett. 120, 140401 (2018).

317. Autti, S., Eltsov, V. & Volovik, G. Observation of a time quasicrystal and its transition to a superfluid time crystal. Phys. Rev. Lett. 120, 215301 (2018).

## Acknowledgements

We thank A. Alù, D. Bartolo, D. Christodoulides, A. Clerk, A. Edelman, A. Galda, M. Han, K. Husain, T. Kottos, Z. Lu, M. C. Marchetti, M.-A. Miri, B. Roussel, C. Scheibner, D. Schuster, J. Simon and B. van Zuiden. M.F. acknowledges support from a MRSEC-funded Kadanoff–Rice fellowship (DMR-2011854) and the Simons Foundation. R.H. was supported by a Grand-in-Aid for JSPS fellows (grant number 17J01238). V.V. was supported by the Complex Dynamics and Systems Program of the Army Research Office under grant number W911NF-19-1-0268 and the Simons Foundation. This work was partially supported by the University of Chicago Materials Research Science and Engineering Center, which is funded by National Science Foundation under award number DMR-2011854. This work was completed in part with resources provided by the University of Chicago’s Research Computing Center. Some of us benefited from participation in the KITP programme on Symmetry, Thermodynamics and Topology in Active Matter supported by grant number NSF PHY-1748958.

## Author information

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### Contributions

M.F., R.H., P.B.L. and V.V. designed the research, performed the research, and wrote the paper.

### Corresponding author

Correspondence to Vincenzo Vitelli.

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## Extended data figures and tables

### Extended Data Fig. 1 Codimensions of eigenvalue degeneracies.

This graph gives the codimension (codim) of twofold degeneracies of eigenvalues in different matrix spaces; see ref. 194. These degeneracies can be exceptional points (EP) or diabolic points (DP, also known as Dirac points). An identical graph can be drawn by replacing ‘real symmetric’ with ‘purely imaginary symmetric’, ‘Hermitian’ with ‘anti-Hermitian’ and ‘real’ with ‘imaginary’.

### Extended Data Fig. 2 Many-body suppression of noise-activated chirality inversions.

a, A change in the sign of the angle Δϕ between the order parameters vA and vB (in blue and red) flips the chirality (clockwise or anticlockwise) of the chiral phase. Qualitatively, the two steady-state values ±Δϕc towards which the system relax correspond to the minima of an effective potential Uϕ), with a barrier ΔU separating these minima. The lifetime of the chiral phase is the average time τ separating two flips of chirality (represented in green), namely the Kramers escape time required to jump from one minimum to the other under the effect of noise. b, The standard deviations quantifying the fluctuations of the order parameters va in the chiral phase decrease approximately as $$1/\sqrt{N}$$ with the number of agents N. The grey lines are equally spaced $$1/\sqrt{N}$$ curves and are meant as a guide to the eye (not a fit). The data are obtained from simulations of the Kuramoto model equation (2) with JAA = JBB = 1,  JAB = 1,  JBA = −1.1, η = 8 × 10−2 and all-to-all couplings. The total duration is Tsimt = 4,000 with δt = 0.5, over which the standard deviation is computed.

### Extended Data Fig. 3 Effect of non-conservative forces.

In this simplified pictorial representation, the order parameter (represented by a ball) evolves in a potential-energy landscape shaped like a sombrero. In a conservative system, the order parameter would relax straight to the bottom of the potential (dashed blue line). Here, transverse non-conservative forces push the order parameter in the direction defined by the bottom of the potential, leading to a curved trajectory (red continuous line) starting from the same initial condition. In the systems we considered, the non-conservative forces arise from the non-reciprocal coupling between two order parameters. This aspect is not captured by this simplified picture.

### Extended Data Fig. 4 Phase diagram of the PT-symmetric non-reciprocal Kuramoto model and exceptional point in the spectrum of the Jacobian.

a, Phase diagram computed numerically from equation (15). The states are defined in Extended Data Table 1. b, The two most unstable eigenvalues λi = σi + iωi of L coalesce at  j+ ≈ 0.007. This value coincides with the transition from travelling waves (TW) to coherent states, marked by a red dashed line. Note that this coalescence occurs at λ = 0 (not at finite frequency nor at finite growth rate). The corresponding eigenvectors become collinear (this can be verified, for instance, by computing the determinant of the matrix of eigenvectors, that vanishes at the exceptional point). The imaginary parts ωi (not shown) are all zero. We have set  jAA = jBB = 1, ΔA = ΔB = 0.25 and ωA = ωB = 0. In b,  j = 0.1 (a similar behaviour is observed for neighbouring values of  j).

### Extended Data Fig. 5 Hysteresis in the chiral Kuramoto model.

When chirality is explicitly broken, exceptional points have codimension two, that is, they are typically points in a two-dimensional parameter space. a, We plot the frequency Ω of the steady state of the Kuramoto model with explicitly broken PT symmetry as a function of the difference Δω = ωA − ωB between the two communities (also called detuning) and the deviation $${\rm{\delta }}{j}_{-}={j}_{-}-{j}_{-}^{{\rm{E}}{\rm{P}}}$$ of the non-reciprocal part  j of the coupling between the communities from its value $${j}_{-}^{{\rm{EP}}}$$ at the exceptional point. The system exhibits a region where two possible steady states with different properties coexist (the two steady states are the continuation of the clockwise and anticlockwise chiral phases present in the PT-symmetric case Δω = 0). This region (red triangle) starts at the exceptional point (red point) and its size increases with the amount of non-reciprocity (here $${j}_{-}^{{\rm{EP}}}\approx 0.2915 > 0$$). The system exhibits hysteresis in the coexistence region (red points). be, Slices from a at fixed δj (marked by green dotted lines in a). After the exceptional point, there is hysteresis/first-order (discontinuous) behaviour. In d, the hysteresis curve bends outwards near the transition. This is due to the oscillation of the norm of the order parameter (which we refer to as swap or periodic synchronization elsewhere) for large enough δj. This additional complication does not occur for moderate values of δj, such as in c. The solution of the dynamical system equation (15) were computed along lines at fixed δj, starting at large |δω| (in a region without phase coexistence) from a random initial condition. The solution (after convergence) was used as an initial value for the next point in the line with fixed δj. This procedure was carried out two times, starting from positive and negative large |δω|. We have set  j+ = 0.08,  jAA = jBB = 1, ΔA = ΔB = 0.25, ωA = ωB = Δω/2.

### Extended Data Fig. 6 Non-reciprocal pattern formation.

We show a space-time density plot of the field u1(xt) in different phases, as well as snapshots of the fields u1(xt) and u2(xt) at time t = 200. a, We observe a disordered phase where both field vanish. b, An aligned phase where both patterns are static and in phase (superimposed). c, An antialigned phase where the patterns are static and completely out of phase. d, A chiral phase where the patterns move at constant velocity, either to the left or to the right (spontaneously breaking parity), and in which the fields have a finite phase difference, usually neither zero nor π. e, A swap phase where the patterns essentially jump by a phase π every period. f, A mix of the chiral and swap behaviours (as in the chiral phase (d), there is a spontaneously broken symmetry between left and right movers). The fields are obtained by direct numerical simulation of the coupled Swift–Hohenberg equations on a one-dimensional domain of size 2L with periodic boundary conditions, starting from random initial conditions. The simulations are performed using the open-source pseudospectral solver Dedalus219. We have used g = 0.25 in all cases. In a, r11 = r22 = −0.5 and r+ = r = 0.00. In the other cases, we have set r11 = r22 = 0.5 (bf) and r+ = 0.50, r = 0.00 (b); r+ = −0.50, r = 0.00 (c); r+ = 0.00, r = 0.25 (d); r+ = 0.87, r = 1.00 (e); r+ = 0.85, r = 1.00 (f).

### Extended Data Fig. 7 Exceptional point in directional interface growth.

The spectrum of the Jacobian L corresponding to equation (20) exhibits an exceptional point at the transition between static patterns and travelling patterns with spontaneous parity breaking (that is, the patterns travel with equal probability to the left or to the right). The two most unstable eigenvalues λi = σi + iωi of L coalesce at μ1 ≈ 0.064 (red circle). This value coincides with the transition from a constant solution to travelling waves, marked by a red dashed line. The coalescence occurs at λ = 0 (not at finite frequency nor at finite growth rate), and the corresponding eigenvectors become collinear. Note that another exceptional point occurs near μ1 ≈ 0.014 (green circle), but with a strictly negative growth rate: this does not correspond to a bifurcation. We also show the dephasing Δϕ = 2ϕ1 − ϕ2 between the amplitudes, which undergoes a pitchfork bifurcation; the direction of motion of the pattern is set by Δϕ. We have set α = β = γ = δ = 1, ε = +1 and μ2 = −0.1.

## Supplementary information

### Supplementary Information

This file contains derivations of the results presented in the main text and additional discussions.

### Supplementary Video 1

Demonstration with programmable robots. See Sec. XIV of the Supplementary Notes.

### Supplementary Video 2

Evolution of the order parameters vA (blue) and vB (red) in the time-dependent phases (chiral, swap, chiral+swap) computed from the dynamical system Eq. (S84).

### Supplementary Video 3

Molecular dynamics simulation of the microscopic non-reciprocal flocking model. See Sec. V B of the Supplementary Notes. In the bottom part, we show the instantaneous order parameter obtained by averaging the individual directions of the self-propelled particles.

### Supplementary Video 4

Numerical simulation of the hydrodynamic field theory showing pattern formation at fixed density when the incompressibility constraint is not enforced. See Sec. XV of the Supplementary Notes.

### Supplementary Video 5

Numerical simulation of the hydrodynamic field theory showing pattern formation with the incom- pressibility constraint enforced. See Sec. XV B of the Supplementary Notes.

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Fruchart, M., Hanai, R., Littlewood, P.B. et al. Non-reciprocal phase transitions. Nature 592, 363–369 (2021). https://doi.org/10.1038/s41586-021-03375-9

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• DOI: https://doi.org/10.1038/s41586-021-03375-9

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