* [279] K. Yu. Guslienko, S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Effective dipolar boundary conditions for dynamic magnetization in thin magnetic stripes, Phys. Rev. B 66, 132402 (2002).
  * [280] V. Vlaminck and M. Bailleul, Spin-wave transduction at the submicrometer scale: Experiment and modeling, Phys. Rev. B 81, 014425 (2010).
  * [281] F. Zighem, Y. Roussigne, S.-M. Cherif, and P. Moch, Spin wave modelling in arrays of ferromagnetic thin stripes: application to Brillouin light scattering in permalloy, J. Phys.: Condens. Matter 19, 176220 (2007).
  * [282] R. E. De Wames and T. Wolfram, A new type of surface spin wave in ferromagnetic films, Appl. Phys. Lett. 15, 297 (1969).
  * [283] T. Wolfram and R. E. De Wames, Macroscopic and Microscopic Theories of Dipole-Exchange Spin Waves in Thin Films: Case of the Missing Surface States, Phys. Rev. Lett. 24, 1489 (1970).
  * [284] R. L. Stamps and B. Hillebrands, Dipolar interactions and the magnetic behavior of two-dimensional ferromagnetic systems, Phys. Rev. B 44, 12417 (1991).
  * [285] R. L. Stamps, Spin configurations and spin-wave excitations in exchange-coupled bilayers, Phys. Rev. B 49, 339 (1994).
  * [286] M. Kostylev, Non-reciprocity of dipole-exchange spin waves in thin ferromagnetic films, J. Appl. Phys. 113, 053907 (2013).
  * [287] J. Mendil, M. Trassin, Q. Bu, J. Schaab, M. Baumgartner, C. Murer, P. T. Dao, J. Vijayakumar, D. Bracher, C. Bouillet, C. A. F. Vaz, M. Fiebig, and P. Gambardella, Magnetic properties and domain structure of ultrathin yttrium iron garnet/Pt bilayers, Phys. Rev. Mater. 3, 034403 (2019).
  * [288] J. Liu, X-Y. Wei, G. E. W. Bauer, J. Ben Youssef, and B. J. van Wees, Electrically induced strong modulation of magnon transport in ultrathin magnetic insulator films, Phys. Rev. B 103, 214425 (2021).
  * [289] N. I. Polushkin, Combined Electron Resonance Driven by an All-Oscillating Potential of Patterned Magnets, Phys. Rev. Lett. 103, 077201 (2009).
  * [290] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Enhanced Gilbert Damping in Thin Ferromagnetic Films, Phys. Rev. Lett. 88, 117601 (2002).
  * [291] E. Šimánek and B. Heinrich, Gilbert damping in magnetic multilayers, Phys. Rev. B 67, 144418 (2003).
  * [292] Y. Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa, Enhanced dc spin pumping into a fluctuating ferromagnet near ${T}_{C}$, Phys. Rev. B 89, 174417 (2014).
  * [293] G. F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, 2005).
  * [294] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Introduction to quantum noise, measurement, and amplification, Rev. Mod. Phys. 82, 1155 (2010).
  * [295] C. W. Gardiner and M. J. Collett, Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation, Phys. Rev. A 31, 3761 (1985).
  * [296] J. L. Chen, C. P. Liu, T. Liu, Y. Xiao, K. Xia, G. E. W. Bauer, M. Z. Wu, and H. M. Yu, Strong Interlayer Magnon-Magnon Coupling in Magnetic Metal-Insulator Hybrid Nanostructures, Phys. Rev. Lett. 120, 217202 (2018).
  * [297] L. Temdie, V. Castel, C. Dubs, G. Pradhan, J. Solano, H. Majjad, R. Bernard, Y. Henry, M. Bailleul, and V. Vlaminck, High wave vector non-reciprocal spin wave beams, arXiv:2211.05514.
  * [298] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Spin Seebeck insulator, Nat. Mater. 9, 894 (2010).
  * [299] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and S. Maekawa, Theory of magnon-driven spin Seebeck effect, Phys. Rev. B 81, 214418 (2010).
  * [300] H. Adachi, J. Ohe, S. Takahashi, and S. Maekawa, Linear-response theory of spin Seebeck effect in ferromagnetic insulators, Phys. Rev. B 83, 094410 (2011).
  * [301] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Spin caloritronics, Nat. Mater. 11, 391 (2012).
  * [302] L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssef, and B. J. van Wees, Long-distance transport of magnon spin information in a magnetic insulator at room temperature, Nat. Phys. 11, 1022 (2015).
  * [303] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Magnon spin transport driven by the magnon chemical potential in a magnetic insulator, Phys. Rev. B 94, 014412 (2016).
  * [304] J. H. Han, Y. B. Fan, B. C. McGoldrick, J. Finley, J. T. Hou, P. X. Zhang, and L. Q. Liu, Nonreciprocal Transmission of Incoherent Magnons with Asymmetric Diffusion Length, Nano Lett. 21, 7037 (2021).
  * [305] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Magnon spintronics, Nat. Phys. 11, 453 (2015).
  * [306] D. Grundler, Nanomagnonics around the corner, Nat. Nanotechnol. 11, 407 (2016).
  * [307] V. E. Demidov, S. Urazhdin, G. de Loubens, O. Klein, V. Cros, A. Anane, and S. O. Demokritov, Magnetization oscillations and waves driven by pure spin currents, Phys. Rep. 673, 1 (2017).
  * [308] A. Brataas, B. van Wees, O. Klein, G. de Loubens, and M. Viret, Spin insulatronics, Phys. Rep. 885, 1 (2020).
  * [309] A. Barman et al., The 2021 Magnonics Roadmap, J. Phys.: Condens. Matter 33, 413001 (2021).
  * [310] A. V. Chumak, A. A. Serga, and B. Hillebrands, Magnon transistor for all-magnon data processing, Nat. Commun. 5, 4700 (2014).
  * [311] H. Wu, L. Huang, C. Fang, B. S. Yang, C. H. Wan, G. Q. Yu, J. F. Feng, H. X. Wei, and X. F. Han, Magnon Valve Effect between Two Magnetic Insulators, Phys. Rev. Lett. 120, 097205 (2018).
  * [312] J. Cramer, F. Fuhrmann, U. Ritzmann, V. Gall, T. Niizeki, R. Ramos, Z. Y. Qiu, D. Z. Hou, T. Kikkawa, J. Sinova, U. Nowak, E. Saitoh, and M. Kläui, Magnon detection using a ferroic collinear multilayer spin valve, Nat. Commun. 9, 1089 (2018).
  * [313] L. J. Cornelissen, J. Liu, B. J. van Wees, and R. A. Duine, Spin-Current-Controlled Modulation of the Magnon Spin Conductance in a Three-Terminal Magnon Transistor, Phys. Rev. Lett. 120, 097702 (2018).
  * [314] T. Wimmer, M. Althammer, L. Liensberger, N. Vlietstra, S. Geprägs, M. Weiler, R. Gross, and H. Huebl, Spin Transport in a Magnetic Insulator with Zero Effective Damping, Phys. Rev. Lett. 123, 257201 (2019).
  * [315] J. Liu, X.-Y. Wei, G. E. W. Bauer, J. Ben Youssef, and B. J. van Wees, Electrically induced strong modulation of magnon transport in ultrathin magnetic insulator films, Phys. Rev. B 103, 214425 (2021).
  * [316] S. Klingler, V. Amin, S. Geprägs, K. Ganzhorn, H. Maier-Flaig, M. Althammer, H. Huebl, R. Gross, R. D. McMichael, M. D. Stiles, S. T. B. Goennenwein, and M. Weiler, Spin-Torque Excitation of Perpendicular Standing Spin Waves in Coupled YIG/Co Heterostructures, Phys. Rev. Lett. 120, 127201 (2018).
  * [317] X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, Spin transfer torque on magnetic insulators, Europhys. Lett. 96, 17005 (2011).
  * [318] A. Asenjo-García, M. Moreno-Cardoner, A. Albrecht, H. J. Kimble, and D. E. Chang, Exponential Improvement in Photon Storage Fidelities Using Subradiance and “Selective Radiance" in Atomic Arrays, Phys. Rev. X 7, 031024 (2017).
  * [319] R. A. Gallardo, T. Schneider, A. K. Chaurasiya, A. Oelschlägel, S. S. P. K. Arekapudi, A. Roldán-Molina, R. Hübner, K. Lenz, A. Barman, J. Fassbender, J. Lindner, O. Hellwig, and P. Landeros, Reconfigurable Spin-Wave Nonreciprocity Induced by Dipolar Interaction in a Coupled Ferromagnetic Bilayer, Phys. Rev. Appl. 12, 034012 (2019).
  * [320] M. Grassi, M. Geilen, D. Louis, M. Mohseni, T. Brächer, M. Hehn, D. Stoeffler, M. Bailleul, P. Pirro, and Y. Henry, Slow-Wave-Based Nanomagnonic Diode, Phys. Rev. Appl. 14, 024047 (2020).
  * [321] P. Grünberg, Some ways to modify the spin-wave mode spectra of magnetic multilayers (invited), J. Appl. Phys. 57, 3673 (1985).
  * [322] P. Grünberg, R. Schreiber, Y. Pang, M. B. Brodsky, and H. Sowers, Layered Magnetic Structures: Evidence for Antiferromagnetic Coupling of Fe Layers across Cr Interlayers, Phys. Rev. Lett. 57, 2442 (1986).
  * [323] P. X. Zhang and W. Zinn, Spin-wave modes in anti-parallel magnetized ferromagnetic double layers, Phys. Rev. B 35, 5219 (1987).
  * [324] G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange, Phys. Rev. B 39, 4828(R) (1989).
  * [325] M. Vohl, J. Barnaś, and P. Grünberg, Effect of interlayer exchange coupling on spin-wave spectra in magnetic double layers: Theory and experiment, Phys. Rev. B 39, 12003 (1989).
  * [326] Z. Q. Hu, L. Fu, and L. Q. Liu, Tunable Magnonic Chern Bands and Chiral Spin Currents in Magnetic Multilayers, Phys. Rev. Lett. 128, 217201 (2022).
  * [327] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42, 1698 (1979).
  * [328] B. J. Chapman, E. I. Rosenthal, J. Kerckhoff, B. A. Moores, L. R. Vale, J. A. B. Mates, G. C. Hilton, K. Lalumiére, A. Blais, and K. W. Lehnert, Widely Tunable On-Chip Microwave Circulator for Superconducting Quantum Circuits, Phys. Rev. X 7, 041043 (2017).
  * [329] N. R. Bernier, L. D. Tóth, A. Koottandavida, M. A. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, Nonreciprocal reconfigurable microwave optomechanical circuit, Nat. Commun. 8, 604 (2017).
  * [330] G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, Demonstration of Efficient Nonreciprocity in a Microwave Optomechanical Circuit, Phys. Rev. X 7, 031001 (2017).
  * [331] F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Aumentado, J. D. Teufel, and J. Aumentado, Nonreciprocal Microwave Signal Processing with a Field-Programmable Josephson Amplifier, Phys. Rev. Appl. 7, 024028 (2017).
  * [332] M. Shalaby, M. Peccianti, Y. Ozturk, and R. Morandotti, A magnetic non-reciprocal isolator for broadband terahertz operation, Nat. Commun. 4, 1558 (2013).
  * [333] C. L. Hogan, The ferromagnetic Faraday effect at microwave frequencies and its applications, Rev. Mod. Phys. 25, 253 (1953).
  * [334] J. H. Rowen, Ferrites in microwave applications, Bell Syst. Technol. J. 32, 1333 (1953).
  * [335] J. D. Adam, L. E. Davis, G. F. Dionne, E. F. Schloemann, and S. N. Stitzer, Ferrite devices and materials, IEEE Trans. Microwave Theory Tech. 50, 721 (2002).
  * [336] B. K. Kuanr, V. Veerakumar, R. Marson, S. R. Mishra, R. E. Camley, and Z. Celinski, Nonreciprocal microwave devices based on magnetic nanowires, Appl. Phys. Lett. 94, 202505 (2009).
  * [337] M.-A. Miri, F. Ruesink, E. Verhagen, and A. Alú, Optical Nonreciprocity Based on Optomechanical Coupling, Phys. Rev. Appl. 7, 064014 (2017).
  * [338] Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, Experimental realization of optomechanically induced non-reciprocity, Nat. Photonics 10, 657 (2016).
  * [339] K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, Generalized nonreciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering, Nat. Phys. 13, 465 (2017).
  * [340] A. Metelmann and A. A. Clerk, Nonreciprocal Photon Transmission and Amplification via Reservoir Engineering, Phys. Rev. X 5, 021025 (2015).
  * [341] D. L. Sounas and A. Alú, Non-reciprocal photonics based on time modulation, Nat. Photonics 11, 774 (2017).
  * [342] H. Ramezani, P. K. Jha, Y. Wang, and X. Zhang, Nonreciprocal Localization of Photons, Phys. Rev. Lett. 120, 043901 (2018).
  * [343] F. L. Kien, S. D. Gupta, K. P. Nayak, and K. Hakuta, Nanofiber-mediated radiative transfer between two distant atoms, Phys. Rev. A 72, 063815 (2005).
  * [344] M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, Quantum optical circulator controlled by a single chirally coupled atom, Science 354, 1577 (2016).
  * [345] B. Vermersch, P.-O. Guimond, H. Pichler, and P. Zoller, Quantum State Transfer via Noisy Photonic and Phononic Waveguides, Phys. Rev. Lett. 118, 133601 (2017).
  * [346] P. Schneeweiss, S. Zeiger, T. Hoinkes, A. Rauschenbeutel, and J. Volz, Fiber ring resonator with a nanofiber section for chiral cavity quantum electrodynamics and multimode strong coupling, Opt. Lett. 42, 85 (2017).
  * [347] C. A. Downing, J. C. López Carreño, F. P. Laussy, E. del Valle, and A. I. Fernández-Domínguez, Quasichiral Interactions between Quantum Emitters at the Nanoscale, Phys. Rev. Lett. 122, 057401 (2019).
  * [348] H. J. Qin, R. B. Hollánder, L. Flajšman, F. Hermann, R. Dreyer, G. Woltersdorf, and S. van Dijken, Nanoscale magnonic Fabry-Pérot resonator for low-loss spin-wave manipulation, Nat. Commun. 12, 2293 (2021).
  * [349] S. R. Seshadri, Surface magnetostatic modes of a ferrite slab, Proc. IEEE 58, 506 (1970).
  * [350] M. L. Sokolovskyy, J. W. Klos, S. Mamica, and M. Krawczyk, Calculation of the spin-wave spectra in planar magnonic crystals with metallic overlayers, J. Appl. Phys. 111, 07C515 (2012).
  * [351] M. Mruczkiewicz, M. Krawczyk, G. Gubbiotti, S. Tacchi, Y. A. Filimonov, D. V. Kalyabin, I. V. Lisenkov, and S. A. Nikitov, Nonreciprocity of spin waves in metallized magnonic crystal, New J. Phys. 15 113023 (2013).
  * [352] M. Mruczkiewicz and M. Krawczyk, Nonreciprocal dispersion of spin waves in ferromagnetic thin films covered with a finite-conductivity metal, J. Appl. Phys. 115, 113909 (2014).
  * [353] M. Mruczkiewicz, E. S. Pavlov, S. L. Vysotsky, M. Krawczyk, Y. A. Filimonov, and S. A. Nikitov, Observation of magnonic band gaps in magnonic crystals with nonreciprocal dispersion relation, Phys. Rev. B 90, 174416 (2014).
  * [354] M. Mruczkiewicz, P. Graczyk, P. Lupo, A. Adeyeye, G. Gubbiotti, and M. Krawczyk, Spin-wave nonreciprocity and magnonic band structure in a thin permalloy film induced by dynamical coupling with an array of Ni stripes, Phys. Rev. B 96, 104411 (2017).
  * [355] X. Y. Wei, O. A. Santos, C. H. S. Lusero, G. E. W. Bauer, J. B. Youssef, and B. J. van Wees, Giant magnon spin conductivity approaching the two-dimensional transport regime in ultrathin yttrium iron garnet films, Nat. Mat. 21, 1352 (2022).
  * [356] P. Pincus, Excitation of spin waves in ferromagnets: eddy current and boundary condition effects, Phys. Rev. 118, 658 (1960).
  * [357] M. Kostylev, Strong asymmetry of microwave absorption by bilayer conducting ferromagnetic films in the microstrip-line based broadband ferromagnetic resonance, J. Appl. Phys. 106, 043903 (2009).
  * [358] M. A. Schoen, J. M. Shaw, H. T. Nembach, M. Weiler, and T. J. Silva, Radiative damping in waveguide-based ferromagnetic resonance measured via analysis of perpendicular standing spin waves in sputtered permalloy films, Phys. Rev. B 92, 184417 (2015).
  * [359] Y. Li and W. E. Bailey, Wave-Number-Dependent Gilbert Damping in Metallic Ferromagnets, Phys. Rev. Lett. 116, 117602 (2016).
  * [360] M. Kostylev, Coupling of microwave magnetic dynamics in thin ferromagnetic films to stripline transducers in the geometry of the broadband stripline ferromagnetic resonance, J. Appl. Phys. 119, 013901 (2016).
  * [361] J. W. Rao, S. Kaur, X. L. Fan, D. S. Xue, B. M. Yao, Y. S. Gui, and C. M. Hu, Characterization of the non-resonant radiation damping in coupled cavity photon magnon system, Appl. Phys. Lett. 110, 262404 (2017).
  * [362] S. A. Bunyaev, R. O. Serha, H. Y. Musiienko-Shmarova, A. J. Kreil, P. Frey, D. A. Bozhko, V. I. Vasyuchka, R. V. Verba, M. Kostylev, B. Hillebrands, G. N. Kakazei, and A. A. Serga, Spin-wave relaxation by Eddy Currents in Y3Fe5O12/Pt bilayers and a way to suppress it, Phys. Rev. Appl. 14, 024094 (2020).
  * [363] I. Bertelli, B. G. Simon, T. Yu, J. Aarts, G. E. W. Bauer, Y. M. Blanter, and T. van der Sar, Imaging spin-wave damping underneath metals using electron spins in diamond, Advanced Quantum Technologies 4, 2100094 (2021).
  * [364] I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov, V. V. Bolginov, V. V. Ryazanov, A. A. Golubov, and A. V. Ustinov, Ferromagnet/Superconductor Hybridization for Magnonic Applications, Adv. Funct. Mater. 28, 1802375 (2018).
  * [365] I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov, V. V. Ryazanov, A. A. Golubov, and A. V. Ustinov, Modified dispersion law for spin waves coupled to a superconductor, J. Appl. Phys. 124, 233903 (2018).
  * [366] I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov, P. S. Dzhumaev, O. V. Emelyanova, A. A. Golubov, V. V. Ryazanov, and A. V. Ustinov, Ferromagnet/Superconductor Hybrid Magnonic Metamaterials, Adv. Sci. 6, 1900435 (2019).
  * [367] I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov, A. A. Golubov, V. V. Ryazanov, and A. V. Ustinov, Nonlinear spin waves in ferromagnetic/superconductor hybrids, J. Appl. Phys. 127, 093903 (2020).
  * [368] S. P. Chockalingam, M. Chand, J. Jesudasan, V. Tripathi, and P. Raychaudhuri, Superconducting properties and Hall effect of epitaxial NbN thin films, Phys. Rev. B 77, 214503 (2008).
  * [369] K. G. Fripp, A. V. Shytov, and V. V. Kruglyak, Spin-wave control using dark modes in chiral magnonic resonators, Phys. Rev. B 104, 054437 (2021).
  * [370] J. Lan, W. C. Yu, R. Q. Wu, and J. Xiao, Spin-Wave Diode, Phys. Rev. X 5, 041049 (2015).
  * [371] M. A. Kuznetsov and A. A. Fraerman, Temperature-sensitive spin-wave nonreciprocity induced by interlayer dipolar coupling in ferromagnet/paramagnet and ferromagnet/superconductor hybrid systems, Phys. Rev. B 105, 214401 (2022).
  * [372] M. Silaev, Anderson-Higgs Mass of Magnons in Superconductor-Ferromagnet-Superconductor Systems, Phys. Rev. Appl. 18, L061004 (2022).
  * [373] P. W. Anderson and H. Suhl, Instability in the Motion of Ferromagnets at High Microwave Power Levels, Phys. Rev. 100, 1788 (1955).
  * [374] H. Suhl, The theory of ferromagnetic resonance at high signal powers, J. Phys. Chem. Solids 1, 209 (1957).
  * [375] A. V. Chumak, P. Dhagat, A. Jander, A. A. Serga, and B. Hillebrands, Reverse Doppler effect of magnons with negative group velocity scattered from a moving Bragg grating, Phys. Rev. B 81, 140404(R) (2010).
  * [376] Ya. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Modification of the Landau-Lifshitz equation in the presence of a spin-polarized current in colossal- and giant-magnetoresistive materials, Phys. Rev. B 57, R3213(R) (1998).
  * [377] J. Fernández-Rossier, M. Braun, A. S. Núñez, and A. H. MacDonald, Influence of a uniform current on collective magnetization dynamics in a ferromagnetic metal, Phys. Rev. B 69, 174412 (2004).
  * [378] V. Vlaminck and M. Bailleul, Current-Induced Spin-Wave Doppler Shift, Science 322, 410 (2008).
  * [379] R. J. Doornenbal, A. Roldán-Molina, A. S. Nunez, and R. A. Duine, Spin-Wave Amplification and Lasing Driven by Inhomogeneous Spin-Transfer Torques, Phys. Rev. Lett. 122, 037203 (2019).
  * [380] U. Fano, Effects of Configuration Interaction on Intensities and Phase Shifts, Phys. Rev. 124, 1866 (1961).
  * [381] B. M. Yao, T. Yu, Y. S. Gui, J. W. Rao, Y. T. Zhao, W. Lu, and C.-M. Hu, Coherent control of magnon radiative damping with local photon states, Commun. Phys. 2, 161 (2019).
  * [382] N. Bloembergen and R. V. Pound, Radiation Damping in Magnetic Resonance Experiments, Phys. Rev. 95, 8 (1954).
  * [383] R. W. Sanders, D. Paquette, V. Jaccarino, and S. M. Rezende, Radiation damping in magnetic resonance. II. Continuous-wave antiferromagnetic-resonance experiments, Phys. Rev. B 10, 132 (1974).
  * [384] J. Bourhill, W. C. Yu, V. Vlaminck, G. E. W. Bauer, G. Ruoso, and V. Castel, Experimental generation of circulating cavity magnon polaritons, arXiv:2208.13408.
  * [385] M. A. Ruderman and C. Kittel, Indirect Exchange Coupling of Nuclear Magnetic Moments by Conduction Electrons, Phys. Rev. 96, 99 (1954).
  * [386] T. Kasuya, A Theory of Metallic Ferro- and Antiferromagnetism on Zener’s Model, Prog. Theor. Phys. 16, 45 (1956).
  * [387] K. Yosida, Magnetic Properties of Cu-Mn Alloys, Phys. Rev. 106, 893 (1957).
  * [388] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Nonlocal magnetization dynamics in ferromagnetic heterostructures, Rev. Mod. Phys. 77, 1375 (2005).
  * [389] K. Mølmer, Y. Castin, and J. Dalibard, Monte Carlo wave-function method in quantum optics, J. Opt. Soc. Am. B 10, 524 (1993).
  * [390] C. W. Gardiner, Driving a quantum system with the output field from another driven quantum system, Phys. Rev. Lett. 70, 2269 (1993).
  * [391] M. Harder, Y. Yang, B. M. Yao, C. H. Yu, J. W. Rao, Y. S. Gui, R. L. Stamps, and C.-M. Hu, Level Attraction Due to Dissipative Magnon-Photon Coupling, Phys. Rev. Lett. 121, 137203 (2018).
  * [392] B. M. Yao, T. Yu, X. Zhang, W. Lu, Y. S. Gui, C.-M. Hu, and Y. M. Blanter, The microscopic origin of magnon-photon level attraction by traveling waves: Theory and experiment, Phys. Rev. B 100, 214426 (2019).
  * [393] T. Yu and B.-W. Zeng, Giant microwave sensitivity of a magnetic array by long-range chiral interaction driven skin effect, Phys. Rev. B 105, L180401 (2022).
  * [394] B.-W. Zeng and T. Yu, Radiation-free and non-Hermitian topology inertial defect states of on-chip magnons, Phys. Rev. Research 5, 013003 (2022).
  * [395] Y.-P. Wang and C. -M. Hu, Dissipative couplings in cavity magnonics, J. Appl. Phys. 127, 130901 (2020).
  * [396] M. Harder, B. M. Yao, Y. S. Gui, and C.-M. Hu, Coherent and Dissipative Cavity Magnonics, J. Appl. Phys. 129, 201101 (2021).
  * [397] B. Z. Rameshti, S. V. Kusminskiy, J. A. Haigh, K. Usami, D. Lachance-Quirion, Y. Nakamura, C.-M. Hu, H. X. Tang, G. E. W. Bauer, and Y. M. Blanter, Cavity Magnonics, Phys. Rep. 979, 1 (2022).
  * [398] M. M. Cardoner, D. Plankensteiner, L. Ostermann, D. E. Chang, and H. Ritsch, Subradiance-enhanced excitation transfer between dipole-coupled nanorings of quantum emitters, Phys. Rev. A 100, 023806 (2019).
  * [399] P.-O. Guimond, A. Grankin, D. V. Vasilyev, B. Vermersch, and P. Zoller, Subradiant Bell States in Distant Atomic Arrays, Phys. Rev. Lett. 122, 093601 (2019).
  * [400] G. Buonaiuto, R. Jones, B. Olmos, and I. Lesanovsky, Dynamical creation and detection of entangled many-body states in a chiral atom chain, New J. Phys. 21, 113021 (2019).
  * [401] X. Zhang, C.-L. Zou, N. Zhu, F. Marquardt, L. Jiang, and H. X. Tang, Magnon dark modes and gradient memory, Nat. Commun. 6, 8914 (2015).
  * [402] E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  * [403] N. Hatano and D. R. Nelson, Localization transitions in non-Hermitian quantum mechanics, Phys. Rev. Lett. 77, 570 (1996).
  * [404] S. Y. Yao and Z. Wang, Edge states and topological invariants of non-Hermitian systems, Phys. Rev. Lett. 121, 086803 (2018).
  * [405] S. Y. Yao, F. Song, and Z. Wang, Non-Hermitian chern bands, Phys. Rev. Lett. 121, 136802 (2018).
  * [406] F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems, Phys. Rev. Lett. 121, 026808 (2018).
  * [407] C. H. Lee, L. Li, and J. Gong, Hybrid Higher-Order Skin-Topological Modes in Nonreciprocal Systems, Phys. Rev. Lett. 123, 016805 (2019).
  * [408] Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Adv. Phys. 69, 249 (2020).
  * [409] Y. Xiong, Why does bulk boundary correspondence fail in some non-hermitian topological models, J. Phys. Commun. 2, 035043 (2018).
  * [410] X. Zhu, H. Wang, S. K. Gupta, H. Zhang, B. Xie, M. Lu, and Y. Chen, Photonic non-Hermitian skin effect and non-Bloch bulk-boundary correspondence, Phys. Rev. Research 2, 013280 (2020).
  * [411] L. Zhang, Y. Yang, Y. Ge, Y. J. Guan, Q. Chen, Q. Yan, F. Chen, R. X, Y. Li, D. Jia, S. Q. Yuan, H. X. Sun, H. Chen, and B. Zhang, Acoustic non-Hermitian skin effect from twisted winding topology, Nat Commun. 12, 6297 (2021).
  * [412] D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Non-Hermitian Boundary Modes and Topology, Phys. Rev. Lett. 124, 056802 (2020).
  * [413] D. C. Brody, Biorthogonal quantum mechanics, J. Phys. A 47, 035305 (2014).
  * [414] L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, Non-Hermitian bulk–boundary correspondence in quantum dynamics, Nat. Phys. 16, 761 (2020).
  * [415] J. C. Budich and E. J. Bergholtz, Non-Hermitian Topological Sensors, Phys. Rev. Lett. 125, 180403 (2020).
  * [416] S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, Topological funneling of light, Science 368, 311 (2020).
  * [417] T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits, Nat. Phys. 16, 747 (2020).
  * [418] A. Ghatak, M. Brandenbourger, J. van Wezel, and C. Coulais, Observation of non-Hermitian topology and its bulk–edge correspondence in an active mechanical metamaterial, Proc. Natl. Acad. Sci. 117, 29561 (2020).
  * [419] B. Flebus, R. A. Duine, and H. M. Hurst, Non-Hermitian topology of one-dimensional spin-torque oscillator arrays, Phys. Rev. B 102, 180408(R) (2020).
  * [420] K. Deng and B. Flebus, Non-Hermitian skin effect in magnetic systems, Phys. Rev. B 105, L180406 (2022).
  * [421] C. Kittel, Interaction of Spin Waves and Ultrasonic Waves in Ferromagnetic Crystals, Phys. Rev. 110, 836 (1958).
  * [422] Z. Tian, D. Sander, and J. Kirschner, Nonlinear magnetoelastic coupling of epitaxial layers of Fe, Co, and Ni on Ir(100), Phys. Rev. B 79, 024432 (2009).
  * [423] S. Streib, H. Keshtgar, and G. E. W. Bauer, Damping of Magnetization Dynamics by Phonon Pumping, Phys. Rev. Lett. 121, 027202 (2018).
  * [424] S. Maekawa and M. Tachiki, Surface acoustic attenuation due to surface spin wave in ferro- and antiferromagnets, AIP Conf. Proc. 29, 542 (1976).
  * [425] D. A. Garanin and E. M. Chudnovsky, Thermally activated resonant magnetization tunneling in molecular magnets: ${\mathrm{Mn}}_{12}\mathrm{Ac}$ and others, Phys. Rev. B 56, 11102 (1997).
  * [426] S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, Theoretical analysis of channel drop tunneling processes, Phys. Rev. B 59, 15882 (1999).
  * [427] Y. Xu, Y. Li, R. K. Lee, and A. Yariv, Scattering-theory analysis of waveguide-resonator coupling, Phys. Rev. E 62, 7389 (2000).
  * [428] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Prentice Hall, Englewood Cliffs, N. J., 1963).
  * [429] A. L. Fetter and J. D. Walecka, Quantum Theory of Many Particle Systems (McGraw-Hill, New York, 1971).
  * [430] H. Haug and A. P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors (Springer, Berlin, 1996).
  * [431] M. Tsutsumi, T. Bhattacharyya, and N. Kumagai, Piezoelectric magnetoelastic surface wave guided by interface between semiinfinite piezoelectric and magnetoelastic media, J. Appl. Phys. 46, 5072 (1975).
  * [432] G. Komoriya and G. Thomas, Magnetoelastic surface waves on YIG substrate, J. Appl. Phys. 50, 6459 (1979).
  * [433] Y. Shimizu, K. Hasegawas, and T. Yamada, Nonreciprocity of SAW velocity on a magnetized ferrite substrate, Electron. Commun. Jpn., Part I 63, 1 (1980).
  * [434] K. B. Vlasov, Equations of motion for magnetization in deformed anisotropic media, Sov. Phys. JETP 16, 1505 (1963).
  * [435] R. L. Comstock, Parallel pumping of magnetoelastic waves in ferromagnets, J. Appl. Phys. 35, 2427 (1964).
  * [436] R. Verba, I. Lisenkov, I. Krivorotov, V. Tiberkevich, and A. Slavin, Nonreciprocal Surface Acoustic Waves in Multilayers with Magnetoelastic and Interfacial Dzyaloshinskii-Moriya Interactions, Phys. Rev. Applied 9, 064014 (2018).
  * [437] R. Verba, V. Tiberkevich, and A. Slavin, Wide-Band Nonreciprocity of Surface Acoustic Waves Induced by Magnetoelastic Coupling with a Synthetic Antiferromagnet, Phys. Rev. Applied 12, 054061 (2019).
  * [438] D. Floess and H. Giessen, Nonreciprocal hybrid magnetoplasmonics, Rep. Prog. Phys. 81, 116401 (2018).
  * [439] D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, What is—and what is not—an optical isolator, Nat. Photon. 7, 579 (2013).
  * [440] M. F. Lewis and E. Patterson, Acoustic-Surface-Wave Isolator, Appl. Phys. Lett. 20, 276 (1972).
  * [441] M. Weiler, L. Dreher, C. Heeg, H. Huebl, R. Gross, M. S. Brandt, and S. T. B. Goennenwein, Elastically Driven Ferromagnetic Resonance in Nickel Thin Films, Phys. Rev. Lett. 106, 117601 (2011).
  * [442] M. Weiler, H. Huebl, F. S. Goerg, F. D. Czeschka, R. Gross, and S. T. B. Goennenwein, Spin Pumping with Coherent Elastic Waves, Phys. Rev. Lett. 108, 176601 (2012).
  * [443] R. Sasaki, Y. Nii, Y. Iguchi, and Y. Onose, Nonreciprocal propagation of surface acoustic wave in Ni$/$LiNbO3, Phys. Rev. B 95, 020407(R) (2017).
  * [444] V. G. Bar’yakhtar, V. M. Loktev, and S. M. Ryabchenko, Rotational invariance and magnetoflexural oscillations of ferromagnetic plates and rods, Sov. Phys. JETP. 61, 1040 (1985).
  * [445] L. Dreher, M. Weiler, M. Pernpeintner, H. Huebl, R. Gross, M. S. Brandt, and S. T. B. Goennenwein, Surface acoustic wave driven ferromagnetic resonance in nickel thin films: Theory and experiment, Phys. Rev. B 86, 134415 (2012).
  * [446] Y. Yahagi, B. Harteneck, S. Cabrini, and H. Schmidt, Controlling nanomagnet magnetization dynamics via magnetoelastic coupling, Phys. Rev. B 90, 140405(R) (2014).
  * [447] J. Puebla, M. Xu, B. Rana, K. Yamamoto, S. Maekawa, and Y. Otani, Acoustic ferromagnetic resonance and spin pumping induced by surface acoustic waves, J. Phys. D: Appl. Phys. 53, 264002 (2020).
  * [448] O. S. Latcham, Y. I. Gusieva, A. V. Shytov, O. Y. Gorobets, and V. V. Kruglyak, Controlling acoustic waves using magneto-elastic Fano resonances, Appl. Phys. Lett. 115, 082403 (2019).
  * [449] K. An, A. N. Litvinenko, R. Kohno, A. A. Fuad, V. V. Naletov, L. Vila, U. Ebels, G. de Loubens, H. Hurdequint, N. Beaulieu, J. Ben Youssef, N. Vukadinovic, G. E. W. Bauer, A. N. Slavin, V. S. Tiberkevich, and O. Klein, Coherent long-range transfer of angular momentum between magnon Kittel modes by phonons, Phys. Rev. B 101, 060407(R) (2020).
  * [450] A. Rückriegel and R. A. Duine, Long-Range Phonon Spin Transport in Ferromagnet–Nonmagnetic Insulator Heterostructures, Phys. Rev. Lett. 124, 117201 (2020).
  * [451] K. An, R. Kohno, A. N. Litvinenko, R. L. Seeger, V. V. Naletov, L. Vila, G. de Loubens, J. Ben Youssef, N. Vukadinovic, G. E. W. Bauer, A. N. Slavin, V. S. Tiberkevich, and O. Klein, Bright and Dark States of Two Distant Macrospins Strongly Coupled by Phonons, Phys. Rev. X 12, 011060 (2022).
  * [452] E. Dieulesaint and D. Royer, Elastic Waves in Solids II: Generation, Acousto-Optic Interaction, Applications (Springer, New York, 2000).
  * [453] P. Ventura, M. Solal, P. Dufilie, J. M. Hode, and F. Roux, A new concept in SPUDT design: the RSPUDT (resonant SPUDT), Proc. IEEE Ultrasonic Symp. 1, 1 (1994).
  * [454] H. Nassar, B. Yousefzadeh, R. Fleury, M. Ruzzene, A. Alú, C. Daraio, A. N. Norris, G. Huang, and M. R. Haberman, Nonreciprocity in acoustic and elastic materials, Nat. Rev. Mat. 5, 667 (2020).
  * [455] C. Rasmussen, L. Quan, and A. Alú, Acoustic nonreciprocity, J. Appl. Phys. 129, 210903 (2021).
  * [456] S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, New York, 1995).
  * [457] Y. Imry and R. Landauer, Conductance viewed as transmission, Rev. Mod. Phys. 71, S306 (1999).
  * [458] C. Y. Cai, X.-H. Zhou, W. C. Yu, and T. Yu, Acoustic Frequency Multiplication and Pure Second Harmonic Generation of Phonons by Magnetic Transducers, arXiv:2212.03451.
  * [459] J. Shi, P. Zhang, D. Xiao, and Q. Niu, Proper Definition of Spin Current in Spin-Orbit Coupled Systems, Phys. Rev. Lett. 96, 076604 (2006).
  * [460] M. W. Wu, J. H. Jiang, and M. Q. Weng, Spin dynamics in semiconductors, Phys. Rep. 493, 61 (2010).
  * [461] T. Yu and M. W. Wu, Spin diffusion in ultracold spin-orbit-coupled 40K gas, Phys. Rev. A 92, 013607 (2015).
  * [462] D. Indolese, S. Zihlmann, P. Makk, C. Jünger, K. Thodkar, and C. Schönenberger, Wideband and On-Chip Excitation for Dynamical Spin Injection into Graphene, Phys. Rev. Applied 10, 044053 (2018).
  * [463] H. Uchida, Y. Masuda, R. Fujikawa, A. V. Baryshev, and M. Inoue, Large enhancement of Faraday rotation by localized surface plasmon resonance in Au nanoparticles embedded in Bi:YIG film, J. Magn. Magn. Mat. 321, 843 (2009).
  * [464] E. Melander, E. Östman, J. Keller, J. Schmidt, E. T. Papaioannou, V. Kapaklis, U. B. Arnalds, B. Caballero, A. García-Martín, J. C. Cuevas, and B. Hjórvarsson, Influence of the magnetic field on the plasmonic properties of transparent Ni anti-dot arrays, Appl. Phys. Lett. 101, 063107 (2012).
  * [465] L. P. Pitaevskii, Electric Forces in a Transparent Dispersive Medium, JETP 12, 1008 (1961).
  * [466] R. Hertel, Theory of the inverse Faraday effect in metals, J. Magn. Magn. Mater. 303, L1 (2006).
  * [467] K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, Optical momentum and angular momentum in complex media: from the Abraham–Minkowski debate to unusual properties of surface plasmon-polaritons, New J. Phys. 19, 123014 (2017).
  * [468] M. Battiato, G. Barbalinardo, and P. M. Oppeneer, Quantum theory of the inverse Faraday effect, Phys. Rev. B 89, 014413 (2014).
  * [469] A. Qaiumzadeh, G. E. W. Bauer, and A. Brataas, Manipulation of ferromagnets via the spin-selective optical Stark effect, Phys. Rev. B 88, 064416 (2013).
  * [470] P. S. Pershan, J. P. van der Ziel, and L. D. Malmstrom, Theoretical Discussion of the Inverse Faraday Effect, Raman Scattering, and Related Phenomena, Phys. Rev. 143, 574 (1966).
  * [471] Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  * [472] A. Kirilyuk, A. V. Kimel, and T. Rasing, Ultrafast optical manipulation of magnetic order, Rev. Mod. Phys. 82, 2731 (2010).
  * [473] T. Valet and A. Fert, Theory of the perpendicular magnetoresistance in magnetic multilayers, Phys. Rev. B 48, 7099 (1993).
  * [474] M. S. Ukhtary, M. Maruoka, and R. Saito, Planar rotation of electric field induced by edge-plasmon in a graphene nanoribbon, Phys. Rev. B 100, 155432 (2019).
  * [475] C. G. de Oliveira and J. Tiomno, Representations of Dirac equation in general relativity, Nuovo Cimento 24, 672 (1962).
  * [476] B. Mashhoon, Neutron interferometry in a rotating frame of reference, Phys. Rev. Lett. 61, 2639 (1988).
  * [477] F. W. Hehl and W.-T. Ni, Inertial effects of a Dirac particle, Phys. Rev. D 42, 2045 (1990).
  * [478] M. Matsuo, J. Ieda, and S. Maekawa, Mechanical generation of spin current, Front. Phys. 3, 54 (2015).
  * [479] S. J. Barnett, Magnetization by Rotation, Phys. Rev. 6, 239 (1915).
  * [480] S. J. Barnett, Gyromagnetic and Electron-Inertia Effects, Rev. Mod. Phys. 7, 129 (1935).
  * [481] S. Bretzel, G. E. W. Bauer, Y. Tserkovnyak, and A. Brataas, Barnett effect in thin magnetic films and nanostructures, Appl. Phys. Lett. 95, 122504 (2009).
  * [482] M. Matsuo, J. Ieda, K. Harii, E. Saitoh, and S. Maekawa, Mechanical generation of spin current by spin-rotation coupling, Phys. Rev. B 87, 180402(R) (2013).
  * [483] M. Matsuo, Y. Ohnuma, and S. Maekawa, Theory of spin hydrodynamic generation, Phys. Rev. B 96, 020401(R) (2017).
  * [484] J. Fujimoto and M. Matsuo, Magnon current generation by dynamical distortion, Phys. Rev. B 102, 020406(R) (2020).
  * [485] J. Fujimoto, W. Koshibae, M. Matsuo, and S. Maekawa, Zeeman coupling and Dzyaloshinskii-Moriya interaction driven by electric current vorticity, Phys. Rev. B 103, L220402 (2021).
  * [486] W. Izumida, R. Okuyama, K. Sato, T. Kato, and M. Matsuo, Einstein–de Haas Nanorotor, Phys. Rev. Lett. 128, 017701 (2022).
  * [487] R. Takahashi, M. Matsuo, M. Ono, K. Harii, H. Chudo, S. Okayasu, J. Ieda, S. Takahashi, S. Maekawa, and E. Saitoh, Spin hydrodynamic generation, Nat. Phys. 12, 52 (2016).
  * [488] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87, 1213 (2015).
  * [489] M. I. Dyakonov and V. I. Perel, Current-induced spin orientation of electrons in semiconductors, Phys. Lett. A 35, 459 (1971).
  * [490] J. E. Hirsch, Spin Hall Effect, Phys. Rev. Lett. 83, 1834 (1999).
  * [491] H. J. Jiao and G. E. W. Bauer, Spin Backflow and ac Voltage Generation by Spin Pumping and the Inverse Spin Hall Effect, Phys. Rev. Lett. 110, 217602 (2013).
  * [492] C. Hahn, G. de Loubens, M. Viret, O. Klein, V. V. Naletov, and J. Ben Youssef, Detection of Microwave Spin Pumping Using the Inverse Spin Hall Effect, Phys. Rev. Lett. 111, 217204 (2013).
  * [493] D. H. Wei, M. Obstbaum, M. Ribow, C. H. Back, and G. Woltersdorf, Spin Hall voltages from a.c. and d.c. spin currents, Nat. Commun. 5, 3768 (2014).
  * [494] D. Kobayashi, T. Yoshikawa, M. Matsuo, R. Iguchi, S. Maekawa, E. Saitoh, and Y. Nozaki, Spin Current Generation Using a Surface Acoustic Wave Generated via Spin-Rotation Coupling, Phys. Rev. Lett. 119, 077202 (2017).
  * [495] C. Gardiner and P. Zoller, Quantum Noise (Springer-Verlarg, Berlin, 2000).
  * [496] J. H. Franken, H. J. M. Swagten, and B. Koopmans, Shift registers based on magnetic domain wall ratchets with perpendicular anisotropy, Nat. Nanotech. 7, 499 (2012).
  * [497] E. Raymenants, O. Bultynck, D. Wan, T. Devolder, K. Garello, L. Souriau, A. Thiam, D. Tsvetanova, Y. Canvel, D. E. Nikonov, I. A. Young, M. Heyns, B. Soree, I. Asselberghs, I. Radu, S. Couet, and V. D. Nguyen, Nanoscale domain wall devices with magnetic tunnel junction read and write, Nat. Electron. 4, 392 (2021).
  * [498] A. V. Chumak et al., Roadmap on spin-wave computing, IEEE Trans. Magn. 58, 0800172 (2022).
  * [499] S. Park and B.-J. Yang, Thermal Hall effect from a two-dimensional Schwinger boson gas with Rashba spin-orbit interaction: Application to ferromagnets with in-plane Dzyaloshinskii-Moriya interaction, Phys. Rev. B 102, 214421 (2020).
  * [500] H. Lee, J. H. Han, and P. A. Lee, Thermal Hall effect of spins in a paramagnet, Phys. Rev. B 91, 125413 (2015).
  * [501] W. B. Wang, M. W. Daniels, Z. L. Liao, Y. F. Zhao, J. Wang, G. Koster, G. Rijnders, C.-Z. Chang, D. Xiao, and W. D. Wu, Spin chirality fluctuation in two-dimensional ferromagnets with perpendicular magnetic anisotropy, Nat. Mater. 18, 1054 (2019).
  * [502] K. K. Kolincio, M. Hirschberger, J. Masell, S. Gao, A. Kikkawa, Y. Taguchi, T.-H. Arima, N. Nagaosa, and Y. Tokura, Large Hall and Nernst responses from thermally induced spin chirality in a spin-trimer ferromagnet, Proc. Natl. Acad. Sci. 118, 17 (2021).
  * [503] C. Carnahan, Y. H. Zhang, and D. Xiao, Thermal Hall effect of chiral spin fluctuations, Phys. Rev. B 103, 224419 (2021).
  * [504] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Orbitronics: the intrinsic orbital current in p-doped silicon, Phys. Rev. Lett. 95, 066601 (2005).
  * [505] V. T. Phong, Z. Addison, S. Ahn, H. Min, R. Agarwal, and E. J. Mele, Optically controlled orbitronics on a triangular lattice, Phys. Rev. Lett. 123, 236403 (2019).
  * [506] D. Go, D. Jo, H. -W. Lee, M. Kläui, and Y. Mokrousov, Orbitronics: Orbital currents in solids, Europhys. Lett. 135, 37001 (2021).
  * [507] D. Go and H. -W. Lee, Orbital torque: Torque generation by orbital current injection, Phys. Rev. Research 2, 013177 (2020).
  * [508] S. Lee, M. Kang, D. Go, D. Kim, J. Kang, T. Lee, G. Lee, J. Kang, N. Jong Lee, Y. Mokrousov, S. Kim, K. Kim, K. Lee, and B. Park, Efficient conversion of orbital Hall current to spin current for spin-orbit torque switching, Commun. Phys. 4, 234 (2021).
  * [509] D. Lee, D. Go, H. Park, W. Jeong, H. Ko, D. Yun, D. Jo, S. Lee, G. Go, J. Hyun Oh, K. Kim, B. Park, B. Min, H. Cheol Koo, H. Lee, O. Lee, and K. Lee, Orbital torque in magnetic bilayers, Nat. Commun. 12, 6710 (2021).
  * [510] J. Park, C. H. Kim, J. Rhim, and J. Han, Orbital Rashba effect and its detection by circular dichroism angle-resolved photoemission spectroscopy, Phys. Rev. B 85, 195401 (2012).
  * [511] H. An, Y. Kageyama, Y. Kanno, N. Enishi, and K. Ando, Spin–torque generator engineered by natural oxidation of Cu, Nat. Commun. 7, 13069 (2016).
  * [512] X. Chen, Y. Liu, G. Yang, H. Shi, C. Hu, M. Li, and H. Zeng, Giant antidamping orbital torque originating from the orbital Rashba-Edelstein effect in ferromagnetic heterostructures, Nat. Commun. 9, 2569 (2018).
  * [513] S. Ding, A. Ross, D. Go, L. Baldrati, Z. Ren, F. Freimuth, S. Becker, F. Kammerbauer, J. Yang, G. Jakob, Y. Mokrousov, and M. Kläui, Harnessing orbital-to-spin conversion of interfacial orbital currents for efficient spin-orbit torques, Phys. Rev. Lett. 125, 177201 (2020).
  * [514] S. Ding, Z. Liang, D. Go, C. Yun, M. Xue, Z. Liu, S. Becker, W. Yang, H. Du, C. Wang, Y. Yang, G. Jakob, M. Kläui, Y. Mokrousov, and J. Yang, Observation of the orbital Rashba-Edelstein magnetoresistance, Phys. Rev. Lett. 128, 067201 (2022).
  * [515] M. Padgett, J. Courtial, and L. Allen, Light’s Orbital Angular Momentum, Physics Today 57, 35 (2004).
  * [516] G. E. W. Bauer, R. Iguchi, and K.-i. Uchida, Theory of Transport in Ferroelectric Capacitors, Phys. Rev. Lett. 126, 187603 (2021).
  * [517] P. Tang, R. Iguchi, K.-i. Uchida, and G. E. W. Bauer, Thermoelectric Polarization Transport in Ferroelectric Ballistic Point Contacts, Phys. Rev. Lett. 128, 047601 (2022).
  * [518] P. Tang, R. Iguchi, K.-i. Uchida, and G. E. W. Bauer, Excitations of the ferroelectric order, Phys. Rev. B 106, L081105 (2022).
  * [519] X.-H. Zhou, P. Tang, R. L. Rodríguez-Suárez, S. M. Rezende, G. E. W. Bauer, and T. Yu, Surface Ferron Excitations in Ferroelectrics and Their Directional Routing, arXiv:2212.01047.
  * [520] T. Yu, D. M. Kennes, A. Rubio, and M. A. Sentef, Nematicity Arising from a Chiral Superconducting Ground State in Magic-Angle Twisted Bilayer Graphene under In-Plane Magnetic Fields, Phys. Rev. Lett. 127, 127001 (2021).
  * [521] R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Spin pumping and spin-transfer torques in antiferromagnets, Phys. Rev. Lett. 113, 057601 (2014).
  * [522] M. W. Daniels, R. Cheng, W. Yu, J. Xiao, and D. Xiao, Nonabelian magnonics in antiferromagnets, Phys. Rev. B 98, 134450 (2018).
  * [523] Y. Nambu, J. Barker, Y. Okino, T. Kikkawa, Y. Shiomi, M. Enderle, T. Weber, B. Winn, M. Graves-Brook, J. M. Tranquada, T. Ziman, M. Fujita, G. E. W. Bauer, E. Saitoh, and K. Kakurai, Observation of magnon polarization, Phys. Rev. Lett. 125, 027201 (2020).
  * [524] C. Kim, S. Lee, H. Kim, J. Park, K. Moon, J. Park, J. Yuk, K. Lee, B. Park, S. Kim, K. Kim, and C. Hwang, Distinct handedness of spin wave across the compensation temperatures of ferrimagnets, Nat. Mater. 19, 980 (2020).
  * [525] Y. Liu, Z. Xu, L. Liu, K. Zhang, Y. Meng, Y. Sun, P. Gao, H.-W. Zhao, Q. Niu, and J. Li, Switching magnon chirality in artificial ferrimagnet, Nat. Commun. 13, 1264 (2022).
  * [526] R. Cheng, M. W. Daniels, J. -G. Zhu, and D. Xiao, Antiferromagnetic spin wave field-effect transistor. Sci. Rep. 6, 24223 (2016).
  * [527] W. Yu, J. Lan, and J. Xiao, Magnetic logic gate based on polarized spin waves, Phys. Rev. Appl. 13, 024055 (2020).
  * [528] C. Jia, M. Chen, A. F. Schäffer, and J. Berakdar, Chiral logic computing with twisted antiferromagnetic magnon modes, npj Comput. Mater. 7, 101 (2021).
  * [529] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Electric field effect in atomically thin carbon films, Science 306, 5696 (2004).
  * [530] S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev, and A. Kis, 2D transition metal dichalcogenides, Nat. Rev. Mater. 2, 17033 (2017).
  * [531] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals, Nature 546, 265 (2017).
  * [532] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit, Nature 546, 270 (2017).
  * [533] Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun, Y. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen, and Y. Zhang, Gate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2, Nature 563, 94 (2018).
  * [534] A. Bedoya-Pinto, J. -R. Ji, A. K. Pandeya, P. Gargiani, M. Valvidares, P. Sessi, J. M. Taylor, F. Radu, K. Chang, and S. S. P. Parkin, Intrinsic 2D-XY ferromagnetism in a van der Waals monolayer, Science 374, 616 (2021).
  * [535] J. Prokop, W. X. Tang, Y. Zhang, I. Tudosa, T. R. F. Peixoto, K. Zakeri, and J. Kirschner, Magnons in a ferromagnetic monolayer, Phys. Rev. Lett. 102, 177206 (2009).
  * [536] W. Xing, L. Qiu, X. Wang, Y. Yao, Y. Ma, R. Cai, S. Jia, X. C. Xie, and W. Han, Magnon transport in quasi-two-dimensional van der Waals antiferromagnets, Phys. Rev. X 9, 011026 (2019).
  * [537] J. Cenker, B. Huang, N. Suri, P. Thijssen, A. Miller, T. Song, T. Taniguchi, K. Watanabe, M. A. McGuire, D. Xiao, and X. Xu, Direct observation of two-dimensional magnons in atomically thin CrI3, Nat. Phys. 17, 20 (2021).
  * [538] T. T. Mai, K. F. Garrity, A. McCreary, J. Argo, J. R. Simpson, V. Doan-Nguyen, R. V. Aguilar, and A. R. H. Walker, Magnon-phonon hybridization in 2D antiferromagnet MnPSe3, Sci. Adv. 7, eabj3106 (2021).
  * [539] Y. Wu, S. Zhang, J. Zhang, W. Wang, Y. L. Zhu, J. Hu, G. Yin, K. Wong, C. Fang, C. Wan, X. Han, Q. Shao, T. Taniguchi, K. Watanabe, J. Zang, Z. Mao, X. Zhang, and K. L. Wang, Néel-type skyrmion in WTe2/Fe3GeTe2 van der Waals heterostructure, Nat. Commun. 11, 3860 (2020).
  * [540] T. -E. Park, L. Peng, J. Liang, A. Hallal, F. S. Yasin, X. Zhang, K. M. Song, S. J. Kim, K. Kim, M. Weigand, G. Schütz, S. Finizio, J. Raabe, K. Garcia, J. Xia, Y. Zhou, M. Ezawa, X. Liu, J. Chang, H. C. Koo, Y. D. Kim, M. Chshiev, A. Fert, H. Yang, X. Yu, and S. Woo, Néel-type skyrmions and their current-induced motion in van der Waals ferromagnet-based heterostructures, Phys. Rev. B 103, 104410 (2021).
  * [541] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature 556, 43 (2018).
  * [542] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature 556, 80 (2018).
  * [543] T. Srivastava, M. Schott, R. Juge, V. Křižáková, M. Belmeguenai, Y. Roussigné, A. Bernand-Mantel, L. Ranno, S. Pizzini, S.-M. Chérif, A. Stashkevich, S. Auffret, O. Boulle, G. Gaudin, M. Chshiev, C. Baraduc, and H. Béa, Large-voltage tuning of Dzyaloshinskii–Moriya interactions: A route toward dynamic control of skyrmion chirality, Nano Lett. 18, 4871 (2018).
  * [544] J. Liang, Q. Cui, and H. Yang, Electrically switchable Rashba-type Dzyaloshinskii-Moriya interaction and skyrmion in two-dimensional magnetoelectric multiferroics, Phys. Rev. B 102, 220409(R) (2020).
  * [545] G. Chen, M. Robertson, M. Hoffmann, C. Ophus, A. L. F. Cauduro, R. L. Conte, H. Ding, R. Wiesendanger, S. Blügel, A. K. Schmid, and K. Liu, Observation of hydrogen-induced Dzyaloshinskii-Moriya interaction and reversible switching of magnetic chirality, Phys. Rev. X 11, 021015 (2021).
  * [546] R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).
  * [547] N. Gisin and R. Thew, Quantum communication, Nat. Photon. 3, 165 (2007).
  * [548] A. Bienfait, J. J. Pla, Y. Kubo, M. Stern, X. Zhou, C. C. Lo, C. D. Weis, T. Schenkel, M. L. W. Thewalt, D. Vion, D. Esteve, B. Julsgaard, K. Mølmer, J. J. L. Morton, and P. Bertet, Reaching the quantum limit of sensitivity in electron spin resonance, Nat. Nano. 11, 253 (2016).
  * [549] F. Arute et al., Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  * [550] N. Crescini, D. Alesini, C. Braggio, G. Carugno, D. D’Agostino, D. Di Gioacchino, P. Falferi, U. Gambardella, C. Gatti, G. Iannone, C. Ligi, A. Lombardi, A. Ortolan, R. Pengo, G. Ruoso, and L. Taffarello, Axion search with a quantum-limited ferromagnetic haloscope, Phys. Rev. Lett. 124, 171801 (2020).
  * [551] D. D. Awschalom, L. C. Bassett, A. S. Dzurak, E. L. Hu, and J. R. Petta, Quantum spintronics: engineering and manipulating atom-like spins in semiconductors, Science 339, 1174 (2013).
  * [552] M. He, H. Sun, and Q. L. He, Topological insulator: Spintronics and quantum computations, Front. Phys. 14, 1 (2019).
  * [553] H. Y. Yuan, Y. Cao, A. Kamra, R. A. Duine, and P. Yan, Quantum magnonics: when magnon spintronics meets quantum information science, Phys. Rep. 965, 1 (2022).
  * [554] M. Elyasi, Y. M. Blanter, and G. E. W. Bauer, Resources of nonlinear cavity magnonics for quantum information, Phys. Rev. B 101, 054402 (2020).
  * [555] S. Sharma, V. A. S. V. Bittencourt, A. D. Karenowska, and S. V. Kusminskiy, Spin cat states in ferromagnetic insulators, Phys. Rev. B 103, L100403 (2021).
  * [556] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, Coherent coupling between a ferromagnetic magnon and a superconducting qubit, Science 349, 405 (2015).
  * [557] D. Lachance-Quirion, S. P. Wolski, Y. Tabuchi, S. Kono, K. Usami, and Y. Nakamura, Entanglement-based single-shot detection of a single magnon with a superconducting qubit, Science 367, 425 (2020).
  * [558] K. J. Satzinger, Y. P. Zhong, H.-S. Chang, G. A. Peairs, A. Bienfait, M.-H. Chou, A. Y. Cleland, C. R. Conner, É. Dumur, J. Grebel, I. Gutierrez, B. H. November, R. G. Povey, S. J. Whiteley, D. D. Awschalom, D. I. Schuster, and A. N. Cleland, Quantum control of surface acoustic-wave phonons, Nature 563, 661 (2018).
  * [559] A. Bienfait, K. J. Satzinger, Y. P. Zhong, H.-S. Chang, M.-H. Chou, C. R. Conner, É. Dumur, J. Grebel, G. A. Peairs, R. G. Povey, and A. N. Cleland, Phonon-mediated quantum state transfer and remote qubit entanglement, Science 364, 368 (2019).
  * [560] D. E. Chang, J. S. Douglas, A. González-Tudela, C.-L. Hung, and H. J. Kimble, Colloquium: Quantum matter built from nanoscopic lattices of atoms and photons, Rev. Mod. Phys. 90, 031002 (2018).
  * [561] H. P. Zhan, L. H. Sun, and H. T. Tan, Chirality-induced one-way quantum steering between two waveguide-mediated ferrimagnetic microspheres, Phys. Rev. B 106, 104432 (2022).
  * [562] J. Zou, S. K. Kim, and Y. Tserkovnyak, Tuning entanglement by squeezing magnons in anisotropic magnets, Phys. Rev. B 101, 014416 (2020).
  * [563] Y.-L. Ren, S.-L. Ma, and F.-L. Li, Chiral coupling between a ferromagnetic magnon and a superconducting qubit, Phys. Rev. A 106, 053714 (2022).


††thanks: Correspondence should be addressed to P.S.

]Department of Psychology & Tsinghua Laboratory of Brain and Intelligence,
Tsinghua University, Beijing, 100084, China. Also at ]Laboratory of Advanced
Computing and Storage, Central Research Institute, 2012 Laboratories, Huawei
Technologies Co. Ltd., Beijing, 100084, China.

]Department of Psychology & Tsinghua Laboratory of Brain and Intelligence,
Tsinghua University, Beijing, 100084, China.

# Laplacian dynamics of convergent and divergent swarm behaviors

Yang Tian<EMAIL_ADDRESS>[  [  Yunhui Xu
<EMAIL_ADDRESS>Department of Physics, Tsinghua University,
Beijing, 100084, China.  Pei Sun<EMAIL_ADDRESS>[

###### Abstract

Swarming phenomena are ubiquitous in various physical, biological, and social
systems, where simple local interactions between individual units lead to
complex global patterns. A common feature of diverse swarming phenomena is
that the units exhibit either convergent or divergent evolution in their
behaviors, i.e., becoming increasingly similar or distinct, respectively. The
associated dynamics changes across time, leading to complex consequences on a
global scale. In this study, we propose a generalized Laplacian dynamics model
to describe both convergent and divergent swarm behaviors, where the trends of
convergence and divergence compete with each other and jointly determine the
evolution of global patterns. We empirically observe non-trivial phase-
transition-like phenomena between the convergent and divergent evolution
phases, which are controlled by local interaction properties. We also propose
a conjecture regarding the underlying phase transition mechanisms and outline
the main theoretical difficulties for testing this conjecture. Overall, our
framework may serve as a minimal model of swarm behaviors and their intricate
phase transition dynamics.

††preprint: APS/123-QED

## Background.

Swarm behaviors evolve dynamically with time [1]. The convergent and divergent
evolution processes of these behaviors exist across different scales. For
example, in similar environments, multiple species that interact with each
other may evolve along converging pathways [2, 3, 4], while the closely
related populations of a species may evolve along diverging pathways if they
experience distinct selective pressures [5, 6, 7]. As another example, people
may exhibit convergent social behaviors (e.g., opinions or consuming
behaviors) if emergencies occur (e.g., epidemic or social turmoil), while
these social behaviors diverge and become increasingly individualized after
emergencies end [8, 9, 10].

Although the universality of convergent and divergent swarm behaviors ensures
its ubiquity in the world, it does propose critical challenges for analytic
modeling due to the rich connotations and diverse details of swarm behaviors.
It is unclear whether the convergent and divergent dynamics of organic
evolution, the convergent and divergent social behaviors of people, and any
other similar phenomenon can be characterized by a unified and simple model.
While specialized models [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21],
statistical analyses [22, 23], and data-driven methods [24] have achieved
essential progress in studying swarming phenomena, advances in describing the
convergent and divergent evolution processes of swarm behaviors remain
limited.

## Our work.

In this study, we propose a simple model of convergent and divergent swarm
behaviors and describe the dynamic switching of convergent and divergent
evolution across time. This framework may serve as a minimal model to
characterize diverse real swarming phenomena. In addition, our model can be
enriched by considering more agent heterogeneity and local interaction rules
(e.g., memory and preference). Our computational experiments of this model
suggest phase-transition-like phenomena between convergent and divergent
evolution phases, which are controlled by the balance between convergent and
divergent tendencies in local interactions. We formalize a conjecture on the
latent phase transition mechanisms and summarize theoretical difficulties for
verifying this conjecture.

## Local dynamics of convergent evolution.

To reach at a balance between the universality to fit in with different
swarming phenomena and the simpleness to support analytic derivations, we
consider a $n$-dimensional abstract state space $S\subset\mathbb{R}^{n}$,
where each element $\mathbf{s}\in S$ denotes an abstract state that can be
used to model any target properties of system units (e.g., spatial
coordinates, opinions, or genetic characters). A swarm model updates the state
of each unit $\sigma_{i}\in V$ at moment $t\in T$ according to its current
state, denoted by
$\mathbf{s}_{i}\left(t\right)=\mathbf{s}\left(\sigma_{i}\left(t\right)\right)$,
and its interactions with its neighbor units, included by set $N_{i}$,
following a local rule $\psi$

$\displaystyle\mathbf{s}_{i}\left(t+\varepsilon\right)=\psi\left(\mathbf{s}_{i}\left(t\right),\mathbf{s}_{N_{i}}\left(t\right)\right).$
(1)

Here
$\mathbf{s}_{N_{i}}\left(t\right)=\\{\mathbf{s}_{j}\left(t\right)|\sigma_{j}\in
N_{i}\\}$ stands for the states of neighbor units of unit $i$ at moment $t$
and $\varepsilon$ denotes a minimum time step (see Fig. 1(a) for
illustrations).

The localization of interaction rules is realized by constraining $N_{i}$, the
neighbor set of unit $\sigma_{i}$, as a sub-set of units whose states are
close to $\sigma_{i}$

$\displaystyle
N_{i}=\\{\sigma_{j}|d\left(\mathbf{s}_{i}\left(t\right),\mathbf{s}_{i}\left(t\right)\right)\leq\Delta\\}\subset
V,$ (2)

where $d\left(\cdot,\cdot\right)$ denotes an arbitrary distance function and
$\Delta$ is a parameter that determines the interaction range (see Fig. 1(a)).
Please note that set $N_{i}$ evolves across time according to Eq. (2) as the
unit state is time-dependent.

A convergent evolution process is defined as a global pattern of the system,
during which units become increasingly similar to each other in the state
space. This process also corresponds to the case where the states of all units
approach the global mean state. The non-triviality of this process lies in
that each unit only interacts with its neighbors and, consequently, can never
know the absolute coordinates of all units in the state space. Only the
relative coordinate information obtained through interactions is available for
units at each moment. For each unit $\sigma_{i}$, the relative coordinate of
each of its neighbors in the state space is defined as

$\displaystyle\mathbf{r}_{ij}\left(t\right)=\mathbf{s}_{j}\left(t\right)-\mathbf{s}_{i}\left(t\right).$
(3)

Given this information, unit $\sigma_{i}$ can only approach the time-dependent
local mean state

$\displaystyle\overline{\mathbf{s}}_{i}\left(t\right)=\mathbf{s}_{i}\left(t\right)+\frac{1}{|N_{i}|+1}\sum_{\sigma_{j}\in
N_{i}}\mathbf{r}_{ij}\left(t\right)$ (4)

during convergent evolution. Note that $|N_{i}|+1$ in the denominator of Eq.
(4) is used to deal with the case with $|N_{i}|=0$.

The above definitions have described the convergent evolution process of swarm
behaviors on a local scale. To study its non-trivial manifestation on a global
scale, we need to relate Eq. (2) with specific global characteristics of the
system.

## Global dynamics of convergent evolution.

We notice that Eq. (2) can be equivalently reformulated as

$\displaystyle\overline{\mathbf{s}}_{i}\left(t\right)=\mathbf{s}_{i}\left(t\right)+\frac{1}{|N_{i}|+1}\sum_{\sigma_{j}\in
V}\mathbf{A}_{ij}\left(t\right)\left(\mathbf{s}_{j}\left(t\right)-\mathbf{s}_{i}\left(t\right)\right),$
(5)

where $\mathbf{A}\left(t\right)$ denotes the time-dependent adjacency matrix

$\displaystyle\mathbf{A}_{ij}\left(t\right)=\left(1-\delta_{i,j}\right)\Theta\left[\Delta-d\left(\mathbf{s}_{i}\left(t\right),\mathbf{s}_{j}\left(t\right)\right)\right].$
(6)

In Eq. (6), notion $\delta_{\cdot,\cdot}$ denotes the Kronecker delta function
and $\Theta\left(\cdot\right)$ is the Heaviside step function. By simple
calculation, we can transform Eq. (2) as

$\displaystyle\overline{\mathbf{s}}_{i}\left(t\right)-\mathbf{s}_{i}\left(t\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\operatorname{deg}\left(\sigma_{i}\left(t\right)\right)+1}\left(\sum_{\sigma_{j}\in
V}\mathbf{A}_{ij}\left(t\right)\mathbf{s}_{j}\left(t\right)-\operatorname{deg}\left(\sigma_{i}\left(t\right)\right)\mathbf{s}_{i}\left(t\right)\right),$
(7)

where $\operatorname{deg}\left(\cdot\right)$ denotes the degree in a graph
defined by the adjacency matrix in Eq. (6).

If we consider the observable global state

$\displaystyle\mathbf{s}\left(t\right)=\left[\mathbf{s}_{1}\left(t\right),\ldots,\mathbf{s}_{|V|}\left(t\right)\right]^{\mathsf{T}},$
(8)

where $\mathsf{T}$ denotes the transpose, we can reformulate Eq. (7) into a
new form

$\displaystyle\overline{\mathbf{s}}_{i}\left(t\right)-\mathbf{s}_{i}\left(t\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\operatorname{deg}\left(\sigma_{i}\left(t\right)\right)+1}\left(\left[\mathbf{A}_{i1}\left(t\right),\ldots,\mathbf{A}_{i|V|}\left(t\right)\right]-\mathbf{q}_{i}\left(t\right)\right)\mathbf{s}\left(t\right)$
(9)

using vector $\mathbf{q}_{i}\left(t\right)$

$\displaystyle\mathbf{q}_{i}\left(t\right)=\left[\delta_{i,1}\operatorname{deg}\left(\sigma_{1}\left(t\right)\right),\ldots,\delta_{i,|V|}\operatorname{deg}\left(\sigma_{|V|}\left(t\right)\right)\right].$
(10)

After defining the observable of all local mean states

$\displaystyle\overline{\mathbf{s}}\left(t\right)=\left[\overline{\mathbf{s}}_{1}\left(t\right),\ldots,\overline{\mathbf{s}}_{|V|}\left(t\right)\right]^{\mathsf{T}}$
(11)

and the observable of the inverse of the degrees of all units

$\displaystyle\mathbf{q}\left(t\right)=\operatorname{diag}\left(\left[\frac{1}{\operatorname{deg}\left(\sigma_{1}\left(t\right)\right)+1},\ldots,\frac{1}{\operatorname{deg}\left(\sigma_{|V|}\left(t\right)\right)+1}\right]\right)$
(12)

using the diagonal $\operatorname{diag}\left(\cdot\right)$, we can eventually
derive a global version of Eq. (4)

$\displaystyle\overline{\mathbf{s}}\left(t\right)-\mathbf{s}\left(t\right)=$
$\displaystyle-\mathbf{q}\left(t\right)\mathbf{L}\left(t\right)\mathbf{s}\left(t\right),$
(13)

where $\mathbf{L}\left(t\right)$ is the time-dependent Laplacian of the
corresponding graph

$\displaystyle\mathbf{L}_{ij}\left(t\right)=\delta_{i,j}\sum_{k}\mathbf{A}_{ik}\left(t\right)-\mathbf{A}_{ij}\left(t\right).$
(14)

Please see Fig. 1(b) for illustrations.

Now, let us consider the global dynamics characterized by Eq. (13). We define
an abstract velocity vector to quantify the intrinsic degrees of the
willingness of units to exhibit convergent evolution in the state space

$\displaystyle\mathbf{v}_{c}\left(t\right)=\left[\mathbf{v}_{c,1}\left(t\right),\ldots,\mathbf{v}_{c,|V|}\left(t\right)\right]^{\mathsf{T}}.$
(15)

Under the assumption of time continuity, the dynamics for
$\mathbf{s}\left(t\right)$ to approach $\overline{\mathbf{s}}\left(t\right)$
during convergent evolution is formalized as

$\displaystyle\frac{\partial}{\partial
t}\mathbf{s}\left(t\right)=-\mathbf{v}_{c}\left(t\right)\odot\mathbf{q}\left(t\right)\mathbf{L}\left(t\right)\mathbf{s}\left(t\right),$
(16)

where local topology effects are described by $\mathbf{q}\left(t\right)$
(i.e., how the neighbors of a unit affect its evolution), and $\odot$ denotes
the Schur product.

## Global dynamics of divergent evolution.

Then we turn to analyze the divergent evolution processes of swarm behaviors.
At the first glance, one only needs to consider the intrinsic degrees of the
willingness of units to exhibit divergent evolution

$\displaystyle\mathbf{v}_{d}\left(t\right)=\left[\mathbf{v}_{d,1}\left(t\right),\ldots,\mathbf{v}_{d,|V|}\left(t\right)\right]^{\mathsf{T}}$
(17)

and assume that units depart from associated time-dependent local mean states
by moving in a direction opposite to Eq. (16). Under this condition, the joint
dynamics of convergent and divergent evolution is nothing more but a Newtonian
motion in the state space, $\frac{\partial}{\partial
t}\mathbf{s}\left(t\right)=-\left[\mathbf{v}_{c}\left(t\right)-\mathbf{v}_{d}\left(t\right)\right]\odot\mathbf{q}\left(t\right)\mathbf{L}\left(t\right)\mathbf{s}\left(t\right)$.

Figure 1: Conceptual illustrations of the Laplacian dynamics of convergent and
divergent swarm behaviors and the computational experiments with regular
initialization. (a) On a local scale, the dynamics of each unit is determined
by the interactions between it and its neighbors. (b) Based on the underlying
graph structure, the local dynamics of every unit can be related to the global
dynamics. (c) The joint dynamics on a global scale is shaped by local topology
effects, the tendencies of convergent evolution, and the tendencies of
divergent evolution. (d-l) The results of computational experiments with
regular initialization and different parameter settings are visualized. A
homogeneous system with uniform motions is defined on $30$ units, whose states
are defined as $1$-dimensional to offer a clear vision. In the regular
initialization, these units are divided equally into $3$ groups. The distance
between units within each group is set as $2$ and the distance between
adjacent groups is set as $10$. The experiments run $10^{5}$ iterations under
each condition.

However, it is unreasonable for our framework to be such trivial because
Newtonian mechanics may be unpractical in abstract state space. The
information of the mentioned opposite direction can be unavailable in real
cases. For instance, if the state space describes the preference of drinks, it
is impossible to precisely define an opposite direction of preferring coffee,
i.e., it is difficult for units to tell preferring which drink is opposite to
preferring coffee. This is because the definition of an opposite drink of
coffee is ill-posed, at least in most situations. To describe these non-
negligible cases where Newtonian mechanics is unacceptable, we need to
consider a weaker assumption.

Figure 2: The computational experiments with random initialization. (a-c) The
results of the computational experiments are derived based on a homogeneous
system of $30$ units with uniform motions in a $1$-dimensional state space. In
the random initialization, each unit is assigned with an initial coordinate in
the state space that is randomly selected following a uniform distribution in
$\left[0,100\right]$. The experiments run $10^{5}$ iterations under each
condition. Figure 3: The eigenvalue spectrum of matrix
$\mathbf{H}\left(t\right)$ and the average $k$-distance in state space. (a-c)
The observable of matrix $\mathbf{H}\left(t\right)$ is derived using the
homogeneous systems generated in Fig. 1(f), Fig. 1(l), and Fig. 2(b), where we
measure the $m$-th standardized moments of the eigenvalue spectrum. The
measured results of $m\in\\{3,\ldots,12\\}$ are marked by colors, where the
color changes from blue to green as $m$ increases. (d-f) The average
$k$-distances in state space, where $k\in\\{2,3,4\\}$, are measured as the
functions of time. The color changes from blue to green as $k$ increases.

Our idea arises from the relation between $\mathbf{L}\left(t\right)$ and
$-\mathbf{L}^{\dagger}\left(t\right)$, where $\dagger$ denotes the
Moore–Penrose pseudoinverse [25]. To understand their relation, let us
consider an instance where the covariance matrix units is defined as
$\mathbf{C}\left(t\right)=\mathbf{L}\left(t\right)$. Two units are expected to
evolve inversely (i.e., they have a negative covariance) if they are connected
by an edge in the graph defined by Eq. (6). In this case, adjusting edges is
equivalent to adjusting the differences (i.e., gradients) between units.
Therefore, Eq. (16) can utilize $-\mathbf{L}\left(t\right)$ in the derivative
to minimize those differences. If we further consider the associate precision
matrix
$\mathbf{Q}\left(t\right):=\mathbf{C}^{\dagger}\left(t\right)=\mathbf{L}^{\dagger}\left(t\right)$,
we can relate the partial correlation [26, 27] between units $\sigma_{i}$ and
$\sigma_{j}$ with $\mathbf{L}^{\dagger}\left(t\right)$

$\displaystyle\operatorname{corr}\left(\sigma_{i},\sigma_{j}|V\setminus\\{\sigma_{i},\sigma_{j}\\}\right)=$
$\displaystyle-\frac{\mathbf{Q}_{ij}\left(t\right)}{\sqrt{\mathbf{Q}_{ii}\left(t\right)\mathbf{Q}_{jj}\left(t\right)}},$
(18) $\displaystyle=$
$\displaystyle-\frac{\mathbf{L}^{\dagger}_{ij}\left(t\right)}{\sqrt{\mathbf{L}^{\dagger}_{ii}\left(t\right)\mathbf{L}^{\dagger}_{jj}\left(t\right)}}.$
(19)

Based on Eq. (19), two units are expected to evolve consistently (i.e., have a
positive partial correlation) if they correspond to a negative value in
$\mathbf{L}^{\dagger}\left(t\right)$ (i.e., a positive value in
$-\mathbf{L}^{\dagger}\left(t\right)$). In this case, adjusting edges is
equivalent to controlling similarities between units. During divergent
evolution, units are expected to minimize their similarities, which can be
realized by the term $\mathbf{L}^{\dagger}\left(t\right)$ in the derivative of
the following dynamics

$\displaystyle\frac{\partial}{\partial
t}\mathbf{s}\left(t\right)=\mathbf{v}_{d}\left(t\right)\odot\mathbf{q}^{-1}\left(t\right)\mathbf{L}^{\dagger}\left(t\right)\mathbf{s}\left(t\right).$
(20)

In Eq. (20), we use $\mathbf{q}^{-1}\left(t\right)$, the inverse matrix of
$\mathbf{q}\left(t\right)$, to characterize local topology effects.

## Joint global dynamics of convergent and divergent evolution.

Now, we have the opportunity to define the joint global dynamics of convergent
and divergent evolution based on Eq. (16) and Eq. (20). Specifically, the
joint dynamics is given as

$\displaystyle\frac{\partial}{\partial t}\mathbf{s}\left(t\right)=$
$\displaystyle\big{[}\mathbf{v}_{d}\left(t\right)\odot\mathbf{q}^{-1}\left(t\right)\mathbf{L}^{\dagger}\left(t\right)$
$\displaystyle-\mathbf{v}_{c}\left(t\right)\odot\mathbf{q}\left(t\right)\mathbf{L}\left(t\right)\big{]}\mathbf{s}\left(t\right),$
(21)

which is essentially a type of generalized Laplacian dynamics with more
complicated details (see Refs. [28, 29, 30] for the elementary Laplacian
dynamics). See Fig. 1(c) for illustrations. Although the idea underlying Eq.
(21) is natural and simple, analyzing the dynamic properties of Eq. (21) is
intricate because the competition between convergent and divergent evolution
is characterized by the non-trivial interaction between
$\mathbf{L}\left(t\right)$ and $\mathbf{L}^{\dagger}\left(t\right)$ under the
effects of local topology. To offer a clear vision, we primarily consider a
special case of Eq. (21) in our subsequent analysis.

Figure 4: The phase space of convergent and divergent evolution. (a-f) Given
each condition of $\Delta\in\\{3,9,15,21,27,33\\}$, we generate a homogeneous
system with uniform motions and regular initialization for every
$\left(x,y\right)\in\left[0.001,0.2\right]^{2}$ and run the experiment for
$10^{5}$ iterations. The generated results support the measurement of ratio
$\frac{\mathbb{E}\left(d_{k}\left(t\right)\right)}{\mathbb{E}\left(d_{k}\left(0\right)\right)}$
($k=3$). The heat maps of this ratio under all conditions are shown in (a-f),
where different color maps are used to distinguish between different settings
of $\Delta$. (g-l) Ratio
$\frac{\mathbb{E}\left(d_{k}\left(t\right)\right)}{\mathbb{E}\left(d_{k}\left(0\right)\right)}$
is shown as a function of $\frac{x}{y}$ under each experiment condition, where
color maps are consistent with (a-f).

## A special case on a homogeneous system with uniform motions.

The considered special case of Eq. (21) is a homogeneous system of units that
exhibits uniform motions in the state space. To create uniform motions (i.e.,
with constant velocities) and make the system homogeneous (here, being
homogeneous refers to the case where all units share the same intrinsic
properties), we need to define

$\displaystyle\mathbf{v}_{c}\left(t\right)\equiv
x\mathbf{1}\mathbf{1}^{\mathsf{T}},\;\exists x\in\mathbb{R},\;\forall t\in T,$
(22) $\displaystyle\mathbf{v}_{d}\left(t\right)\equiv
y\mathbf{1}\mathbf{1}^{\mathsf{T}},\;\exists y\in\mathbb{R},\;\forall t\in T.$
(23)

Therefore, in a homogeneous system with uniform motions, Eq. (21) reduces to a
simple form

$\displaystyle\frac{\partial}{\partial t}\mathbf{s}\left(t\right)$
$\displaystyle=\left[y\mathbf{q}^{-1}\left(t\right)\mathbf{L}^{\dagger}\left(t\right)-x\mathbf{q}\left(t\right)\mathbf{L}\left(t\right)\right]\mathbf{s}\left(t\right),$
(24) $\displaystyle:=\mathbf{H}\left(t\right)\mathbf{s}\left(t\right)$ (25)

which is easier to analyze. In computational experiments, we apply the
following numerical iteration

$\displaystyle\mathbf{s}\left(t+\varepsilon\right)$
$\displaystyle=\varepsilon\mathbf{H}\left(t\right)\mathbf{s}\left(t\right)+\mathbf{s}\left(t\right)$
(26)

to derive the results of Eq. (25). Note that Eq. (26) only serves as a simple
illustration. One can also consider more complicated numerical approaches.

For the sake of visualization, we mainly consider a $1$-dimensional state
space in our work, yet a state space with higher dimensions can be directly
analyzed without any modification. In Figs. 1(d-l), we present diverse
instances of this homogeneous system with different parameter settings, where
a regular initialization condition of the system is applied to offer a clear
vision (i.e., initialized units are uniformly distributed and form clear
cluster structures in the state space). As shown in Figs. 1(d-i), different
ratios between $x$ and $y$ in Eq. (26) create distinct trends of convergent
and divergent swarm behaviors. A larger $x$ can lead to convergent evolution
while a larger $y$ makes units divergent. As suggested by Figs. 1(j-l), the
change of local interaction range controlled by a varying $\Delta$ in Eq. (2)
leads to distinct final patterns in the state space after evolution. A larger
local interaction range can amplify the trends of convergent and divergent
evolution because more units are involved. These phenomena can also be
observed in Fig. 2, where a random initialization condition is applied to the
system (i.e., the initial states of units are randomly sampled following a
specific distribution). These results provide qualitative illustrations of the
global dynamics of swarm behaviors and suggest how the dynamics is shaped by
the competition between convergent and divergent evolution trends or by the
local interaction range.

Figure 5: The non-trivial properties of the Laplacian dynamics of convergent
and divergent swarm behaviors. (a-b) The local topology effects (LTE) on the
eigenvalues and eigenvectors of $x\mathbf{L}\left(t\right)$ and
$y\mathbf{L}^{\dagger}\left(t\right)$ are shown as the functions of $t$ based
on the homogeneous system generated in Fig. 1(l) (also see the inserted sub-
plot in Fig. 5(c)). (c) The deviation of $\mathbf{H}\left(t\right)$ from being
commutative is measured by the proposed commutator error (CE), where the
applied homogeneous system is shown in the inserted sub-plot for reference.

To quantitatively analyze the convergent and divergent evolution of swarm
behaviors, we study the properties of the eigenvalue spectrum of
$\mathbf{H}\left(t\right)$. Specifically, we analyze the standardized moments
of the eigenvalue spectrum to reflect the latent dynamics of the evolution
process (note that the first and the second standardized moment are not
considered because they are constant [31]). Meanwhile, we consider
$d_{k}\left(\mathbf{s}_{i}\left(t\right)\right)$, the distance between state
$\mathbf{s}_{i}\left(t\right)$ and its $k$-nearest state at moment $t$ given a
distance function $d\left(\cdot,\cdot\right)$ in Eq. (2). We define the
average $k$-nearest distance at moment $t$ as

$\displaystyle\mathbb{E}\left(d_{k}\left(t\right)\right)=\frac{1}{|V|}\sum_{\sigma_{i}\in
V}d_{k}\left(\mathbf{s}_{i}\left(t\right)\right)$ (27)

to characterize the global state of the system at moment $t$. In Fig. 3, these
two kinds of metrics are calculated on three representative instances of the
homogeneous system in Eq. (25). As shown in Fig. 3, distance
$\mathbb{E}\left(d_{k}\left(t\right)\right)$ exhibits distinct changes during
convergent and divergent evolution (we define $d\left(\cdot,\cdot\right)$ as
the Euclidean distance for convenience). Before the dynamics converges to its
limit, the standardized moments of the eigenvalue spectrum dramatically
increase during divergent evolution while decreasing during divergent
evolution.

The results shown in Fig. 3 inspire us to distinguish between convergent and
divergent phases using
$\frac{\mathbb{E}\left(d_{k}\left(t\right)\right)}{\mathbb{E}\left(d_{k}\left(0\right)\right)}$,
the relative ratio between the average $k$-nearest distances at moment $0$ and
moment $t$. This ratio is larger than one in the divergent phase while it is
smaller than one in the convergent phase. In Fig. 4, we analyze the phase
space of the homogeneous system in terms of
$\frac{\mathbb{E}\left(d_{k}\left(t\right)\right)}{\mathbb{E}\left(d_{k}\left(0\right)\right)}$
under different conditions of $x$, $y$, and $\Delta$. As shown in Figs.
4(a-f), convergent and divergent phases share a non-trivial and relatively
blurry boundary on the plane of $\left(x,y\right)$, which changes across
different local interaction range conditions. Phase-transition-like phenomena
can be observed if we consider
$\frac{\mathbb{E}\left(d_{k}\left(t\right)\right)}{\mathbb{E}\left(d_{k}\left(0\right)\right)}$
as an order parameter and treat $\frac{x}{y}$ as a control parameter in Figs.
4(g-l). As $\frac{x}{y}$ increases, a kind of transition from the divergent
phase to the convergent phase occurs. However, it remains elusive whether
these phenomena really arise from specific underlying phase transitions or
from other unknown mechanisms. We cannot confirm the detailed properties of
these potential phase transitions (e.g., the order of transition) yet. Below,
we summarize the main difficulties for theoretically verifying the latent
phase transition to suggest possible directions for future explorations.

## Mathematical challenges for studying homogeneous system with uniform
motions.

The main challenges for studying the latent phase transition arise from the
non-triviality of predicting the limiting behaviors of the homogeneous system
(i.e., the classification of convergent and divergent phases depends on the
limit state of the system after the evolution process converges).

Although the motions of units in the state space have been constrained as
uniform, the homogeneous system in Eq. (25) is shaped by complicated local
topology effects, which is in contrast to a more theoretically favorable case
where local topology effects are completely excluded

$\displaystyle\frac{\partial}{\partial t}\mathbf{s}\left(t\right)$
$\displaystyle=\left[y\mathbf{L}^{\dagger}\left(t\right)-x\mathbf{L}\left(t\right)\right]\mathbf{s}\left(t\right):=\mathbf{M}\left(t\right)\mathbf{s}\left(t\right).$
(28)

In such an ideal case, we have the opportunity to express
$0,\omega_{2}\left(t\right)\leq\ldots\leq\omega_{|V|}\left(t\right)$, the
eigenvalues of matrix $\mathbf{M}\left(t\right)$, based on the eigenvalues
$0\leq\lambda_{2}\left(t\right)\leq\ldots\leq\lambda_{|V|}\left(t\right)$ of
matrix $\mathbf{L}\left(t\right)$

$\displaystyle\omega_{i}\left(t\right)=\frac{y}{\lambda_{i}\left(t\right)}-x\lambda_{i}\left(t\right),\;\forall
i\geq 2.$ (29)

Meanwhile, matrix $\mathbf{M}\left(t\right)$ shares the same group of
eigenvectors,
$\mathbf{1},\mathbf{u}_{2}\left(t\right),\ldots,\mathbf{u}_{|V|}\left(t\right)$,
as matrix $\mathbf{L}\left(t\right)$. Eq. (29) is derived from the fact that
the eigenvalues and eigenvectors of matrix $\mathbf{M}\left(t\right)$ are
$0,\frac{1}{\lambda_{2}\left(t\right)},\ldots,\frac{1}{\lambda_{|V|}\left(t\right)}$
and
$\mathbf{1},\mathbf{u}_{2}\left(t\right),\ldots,\mathbf{u}_{|V|}\left(t\right)$,
respectively [32, 33]. The explicit eigen-decomposition of
$\mathbf{M}\left(t\right)$ provides rich information about the evolution
dynamics, making it possible to analyze Eq. (28) following the dynamic system
theory [33]. However, this possibility vanishes due to the local topology
effects in Eq. (25). To quantify these non-trivial effects, we can measure the
cosine distance between the vector of eigenvalues of
$y\mathbf{L}^{\dagger}\left(t\right)$ and that of
$y\mathbf{q}^{-1}\left(t\right)\mathbf{L}^{\dagger}\left(t\right)$ (similarly,
between the vector of eigenvalues of $x\mathbf{L}\left(t\right)$ and that of
$x\mathbf{q}\left(t\right)\mathbf{L}^{\dagger}\left(t\right)$). Meanwhile, we
can calculate the average cosine distance between the eigenvectors of
$y\mathbf{L}^{\dagger}\left(t\right)$ and those of
$y\mathbf{q}^{-1}\left(t\right)\mathbf{L}^{\dagger}\left(t\right)$ (similarly,
between the eigenvectors of $x\mathbf{L}\left(t\right)$ and those of
$x\mathbf{q}\left(t\right)\mathbf{L}^{\dagger}\left(t\right)$). These cosine
distances reflect the local topology effects on the eigenvalues and
eigenvectors of $x\mathbf{L}\left(t\right)$ and
$y\mathbf{L}^{\dagger}\left(t\right)$. In Figs. 5(a-b), we present an instance
calculated on a representative homogeneous system, suggesting that these non-
negligible local topology effects are volatile until the evolution dynamics
converges. Therefore, Eq. (29) departs from the actual properties of Eq. (25)
in most cases.

Moreover, another challenge arises from that $\mathbf{H}\left(t\right)$ in Eq.
(25) is not commutative until the evolution dynamics converges to its limit.
We can measure the deviation of $\mathbf{H}\left(t\right)$ from being
commutative based on

$\displaystyle\chi\left(t,t^{\prime}\right)=\sum_{i,j}|\left[\mathbf{H}\left(t\right),\mathbf{H}\left(t^{\prime}\right)\right]_{ij}|,$
(30)

where $\left[\cdot,\cdot\right]$ denotes the commutator and
$\left[\cdot,\cdot\right]_{ij}$ is the $\left(i,j\right)$-th element of the
commutator. We refer to $\chi\left(\cdot,\cdot\right)$ as the commutator
error. In Fig. 5(c), we calculate an instance of
$\chi\left(\cdot,\cdot\right)$ by setting $t^{\prime}=t+\varepsilon$. The
derived results suggest that $\mathbf{H}\left(t\right)$ becomes trivially
commutative only after it arrives at its limit. The absence of a commutative
$\mathbf{H}\left(t\right)$ in most cases makes it invalid to express the
solution of Eq. (25) as
$\mathbf{s}\left(t\right)=\exp\left(\int_{0}^{t}\mathbf{H}\left(\tau\right)\mathsf{d}\tau\right)\mathbf{s}\left(0\right)$,
rejecting the possibility of simplifying Eq. (25).

In sum, the above properties create obstacles for theoretically analyzing the
homogeneous system in Eq. (25). The situation becomes even more non-trivial if
we relax the constraints on $\mathbf{v}_{c}$ and $\mathbf{v}_{d}$ to consider
heterogeneous cases. These challenges may serve as valuable directions for
future theoretical studies.

## Conclusion.

In this study, we propose a minimal model to describe the convergent and
divergent evolution of swarm behaviors. Despite its mathematical simplicity,
the model exhibits abundant non-trivial behaviors and suggests the potential
for phase transitions or similar phenomena between convergent and divergent
evolution controlled by local interactions. The non-triviality of the proposed
model may inspire new mathematical techniques in analyzing the physics of
swarming phenomena. Moreover, the simplicity of the proposed model in
computational aspects enables it to serve as a foundation for developing
large-scale numerical simulations of swarm behaviors.

## Acknowledgements.

This project is supported by the Artificial and General Intelligence Research
Program of Guo Qiang Research Institute at Tsinghua University (2020GQG1017)
as well as the Tsinghua University Initiative Scientific Research Program.

## References

  * Gazi [2013] V. Gazi, On lagrangian dynamics based modeling of swarm behavior, Physica D: Nonlinear Phenomena 260, 159 (2013).
  * Khatri  _et al._ [2009] B. S. Khatri, T. C. McLeish, and R. P. Sear, Statistical mechanics of convergent evolution in spatial patterning, Proceedings of the National Academy of Sciences 106, 9564 (2009).
  * Powell and Mariscal [2015] R. Powell and C. Mariscal, Convergent evolution as natural experiment: the tape of life reconsidered, Interface focus 5, 20150040 (2015).
  * Stayton [2015] C. T. Stayton, What does convergent evolution mean? the interpretation of convergence and its implications in the search for limits to evolution, Interface focus 5, 20150039 (2015).
  * Sitarz and Maksymowicz [2005] M. Sitarz and A. Maksymowicz, Divergent evolution paths of different genetic families in the penna model, International Journal of Modern Physics C 16, 1917 (2005).
  * Vogwill  _et al._ [2016] T. Vogwill, R. L. Phillips, D. R. Gifford, and R. C. MacLean, Divergent evolution peaks under intermediate population bottlenecks during bacterial experimental evolution, Proceedings of the Royal Society B: Biological Sciences 283, 20160749 (2016).
  * Zeldovich and Shakhnovich [2008] K. B. Zeldovich and E. I. Shakhnovich, Understanding protein evolution: from protein physics to darwinian selection, Annu. Rev. Phys. Chem. 59, 105 (2008).
  * Flierl  _et al._ [1999] G. Flierl, D. Grünbaum, S. Levins, and D. Olson, From individuals to aggregations: the interplay between behavior and physics, Journal of Theoretical biology 196, 397 (1999).
  * Castellano  _et al._ [2009] C. Castellano, S. Fortunato, and V. Loreto, Statistical physics of social dynamics, Reviews of modern physics 81, 591 (2009).
  * Takayasu  _et al._ [2015] M. Takayasu, K. Sato, Y. Sano, K. Yamada, W. Miura, and H. Takayasu, Rumor diffusion and convergence during the 3.11 earthquake: a twitter case study, PLoS one 10, e0121443 (2015).
  * Tanaka [2007] D. Tanaka, General chemotactic model of oscillators, Physical review letters 99, 134103 (2007).
  * Hindes  _et al._ [2016] J. Hindes, K. Szwaykowska, and I. B. Schwartz, Hybrid dynamics in delay-coupled swarms with mothership networks, Physical Review E 94, 032306 (2016).
  * Piwowarczyk  _et al._ [2019] R. Piwowarczyk, M. Selin, T. Ihle, and G. Volpe, Influence of sensorial delay on clustering and swarming, Physical Review E 100, 012607 (2019).
  * Czirók  _et al._ [2001] A. Czirók, M. Matsushita, and T. Vicsek, Theory of periodic swarming of bacteria: Application to proteus mirabilis, Physical Review E 63, 031915 (2001).
  * Yu  _et al._ [2013] W. Yu, G. Chen, M. Cao, J. Lü, and H.-T. Zhang, Swarming behaviors in multi-agent systems with nonlinear dynamics, Chaos: An Interdisciplinary Journal of Nonlinear Science 23, 043118 (2013).
  * Wang  _et al._ [2021] G. Wang, T. V. Phan, S. Li, M. Wombacher, J. Qu, Y. Peng, G. Chen, D. I. Goldman, S. A. Levin, R. H. Austin, _et al._ , Emergent field-driven robot swarm states, Physical review letters 126, 108002 (2021).
  * Yang  _et al._ [2010] Y. Yang, V. Marceau, and G. Gompper, Swarm behavior of self-propelled rods and swimming flagella, Physical Review E 82, 031904 (2010).
  * Thutupalli  _et al._ [2011] S. Thutupalli, R. Seemann, and S. Herminghaus, Swarming behavior of simple model squirmers, New Journal of Physics 13, 073021 (2011).
  * Iwasa  _et al._ [2012] M. Iwasa, K. Iida, and D. Tanaka, Various collective behavior in swarm oscillator model, Physics Letters A 376, 2117 (2012).
  * Iwasa and Tanaka [2017] M. Iwasa and D. Tanaka, Mechanism underlying the diverse collective behavior in the swarm oscillator model, Physics Letters A 381, 3054 (2017).
  * Huepe and Aldana [2008] C. Huepe and M. Aldana, New tools for characterizing swarming systems: A comparison of minimal models, Physica A: Statistical Mechanics and its Applications 387, 2809 (2008).
  * Attanasi  _et al._ [2014] A. Attanasi, A. Cavagna, L. Del Castello, I. Giardina, S. Melillo, L. Parisi, O. Pohl, B. Rossaro, E. Shen, E. Silvestri, _et al._ , Finite-size scaling as a way to probe near-criticality in natural swarms, Physical review letters 113, 238102 (2014).
  * Jose  _et al._ [2022] A. Jose, G. Ariel, and A. Be’Er, Physical characteristics of mixed-species swarming colonies, Physical Review E 105, 064404 (2022).
  * Xue  _et al._ [2023] T. Xue, X. Li, X. Chen, L. Chen, and Z. Han, Machine learning phases in swarming systems, Machine Learning: Science and Technology (2023).
  * Barata and Hussein [2012] J. C. A. Barata and M. S. Hussein, The moore–penrose pseudoinverse: A tutorial review of the theory, Brazilian Journal of Physics 42, 146 (2012).
  * Rue and Held [2005] H. Rue and L. Held,  _Gaussian Markov random fields: theory and applications_ (Chapman and Hall/CRC, 2005).
  * Lawrance [1976] A. Lawrance, On conditional and partial correlation, The American Statistician 30, 146 (1976).
  * Altafini and Lini [2014] C. Altafini and G. Lini, Predictable dynamics of opinion forming for networks with antagonistic interactions, IEEE Transactions on Automatic Control 60, 342 (2014).
  * Fax and Murray [2004] J. A. Fax and R. M. Murray, Information flow and cooperative control of vehicle formations, IEEE transactions on automatic control 49, 1465 (2004).
  * Veerman and Lyons [2020] J. P. Veerman and R. Lyons, A primer on laplacian dynamics in directed graphs, arXiv preprint arXiv:2002.02605 (2020).
  * Larsen and Marx [2005] R. J. Larsen and M. L. Marx,  _An introduction to mathematical statistics_ (Prentice Hall Hoboken, NJ, 2005).
  * Fontan and Altafini [2021] A. Fontan and C. Altafini, On the properties of laplacian pseudoinverses, in  _2021 60th IEEE Conference on Decision and Control (CDC)_ (IEEE, 2021) pp. 5538–5543.
  * Bullo [2020] F. Bullo,  _Lectures on network systems_ , Vol. 1 (Kindle Direct Publishing, 2020).