Abstract
From protein motifs1 to black holes2, topological solitons are pervasive nonlinear excitations that are robust and can be driven by external fields3. So far, existing driving mechanisms all accelerate solitons and antisolitons in opposite directions3,4. Here we introduce a local driving mechanism for solitons that accelerates both solitons and antisolitons in the same direction instead: non-reciprocal driving. To realize this mechanism, we construct an active mechanical metamaterial consisting of non-reciprocally coupled oscillators5,6,7,8 subject to a bistable potential9,10,11,12,13,14. We find that such nonlinearity coaxes non-reciprocal excitations—so-called non-Hermitian skin waves5,6,7,8,15,16,17,18,19,20,21,22, which are typically unstable—into robust one-way (anti)solitons. We harness such non-reciprocal topological solitons by constructing an active waveguide capable of transmitting and filtering unidirectional information. Finally, we illustrate this mechanism in another class of metamaterials that shows the breaking of ‘supersymmetry’23,24 causing only antisolitons to be driven. Our observations and models demonstrate a subtle interplay between non-reciprocity and topological solitons, whereby solitons create their own driving force by locally straining the material. Beyond the scope of our study, non-reciprocal solitons might provide an efficient driving mechanism for robotic locomotion25 and could emerge in other settings, for example, quantum mechanics26,27, optics28,29,30 and soft matter31.
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Data availability
All the data supporting this study are available on the public repository https://uva-hva.gitlab.host/published-projects/non-reciprocal-topological-solitons.
Code availability
All the codes supporting this study are available on the public repository https://uva-hva.gitlab.host/published-projects/non-reciprocal-topological-solitons.
References
Chernodub, M., Hu, S. & Niemi, A. J. Topological solitons and folded proteins. Phys. Rev. E 82, 011916 (2010).
Heidmann, P., Bah, I. & Berti, E. Imaging topological solitons: the microstructure behind the shadow. Phys. Rev. D 107, 084042 (2023).
Dauxois, T. & Peyrard, M. Physics of Solitons (Cambridge Univ. Press, 2006).
Bennett, C. H., Büttiker, M., Landauer, R. & Thomas, H. Kinematics of the forced and overdamped sine-Gordon soliton gas. J. Stat. Phys. 24, 419–442 (1981).
Brandenbourger, M., Locsin, X., Lerner, E. & Coulais, C. Non-reciprocal robotic metamaterials. Nat. Commun. 10, 4608 (2019).
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, 29561–29568 (2020).
Chen, Y., Li, X., Scheibner, C., Vitelli, V. & Huang, G. Realization of active metamaterials with odd micropolar elasticity. Nat. Commun. 12, 5935 (2021).
Wang, W., Wang, X. & Ma, G. Non-Hermitian morphing of topological modes. Nature 608, 50–55 (2022).
Kochmann, D. M. & Bertoldi, K. Exploiting microstructural instabilities in solids and structures: from metamaterials to structural transitions. Appl. Mech. Rev. 69, 050801 (2017).
Nadkarni, N., Daraio, C. & Kochmann, D. M. Dynamics of periodic mechanical structures containing bistable elastic elements: from elastic to solitary wave propagation. Phys. Rev. E 90, 023204 (2014).
Nadkarni, N., Arrieta, A. F., Chong, C., Kochmann, D. M. & Daraio, C. Unidirectional transition waves in bistable lattices. Phys. Rev. Lett. 116, 244501 (2016).
Nadkarni, N., Daraio, C., Abeyaratne, R. & Kochmann, D. M. Universal energy transport law for dissipative and diffusive phase transitions. Phys. Rev. B 93, 104109 (2016).
Raney, J. R. et al. Stable propagation of mechanical signals in soft media using stored elastic energy. Proc. Natl Acad. Sci. USA 113, 9722–9727 (2016).
Janbaz, S. & Coulais, C. Diffusive kinks turn kirigami into machines. Nat. Commun. 15, 1255 (2024).
Coulais, C., Fleury, R. & van Wezel, J. Topology and broken hermiticity. Nat. Phys. 17, 9–13 (2020).
Bergholtz, E. J., Budich, J. C. & Kunst, F. K. Exceptional topology of non-Hermitian systems. Rev. Mod. Phys. 93, 015005 (2021).
Shankar, S., Souslov, A., Bowick, M. J., Marchetti, M. C. & Vitelli, V. Topological active matter. Nat. Rev. Phys. 4, 380–398 (2022).
Hatano, N. & Nelson, D. R. Localization transitions in non-Hermitian quantum mechanics. Phys. Rev. Lett. 77, 570–573 (1996).
Gong, Z. et al. Topological phases of non-Hermitian systems. Phys. Rev. X 8, 031079 (2018).
Martinez Alvarez, V. M., Barrios Vargas, J. E. & Foa Torres, L. E. F. Non-Hermitian robust edge states in one dimension: anomalous localization and eigenspace condensation at exceptional points. Phys. Rev. B 97, 121401 (2018).
Yao, S. & Wang, Z. Edge states and topological invariants of non-Hermitian systems. Phys. Rev. Lett. 121, 086803 (2018).
Weidemann, S. et al. Topological funneling of light. Science 368, 311–314 (2020).
Chen, B. G. G., Upadhyaya, N. & Vitelli, V. Nonlinear conduction via solitons in a topological mechanical insulator. Proc. Natl Acad. Sci. USA 111, 13004–13009 (2014).
Upadhyaya, N., Chen, B. G. & Vitelli, V. Nuts and bolts of supersymmetry. Phys. Rev. Res. 2, 043098 (2020).
Brandenbourger, M., Scheibner, C., Veenstra, J., Vitelli, V. & Coulais, C. Limit cycles turn active matter into robots. Preprint at arXiv https://doi.org/10.48550/arXiv.2108.08837 (2022).
Meier, E. J., An, F. A. & Gadway, B. Observation of the topological soliton state in the Su-Schrieffer-Heeger model. Nat. Commun. 7, 13986 (2016).
Pucher, S., Liedl, C., Jin, S., Rauschenbeutel, A. & Schneeweiss, P. Atomic spin-controlled non-reciprocal Raman amplification of fibre-guided light. Nat. Photon. 16, 380–383 (2022).
Pernet, N. et al. Gap solitons in a one-dimensional driven-dissipative topological lattice. Nat. Phys. 18, 678–684 (2022).
del Pino, J., Slim, J. J. & Verhagen, E. Non-Hermitian chiral phononics through optomechanically induced squeezing. Nature 606, 82–87 (2022).
Wanjura, C. C. et al. Quadrature nonreciprocity in bosonic networks without breaking time-reversal symmetry. Nat. Phys. 19, 1429–1436 (2023).
Zhao, H., Tai, J. B., Wu, J.-S. & Smalyukh, I. I. Liquid crystal defect structures with möbius strip topology. Nat. Phys. 19, 451–459 (2023).
Kunst, F. K., Edvardsson, E., Budich, J. C. & Bergholtz, E. J. Biorthogonal bulk-boundary correspondence in non-Hermitian systems. Phys. Rev. Lett. 121, 026808 (2018).
McDonald, A., Pereg-Barnea, T. & Clerk, A. A. Phase-dependent chiral transport and effective non-Hermitian dynamics in a bosonic Kitaev-Majorana chain. Phys. Rev. X 8, 041031 (2018).
McDonald, A. & Clerk, A. A. Exponentially-enhanced quantum sensing with non-Hermitian lattice dynamics. Nat. Commun. 11, 5382 (2020).
Helbig, T. et al. Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits. Nat. Phys. 16, 747–750 (2020).
Mathew, J. P., Pino, J. D. & Verhagen, E. Synthetic gauge fields for phonon transport in a nano-optomechanical system. Nat. Nanotech. 15, 198–202 (2020).
Xiao, L. et al. Non-Hermitian bulk-boundary correspondence in quantum dynamics. Nat. Phys. 16, 761–766 (2020).
Bililign, E. S. et al. Motile dislocations knead odd crystals into whorls. Nat. Phys. 18, 212–218 (2021).
Poncet, A. & Bartolo, D. When soft crystals defy Newton’s third law: nonreciprocal mechanics and dislocation motility. Phys. Rev. Lett. 128, 048002 (2022).
Tan, T. H. et al. Odd dynamics of living chiral crystals. Nature 607, 287–293 (2022).
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).
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).
Nagatani, T. The physics of traffic jams. Rep. Prog. Phys. 65, 1331–1386 (2002).
Librandi, G., Tubaldi, E. & Bertoldi, K. Programming nonreciprocity and reversibility in multistable mechanical metamaterials. Nat. Commun. 12, 3454 (2021).
Hwang, M. & Arrieta, A. F. Solitary waves in bistable lattices with stiffness grading: augmenting propagation control. Phys. Rev. E 98, 042205 (2018).
Braverman, L., Scheibner, C., VanSaders, B. & Vitelli, V. Topological defects in solids with odd elasticity. Phys. Rev. Lett. 127, 268001 (2021).
Lee, C. H. & Thomale, R. Anatomy of skin modes and topology in non-Hermitian systems. Phys. Rev. B 99, 201103 (2019).
Deng, B., Wang, P., He, Q., Tournat, V. & Bertoldi, K. Metamaterials with amplitude gaps for elastic solitons. Nat. Commun. 9, 1–8 (2018).
Peyrard, M. & Kruskal, M. D. Kink dynamics in the highly discrete sine-Gordon system. Phys. D: Nonlinear Phenom. 14, 88–102 (1984).
Braun, O. M., Hu, B. & Zeltser, A. Driven kink in the Frenkel-Kontorova model. Phys. Rev. E 62, 4235–4245 (2000).
Kivshar, Y. S. & Malomed, B. A. Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 61, 763–915 (1989).
Kivshar, Y. S., Pelinovsky, D. E., Cretegny, T. & Peyrard, M. Internal modes of solitary waves. Phys. Rev. Lett. 80, 5032–5035 (1998).
Kosevich, A. M. & Kivshar, Y. S. The perturbation-induced evolution of a soliton-antisoliton pair in the sine-Gordon system. Fiz. Nizk. Temp. 12, 1270 (1982).
Krasnov, V. M. Radiative annihilation of a soliton and an antisoliton in the coupled sine-Gordon equation. Phys. Rev. B 85, 134525 (2012).
Guo, X., Guzman, M., Carpentier, D., Bartolo, D. & Coulais, C. Non-orientable order and non-commutative response in frustrated metamaterials. Nature 618, 506–512 (2023).
Baumgartner, C. et al. Supercurrent rectification and magnetochiral effects in symmetric josephson junctions. Nat. Nanotech. 17, 39–44 (2022).
Yang, T. et al. Bifurcation instructed design of multistate machines. Proc. Natl Acad. Sci. USA 120, e2300081120 (2023).
Pinto-Ramos, D., Alfaro-Bittner, K., Clerc, M. G. & Rojas, R. G. Nonreciprocal coupling induced self-assembled localized structures. Phys. Rev. Lett. 126, 194102 (2021).
Zwicker, D. Py-PDE: a python package for solving partial differential equations. J. Open Source Softw. 5, 2158 (2020).
Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nat. Phys. 10, 39–45 (2014).
Zhou, Y., Chen, B. G., Upadhyaya, N. & Vitelli, V. Kink-antikink asymmetry and impurity interactions in topological mechanical chains. Phys. Rev. E. 95, 022202 (2017).
Faddeev, L. D. & Takhtajan, L. A. Hamiltonian Methods in the Theory of Solitons (Springer, 1987).
Acknowledgements
We thank R. Hassing and K. van Nieuwland for technical support and F. van Gorp, J.-S. Caux, J. van Wezel, A. Souslov, J. Binysh and V. Vitelli for insightful discussions. We also thank S. Bouché, Q. Cai and J. Lankhorst for gathering preliminary data in the context of the MSc course ‘Project Academic Skills for Research’ they followed at the University of Amsterdam. We acknowledge funding from the European Research Council under grant agreement no. 852587 and from the Netherlands Organisation for Scientific Research under grant agreement no. VI.Vidi.213.131.3.
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C.C. and J.V. conceptualized and guided the project. J.V. and X.G. designed the samples and experiments. J.V. carried out the experiments. J.V., X.G. and C.C. carried out the numerical simulations. O.G., C.V.M. and A.S. performed the theoretical study. All authors contributed extensively to the interpretation of the data and the production of the manuscript. J.V. and C.C. wrote the main text. J.V. created the figures and videos. All authors contributed to the writing of the Methods and the Supplementary Information.
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Extended data figures and tables
Extended Data Fig. 1 Dependence of the Peierls-Nabarro barrier on the nondimensional amplitude D and initial conditions in the Frenkel-Kontorova model.
(abc) The Peierls-Nabarro barrier, regime of instability and (anti)soliton velocities as the continuum limit is approached as a function of the unnormalized non-reciprocity \(\eta \sqrt{D}\) and damping \(\Gamma \sqrt{D}\). As the discreteness parameter D becomes smaller, the line separating stable from unstable solutions approaches Γ = η as predicted for the continuum. The initial condition used corresponds to the experimentally used soliton with single lattice spacing width. In addition, the Peierls-Nabarro barrier gradually decreases and (d) eventually goes to zero, provided that the initial soliton shape also becomes less discrete49. (e) When the activation amplitude ϕ0 of the experimental initial condition is changed, the Peierls-Nabarro barrier also changes but for large enough amplitudes, it becomes constant. (f) When instead of an initial activation angle, an edge oscillator is initialized with some radial velocity \({\dot{\phi }}_{0}\), the Peierls-Nabarro barrier remains constant.
Extended Data Fig. 2 Calibration of experimental parameters.
(a) The nonlinear potential generated by the periodically spaced magnets, as measured with an Instron torsion testing machine. Red line represents the sinusoidal fit used to calibrate the magnetic potential amplitude B. (b) Instron measurement of the elastic forces experienced by a single oscillator connected to two neighboring oscillators. Red line shows the smoothed data and green dashed lines show linear fits around the two potential minima, denoting the elastic coupling strength κ. (c) Oscillation of a single oscillator elastically coupled to two neighbors, used to measure the viscous damping coefficient γ. (d) The biased potential for different amounts of bias δ. (e) The difference between the potential minima ΔV between the two uneven minima plotted versus the bias δ. A linear fit establishes the relation between the bias and δ the effective external force E it corresponds to.
Extended Data Fig. 3 Stability of the soliton.
(a) Snapshots of a soliton in the unstable regime showing the destabilization of high wavenumber modes, found numerically for η = 1.1 and Γ = 1. (b) Growth rates \({\rm{I}}{\rm{m}}(\varOmega )\) of perturbations around the soliton solution for various wavenumbers given by Eq. (9). The dotted line at \({\rm{I}}{\rm{m}}(\varOmega )\) marks the transition between decaying and growing solutions, with high wavenumbers being the first to become unstable as the threshold of stability η = Γ is crossed. (c) Dependence of \({\rm{I}}{\rm{m}}({\varOmega }_{\pm })\) on the wavenumber k for Γ = 1 and η = 0.5 (red) and η = 1.5 (blue). In the latter case, modes in the regions \({\rm{Im}}({\varOmega }_{+}) > 0\) become unstable at \(k=\pm \Gamma /\sqrt{{\eta }^{2}-{\Gamma }^{2}}\) given by the dashed lines.
Extended Data Fig. 4 Kane-Lubensky chain.
Sketch of the Kane-Lubensky chain and its notation conventions.
Extended Data Fig. 5 Insensitivity of non-reciprocal solitons to boundary conditions.
Although at a linear level, the non-Hermitian skin effect causes the energy spectrum to change radically upon changing boundary conditions, nonreciprocal solitons are insensitive to the boundary as their topological charge protects them from amplifying exponentially in space. (a) Simulation of a single Frenkel-Kontorova soliton driven by non-reciprocity (η = 1.1, Γ = 1.3, D = 1.2) under antiperiodic boundary conditions. (b) Simulation of a Frenkel-Kontorova soliton-antisoliton pair driven by non-reciprocity (η = 1.1, Γ = 1.3, D = 1.2) under periodic boundary conditions. Neither periodic, antiperiodic or the open boundary conditions used in the main text affect the stability and velocity of the (anti)soliton.
Extended Data Fig. 6 Solitons with higher topological charge.
(a) Simulation of a staircase of Frenkel-Kontorova solitons under the influence of non-reciprocity (η = 1.1, Γ = 1.3, D = 1.2). As in the single soliton case, (anti)solitons with higher topological charge travel undisturbed at the same steady state velocity.
Extended Data Fig. 7 Effect of non-reciprocal driving and damping on the collision of sine-Gordon solitons.
(a) In the absence of both driving and damping, solitons and antisolitons pass through each other without interacting. (b) For nonzero damping, soliton and antisoliton annihilate and the resulting non-topological solution dissipates away. (c) With only non-reciprocity turned on, both excitations still pass through each other unhindered but are also rendered unstable. (d) Dissipation and non-reciprocity can balance, giving rise to non-reciprocal breather solutions.
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Veenstra, J., Gamayun, O., Guo, X. et al. Non-reciprocal topological solitons in active metamaterials. Nature 627, 528–533 (2024). https://doi.org/10.1038/s41586-024-07097-6
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DOI: https://doi.org/10.1038/s41586-024-07097-6
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