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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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Fig. 1: Exceptional transitions: examples and mechanism.
Fig. 2: Phase diagrams and active time (quasi)crystals.
Fig. 3: Exceptional-point-enforced pattern formation and topological defects.
Fig. 4: A visual procedure to identify and analyse exceptional transitions.

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).

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  PubMed  Google Scholar 

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

    Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

  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).

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  PubMed  PubMed Central  Google Scholar 

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

    Article  MathSciNet  CAS  MATH  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  Google Scholar 

  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).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  CAS  MATH  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

  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).

    CAS  Google Scholar 

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

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  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).

    Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  ADS  MathSciNet  Google Scholar 

  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).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    CAS  Google Scholar 

  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).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  PubMed  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  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).

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  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).

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Google Scholar 

  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  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).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  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).

    Article  ADS  PubMed  Google Scholar 

  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).

    Article  CAS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  ADS  Google Scholar 

  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).

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

  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).

    Article  ADS  PubMed  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  PubMed  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

  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).

    Article  ADS  MathSciNet  Google Scholar 

  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).

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

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

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  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).

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  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).

    Article  ADS  Google Scholar 

  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).

    Article  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

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

    Article  CAS  Google Scholar 

  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).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  PubMed  PubMed Central  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

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

    Article  CAS  Google Scholar 

  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).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  MATH  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

  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).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  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).

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  PubMed  PubMed Central  Google Scholar 

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

    Article  CAS  Google Scholar 

  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).

    Article  CAS  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  CAS  PubMed  MATH  Google Scholar 

  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).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  Google Scholar 

  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).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  PubMed  MATH  Google Scholar 

  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).

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  MathSciNet  PubMed  MATH  Google Scholar 

  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).

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  MathSciNet  PubMed  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  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).

    Article  ADS  MathSciNet  PubMed  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Google Scholar 

  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).

    Article  MathSciNet  PubMed  PubMed Central  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  PubMed  PubMed Central  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  ADS  Google Scholar 

  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).

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  ADS  CAS  PubMed  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  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).

    Article  CAS  Google Scholar 

  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).

    Article  PubMed  MATH  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

  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).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  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).

    Article  ADS  CAS  PubMed  Google Scholar 

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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.

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

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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.

Extended Data Table 1 An O(2) ‘Rosetta stone’

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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