Strongly nonlinear topological phases of cascaded topoelectrical circuits

Abstract

Circuits provide ideal platforms of topological phases and matter, yet the study of topological circuits in the strongly nonlinear regime, has been lacking. We propose and experimentally demonstrate strongly nonlinear topological phases and transitions in one-dimensional electrical circuits composed of nonlinear capacitors. Nonlinear topological interface modes arise on domain walls of the circuit lattices, whose topological phases are controlled by the amplitudes of nonlinear voltage waves. Experimentally measured topological transition amplitudes are in good agreement with those derived from nonlinear topological band theory. Our prototype paves the way towards flexible metamaterials with amplitude-controlled rich topological phases and is readily extendable to two and three-dimensional systems that allow novel applications.

Graphical abstract

Keywords

strongly nonlinear / Berry phase / topological / electrical

References

[1] M. Hafezi , S. Mittal , J. Fan , A. Migdall , J. M. Taylor . Imaging topological edge states in silicon photonics. Nat. Photonics, 2013, 7(12): 1001

[2] M. S. Kirsch , Y. Zhang , M. Kremer , L. J. Maczewsky , S. K. Ivanov , Y. V. Kartashov , L. Torner , D. Bauer , A. Szameit , M. Heinrich . Nonlinear second-order photonic topological insulators. Nat. Phys., 2021, 17(9): 995

[3] M. C. Rechtsman , J. M. Zeuner , Y. Plotnik , Y. Lumer , D. Podolsky , F. Dreisow , S. Nolte , M. Segev , A. Szameit . Photonic Floquet topological insulators. Nature, 2013, 496(7444): 196

[4] H. Schomerus . Topologically protected midgap states in complex photonic lattices. Opt. Lett., 2013, 38(11): 1912

[5] A. B. Khanikaev , S. Hossein Mousavi , W. K. Tse , M. Kargarian , A. H. MacDonald , G. Shvets . Photonic topological insulators. Nat. Mater., 2013, 12(3): 233

[6] L. Lu , J. D. Joannopoulos , M. Soljačić . Topological photonics. Nat. Photonics, 2014, 8(11): 821

[7] T. Tuloup , R. W. Bomantara , C. H. Lee , J. Gong . Nonlinearity induced topological physics in momentum space and real space. Phys. Rev. B, 2020, 102(11): 115411

[8] R. W. Bomantara , W. Zhao , L. Zhou , J. Gong . Nonlinear Dirac cones. Phys. Rev. B, 2017, 96(12): 121406

[9] J. M. Zeuner , M. C. Rechtsman , Y. Plotnik , Y. Lumer , S. Nolte , M. S. Rudner , M. Segev , A. Szameit . Observation of topological transition in the bulk of a non-Hermitian system. Phys. Rev. Lett., 2015, 115(4): 040402

[10] J. Jiang , J. Ren , Z. Guo , W. Zhu , Y. Long , H. Jiang , H. Chen . Seeing topological winding number and band inversion in photonic dimer chain of split-ring resonators. Phys. Rev. B, 2020, 101(16): 165427

[11] Z. Guo , J. Jiang , H. Jiang , J. Ren , H. Chen . Observation of topological bound states in a double Su−Schrieffer−Heeger chain composed of split ring resonators. Phys. Rev. Res., 2021, 3(1): 013122

[12] Z. Yang , F. Gao , X. Shi , X. Lin , Z. Gao , Y. Chong , B. Zhang . Topological acoustics. Phys. Rev. Lett., 2015, 114(11): 114301

[13] A. Souslov , B. C. van Zuiden , D. Bartolo , V. Vitelli . Topological sound in active-liquid metamaterials. Nat. Phys., 2017, 13(11): 1091

[14] G. Lee , D. Lee , J. Park , Y. Jang , M. Kim , J. Rho . Piezoelectric energy harvesting using mechanical metamaterials and phononic crystals. Commun. Phys., 2022, 5(1): 94

[15] R. Süsstrunk , S. D. Huber . Observation of phononic helical edge states in a mechanical topological insulator. Science, 2015, 349(6243): 47

[16] C. He , X. Ni , H. Ge , X. Sun , Y. Chen , M. Lu , X. Liu , Y. Chen . Acoustic topological insulator and robust one-way sound transport. Nat. Phys., 2016, 12(12): 1124

[17] V. Peano , C. Brendel , M. Schmidt , F. Marquardt . Topological phases of sound and light. Phys. Rev. X, 2015, 5(3): 031011

[18] M. Xiao , G. Ma , Z. Yang , P. Sheng , Z. Q. Zhang , C. T. Chan . Geometric phase and band inversion in periodic acoustic system. Nat. Phys., 2015, 11(3): 240

[19] H. He , C. Qiu , L. Ye , X. Cai , X. Fan , M. Ke , F. Zhang , Z. Liu . Topological negative refraction of surface acoustic waves in a Weyl phononic crystal. Nature, 2018, 560(7716): 61

[20] J. Lu , C. Qiu , L. Ye , X. Fan , M. Ke , F. Zhang , Z. Liu . Observation of topological valley transport of sound in sonic crystals. Nat. Phys., 2017, 13(4): 369

[21] C. L. Kane , T. C. Lubensky . Topological boundary modes in isostatic lattices. Nat. Phys., 2014, 10(1): 39

[22] J. Paulose , B. G. Chen , V. Vitelli . Topological modes bound to dislocations in mechanical metamaterials. Nat. Phys., 2015, 11(2): 153

[23] H. Xiu , H. Liu , A. Poli , G. Wan , K. Sun , E. M. Arruda , X. Mao . Topological transformability and reprogrammability of multistable mechanical metamaterials. Proc. Natl. Acad. Sci. USA, 2022, 119(52): e2211725119

[24] D. Zhou , L. Zhang , X. Mao . Topological edge floppy modes in disordered fiber networks. Phys. Rev. Lett., 2018, 120(6): 068003

[25] M. Fruchart , V. Vitelli . Symmetries and dualities in the theory of elasticity. Phys. Rev. Lett., 2020, 124(24): 248001

[26] J. Ma , D. Zhou , K. Sun , X. Mao , S. Gonella . Edge modes and asymmetric wave transport in topological lattices: Experimental characterization at finite frequencies. Phys. Rev. Lett., 2018, 121(9): 094301

[27] H. Liu , D. Zhou , L. Zhang , D. K. Lubensky , X. Mao . Topological floppy modes in models of epithelial tissues. Soft Matter, 2021, 17(38): 8624

[28] M. Rosa , M. Ruzzene , E. Prodan . Topological gaps by twisting. Commun. Phys., 2021, 4(1): 130

[29] D. Zhou , L. Zhang , X. Mao . Topological boundary floppy modes in quasicrystals. Phys. Rev. X, 2019, 9(2): 021054

[30] Y. Fu , H. Qin . Topological phases and bulk-edge correspondence of magnetized cold plasmas. Nat. Commun., 2021, 12(1): 3924

[31] Y. Fu , H. Qin . The dispersion and propagation of topological Langmuir-cyclotron waves in cold magnetized plasmas. J. Plasma Phys., 2022, 88(4): 835880401

[32] V. V. Albert , L. I. Glazman , L. Jiang . Topological properties of linear circuit lattices. Phys. Rev. Lett., 2015, 114(17): 173902

[33] S. Imhof , C. Berger , F. Bayer , J. Brehm , L. W. Molenkamp , T. Kiessling , F. Schindler , C. H. Lee , M. Greiter , T. Neupert , R. Thomale . Topolectrical circuit realization of topological corner modes. Nat. Phys., 2018, 14(9): 925

[34] J. Ningyuan , C. Owens , A. Sommer , D. Schuster , J. Simon . Time- and site-resolved dynamics in a topological circuit. Phys. Rev. X, 2015, 5(2): 021031

[35] T.GorenK.PlekhanovF.AppasK.L. Hur, Topological Zak phase in strongly coupled LC circuits, Phys. Rev. B 97, 041106(R) (2018)

[36] W. Zhu , S. Hou , Y. Long , H. Chen , J. Ren . Simulating quantum spin Hall effect in the topological Lieb lattice of a linear circuit network. Phys. Rev. B, 2018, 97(7): 075310

[37] M.Serra-GarciaR.SüsstrunkS.D. Huber, Observation of quadrupole transitions and edge mode topology in an LC network, Phys. Rev. B 99, 020304(R) (2019)

[38] T. Helbig , T. Hofmann , S. Imhof , M. Abdelghany , T. Kiessling , L. W. Molenkamp , C. H. Lee , A. Szameit , M. Greiter , R. Thomale . Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits. Nat. Phys., 2020, 16(7): 747

[39] C. H. Lee , S. Imhof , C. Berger , F. Bayer , J. Brehm , L. W. Molenkamp , T. Kiessling , R. Thomale . Topolectrical circuits. Commun. Phys., 2018, 1(1): 39

[40] D. Zou , T. Chen , W. He , J. Bao , C. H. Lee , H. Sun , X. Zhang . Observation of hybrid higher-order skin-topological effect in non-Hermitian topolectrical circuits. Nat. Commun., 2021, 12(1): 7201

[41] T. Hofmann , T. Helbig , F. Schindler , N. Salgo , M. Brzezińska , M. Greiter , T. Kiessling , D. Wolf , A. Vollhardt , A. Kabaši , C. H. Lee , A. Bilušić , R. Thomale , T. Neupert . Reciprocal skin effect and its realization in a topolectrical circuit. Phys. Rev. Res., 2020, 2(2): 023265

[42] H.YangZ.X. LiY.LiuY.CaoP.Yan, Observation of symmetry-protected zero modes in topolectrical circuits, Phys. Rev. Res. 2(2), 022028 (2020) (J)

[43] T. Hofmann , T. Helbig , C. H. Lee , M. Greiter , R. Thomale . Chiral voltage propagation and calibration in a topolectrical Chern circuit. Phys. Rev. Lett., 2019, 122(24): 247702

[44] L. Xiao , T. Deng , K. Wang , G. Zhu , Z. Wang , W. Yi , P. Xue . Non-Hermitian bulk–boundary correspondence in quantum dynamics. Nat. Phys., 2020, 16(7): 761

[45] W. Zhu , Y. Long , H. Chen , J. Ren . Quantum valley Hall effects and spin-valley locking in topological Kane−Mele circuit networks. Phys. Rev. B, 2019, 99(11): 115410

[46] Y. Lumer , Y. Plotnik , M. C. Rechtsman , M. Segev . Self-localized states in photonic topological insulators. Phys. Rev. Lett., 2013, 111(24): 243905

[47] D. Leykam , Y. D. Chong . Edge solitons in nonlinear-photonic topological insulators. Phys. Rev. Lett., 2016, 117(14): 143901

[48] Y.LumerM.C. RechtsmanY.PlotnikM.Segev, Instability of bosonic topological edge states in the presence of interactions, Phys. Rev. A 94, 021801(R) (2016)

[49] Y. Hadad , A. B. Khanikaev , A. Alu . Self-induced topological transitions and edge states supported by nonlinear staggered potentials. Phys. Rev. B, 2016, 93(15): 155112

[50] Y. Hadad , J. C. Soric , A. B. Khanikaev , A. Alù . Self-induced topological protection in nonlinear circuit arrays. Nat. Electron., 2018, 1(3): 178

[51] H.XiuI.FrankelH.LiuK.QianS.SarkarB.C. MacniderZ.ChenN.BoechlerX.Mao, Synthetically non-Hermitian nonlinear wave-like behavior in a topological mechanical metamaterial, arXiv: 2207.09273 (2022)

[52] M. Fruchart , R. Hanai , P. B. Littlewood , V. Vitelli . Non-reciprocal phase transitions. Nature, 2021, 592(7854): 363

[53] Y. Wang , L. J. Lang , C. H. Lee , B. Zhang , Y. D. Chong . Topologically enhanced harmonic generation in a nonlinear transmission line metamaterial. Nat. Commun., 2019, 10(1): 1102

[54] D. Zhou , J. Ma , K. Sun , S. Gonella , X. Mao . Switchable phonon diodes using nonlinear topological Maxwell lattices. Phys. Rev. B, 2020, 101(10): 104106

[55] R. K. Pal , J. Vila , M. Leamy , M. Ruzzene . Amplitude-dependent topological edge states in nonlinear phononic lattices. Phys. Rev. E, 2018, 97(3): 032209

[56] D. Zhou , D. Z. Rocklin , M. J. Leamy , Y. Yao . Topological invariant and anomalous edge modes of strongly nonlinear systems. Nat. Commun., 2022, 13(1): 3379

[57] J. R. Tempelman , K. H. Matlack , A. F. Vakakis . Topological protection in a strongly nonlinear interface lattice. Phys. Rev. B, 2021, 104(17): 174306

[58] J. Vila , G. Paulino , M. Ruzzene . Role of nonlinearities in topological protection: Testing magnetically coupled fidget spinners. Phys. Rev. B, 2019, 99(12): 125116

[59] D. Zhou , J. Zhang . Non-Hermitian topological metamaterials with odd elasticity. Phys. Rev. Res., 2020, 2(2): 023173

[60] W. Cheng , G. Hu . Acoustic skin effect with non-reciprocal Willis materials. Appl. Phys. Lett., 2022, 121(4): 041701

[61] See Supplementary Information for experimental setup and measurement, the nonlinear topological band theory, nonlinear Berry phase, and topological phase transitions.

[62] A.F. Vakakis (Ed.), Normal Modes and Localization in Nonlinear Systems, Springer Dordrecht, 2001

[63] C. Shang , Y. Zheng , B. A. Malomed . Weyl solitons in three-dimensional optical lattices. Phys. Rev. A, 2018, 97(4): 043602

Electronic supplementary materials

The online version contains supplementary material available at https://doi.org/10.1007/s11467-023-1292-4 and https://journal.hep.com.cn/fop/EN/10.1007/s11467-023-1292-4.

Author contributions

All authors contributed extensively to the work presented in this paper. Jijie Tang, and Feng Li carried out the experiments. Fangyuan Ma, and Di Zhou provided the theory and calculations. Di Zhou, and Feng Li wrote the paper and Supplemental Material.

Conflict of interest

The authors declare that they have no conflict of interest.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12102039, 12272040, and 12074446).

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