Parity-dependent skin effects and topological properties in the multilayer nonreciprocal Su−Schrieffer−Heeger structures

Abstract

We concentrate on the skin effects and topological properties in the multilayer non-Hermitian Su−Schrieffer−Heeger (SSH) structure, by taking into account the nonreciprocal couplings between the different sublattices in the unit cells. Following the detailed demonstration of the theoretical method, we find that in this system, the skin effects and topological phase transitions induced by nonreciprocal couplings display the apparent parity effect, following the increase of the layer number of this SSH structure. On the one hand, the skin effect is determined by the parity of the layer number of this SSH system, as well as the parity of the band index of the bulk states. On the other hand, for the topological edge modes, such an interesting parity effect can also be observed clearly. Next, when the parameter disorders are taken into account, the zero-energy edge modes in the odd-layer structures tend to be more robust, whereas the other edge modes are easy to be destroyed. In view of these results, it can be ascertained that the findings in this work promote to understand the influences of nonreciprocal couplings on the skin effects and topological properties in the multilayer SSH lattices.

Graphical abstract

Keywords

multilayer SSH lattice / nonreciprocal couplings / band structure / skin effect

References

[1] C. M. Bender, D. C. Brody, H. F. Jones. Complex extension of quantum mechanics. Phys. Rev. Lett., 2002, 89: 270401

[2] V. V. Konotop, J. Yang, D. A. Zezyulin. Non-linear waves in PT-symmetric systems. Rev. Mod. Phys., 2016, 88: 035002

[3] Y. Ashida, Z. Gong, M. Ueda. Non-Hermitian physics. Adv. Phys., 2020, 69: 249

[4] S. K. Özdemir, S. Rotter, F. Nori, L. Yang. Parity−time symmetry and exceptional points in photonics. Nat. Mater., 2019, 18: 783

[5] C. Wu, N. Liu, G. Chen, S. Jia. Non-Hermiticity-induced topological transitions in long-range Su−Schrieffer−Heeger models. Phys. Rev. A, 2022, 106: 012211

[6] A. Fan, S. D. Liang. Complex energy plane and topological invariant in non-Hermitian systems. Front. Phys., 2022, 17: 33501

[7] J. C. Budich, E. J. Bergholtz. Non-Hermitian topological sensors. Phys. Rev. Lett., 2020, 125: 180403

[8] F. Koch, J. C. Budich. Quantum non-Hermitian topological sensors. Phys. Rev. Research, 2022, 4: 013113

[9] A. McDonald, A. A. Clerk. Exponentially-enhanced quantum sensing with non-Hermitian lattice dynamics. Nat. Commun., 2020, 11: 5382

[10] M. Parto, S. Wittek, H. Hodaei, G. Harari. . Edge-mode lasing in 1D topological active arrays. Phys. Rev. Lett., 2018, 120: 113901

[11] S. Weidemann, M. Kremer, T. Helbig, T. Hofmann. . Topological funneling of light. Science, 2020, 368: 311

[12] S. Garmon, K. Noba. Reservoir-assisted symmetry breaking and coalesced zero-energy modes in an open PT-symmetric Su−Schrieffer−Heeger model. Phys. Rev. A, 2021, 104: 062215

[13] J.-R. Li, L.-L. Zhang, W.-B. Cui, W.-J. Gong. Topological properties in non-Hermitian tetratomic Su−Schrieffer−Heeger lattices. Phys. Rev. Research, 2022, 4: 023009

[14] A. Yoshida, Y. Otaki, R. Otaki, T. Fukui. Edge states, corner states, and flat bands in a two-dimensional PT-symmetric system. Phys. Rev. B, 2019, 100: 125125

[15] A. F. Tzortzakakis, A. Katsaris, N. E. Palaiodi-mopoulos, P. A. Kalozoumis. . Topological edge states of the PT-symmetric Su−Schrieffer−Heeger model: An effective two-state description. Phys. Rev. A, 2022, 106: 023513

[16] X. Zhu, H. Wang, S. K. Gupta, H. Zhang, B. Xie, M. Lu, Y. Chen. Photonic non-Hermitian skin effect and non-Bloch bulk−boundary correspondence. Phys. Rev. Research, 2020, 2: 013280

[17] K. Xu, X. Zhang, K. Luo, R. Yu, D. Li, H. Zhang. Coexistence of topological edge states and skin effects in the non-Hermitian Su−Schrieffer−Heeger model with long-range nonreciprocal hopping in topoelectric realizations. Phys. Rev. B, 2021, 103: 125411

[18] S. M. Rafi-Ul-Islam, B. S. Zhuo, S. Haydar, C. H. Lee, M. B. A. Jalil. Critical hybridization of skin modes in coupled non-Hermitian chains. Phys. Rev. Research, 2022, 4: 013243

[19] R. Lin, T. Tai, L. Li, C. H. Lee. Topological non-Hermitian skin effect. Front. Phys., 2023, 18: 53605

[20] C. M. Bender, S. Boettcher. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett., 1998, 80: 5243

[21] C. M. Bender. Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys., 2007, 70: 947

[22] S. Longhi. Convective and absolute PT-symmetry breaking in tight-binding lattices. Phys. Rev. A, 2013, 88: 052102

[23] F. K. Kunst, E. Edvardsson, J. C. Budich, E. J. Bergholtz. Biorthogonal bulk−boundary correspondence in non-Hermitian systems. Phys. Rev. Lett., 2018, 121: 026808

[24] L. Jin, P. Wang, Z. Song. Su−Schrieffer−Heeger chain with one pair of PT-symmetric defects. Sci. Rep., 2017, 7: 5903

[25] Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, S. Zhang. Spontaneous PT-symmetry breaking in non-Hermitian coupled-cavity array. Phys. Rev. A, 2017, 96: 043810

[26] X. S. Li, Z. Z. Li, L. L. Zhang, W. J. Gong. PT symmetry of the Su−Schrieffer−Heeger model with imaginary boundary potentials and next-nearest-neighboring coupling. J. Phys. Condens. Matter, 2020, 32: 165401

[27] K. Kawabata, Y. Ashida, H. Katsura, M. Ueda. Parity−time-symmetric topological superconductor. Phys. Rev. B, 2018, 98: 085116

[28] M. Klett, H. Cartarius, D. Dast, J. Main, G. Wunner. Relation between PT-symmetry breaking and topologically nontrivial phases in the Su−Schrieffer−Heeger and Kitaev models. Phys. Rev. A, 2017, 95: 053626

[29] L. Jin. Topological phases and edge states in a non-Hermitian trimerized optical lattice. Phys. Rev. A, 2017, 96: 032103

[30] L. L. Zhang, J. R. Li, D. Zhang, T. T. Xu, W. B. Cui, W. J. Gong. PT-symmetric non-Hermitian zigzag-edged ribbon of bilayer photonic graphene. Results in Physics, 2022, 34: 105274

[31] X. M. Zhao, C. X. Guo, S. P. Kou, L. Zhuang, W. M. Liu. Defective Majorana zero modes in a non-Hermitian Kitaev chain. Phys. Rev. B, 2021, 104: 205131

[32] C. Yuce, H. Ramezani. Topological states in a non-Hermitian two-dimensional Su−Schrieffer−Heeger model. Phys. Rev. A, 2019, 100: 032102

[33] L. Feng, R. El-Ganainy, L. Ge. Non-Hermitian photonics based on parity−time symmetry. Nature Photon., 2017, 11: 752

[34] A. Regensburger, C. Bersch, M. A. Miri, G. On-ishchukov, D. N. Christodoulides, U. Peschel. Parity−time synthetic photonic lattices. Nature, 2012, 488: 167

[35] Y. Wu, B. Zhu, S. F. Hu. . Floquet control of the gain and loss in a PT-symmetric optical coupler. Front. Phys., 2017, 12: 121102

[36] C. Chen, Y. Liu, L. Zhao. . Asymmetric nonlinear-mode-conversion in an optical waveguide with PT symmetry. Front. Phys., 2022, 17: 52504

[37] A. Stegmaier, S. Imhof, T. Helbig, T. Hofmann, C. H. Lee, M. Kremer, A. Fritzsche. . Topological defect engineering and PT symmetry in non-Hermitian electrical circuits. Phys. Rev. Lett., 2021, 126: 215302

[38] Z. Lin, J. Schindler, F. M. Ellis, T. Kottos. Experimental observation of the dual behavior of PT-symmetric scattering. Phys. Rev. A, 2012, 85: 050101(R)

[39] L. Lu, J. D. Joannopoulos, M. Soljăcić. Topological photonics. Nature Photon., 2014, 8: 821

[40] M. G. Silveirinha. Topological theory of non-Hermitian photonic systems. Phys. Rev. B, 2019, 99: 125155

[41] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, I. Carusotto. Topological photonics. Rev. Mod. Phys., 2019, 91: 015006

[42] G. Q. Liang, Y. D. Chong. Optical resonator analog of a two-dimensional topological insulator. Phys. Rev. Lett., 2013, 110: 203904

[43] H. Hodaei, M. A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides. . Single mode lasing in transversely multi-moded PT-symmetric microring resonators. Laser Photonics Rev., 2016, 10: 494

[44] Y. Y. Fu, Y. Fei, D. X. Dong. . Photonic spin Hall effect in PT symmetric metamaterials. Front. Phys., 2019, 14: 62601

[45] M. Kang, F. Liu, J. Li. Effective spontaneous PT-symmetry breaking in hybridized metamaterials. Phys. Rev. A, 2013, 87: 053824

[46] Y. Sun, W. Tan, H. Q. Li, J. Li, H. Chen. Experimental demonstration of a coherent perfect absorber with PT phase transition. Phys. Rev. Lett., 2014, 112: 143903

[47] H. Jing, S. K. Ozdemir, X. Y. Lü, J. Zhang, L. Yang, F. Nori. PT-symmetric phonon laser. Phys. Rev. Lett., 2014, 113: 053604

[48] Y. D. Chong, L. Ge, A. D. Stone. PT-symmetry breaking and laser-absorber modes in optical scattering systems. Phys. Rev. Lett., 2011, 106: 093902

[49] S. Yao, Z. Wang. Edge states and topological invariants of non-Hermitian systems. Phys. Rev. Lett., 2018, 121: 086803

[50] S. Yao, F. Song, Z. Wang. Non-Hermitian Chern bands. Phys. Rev. Lett., 2018, 121: 136802

[51] V. M. Martinez Alvarez, J. E. Barrios Vargas, L. E. F. Foa Torres. Non-Hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points. Phys. Rev. B, 2018, 97: 121401(R)

[52] C. H. Lee, R. Thomale. Anatomy of skin modes and topology in non-Hermitian systems. Phys. Rev. B, 2019, 99: 201103

[53] S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, A. Szameit. Topological funneling of light. Science, 2020, 19: 311

[54] Y. Xiong. Why does bulk boundary correspondence fail in some non-Hermitian topological models. J. Phys. Commun., 2018, 2: 035043

[55] T. S. Deng, W. Yi. Non-Bloch topological invariants in a non-Hermitian domain wall system. Phys. Rev. B, 2019, 100: 035102

[56] F. Song, S. Yao, Z. Wang. Non-Hermitian topological invariants in real space. Phys. Rev. Lett., 2019, 123: 246801

[57] K. Yokomizo, S. Murakami. Non-Bloch band theory of non-Hermitian systems. Phys. Rev. Lett., 2019, 123: 066404

[58] L. Xiao, T. S. Deng, K. K. Wang, G. Y. Zhu, Z. Wang, W. Yi, P. Xue. Observation of non-Hermitian bulk−boundary correspondence in quantum dynamics. Nat. Phys., 2020, 16: 761

[59] P. C. Cao, Y. G. Peng, Y. Li, X. F. Zhu. Phase-locking diffusive skin effect. Chin. Phys. Lett., 2022, 39: 057801

[60] A. Ghatak, M. Brandenbourger, J. van Wezel, C. Coulais. Observation of non-Hermitian topology and its bulk−edge correspondence in an active mechanical metamaterial. Proc. Natl. Acad. Sci., 2020, 117: 29561

[61] E. J. Bergholtz, J. C. Budich, F. K. Kunst. Exceptional topology of non-Hermitian systems. Rev. Mod. Phys., 2021, 93: 015005

[62] T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, R. Thomale. Chiral voltage propagation and calibration in a topolectrical Chern circuit. Nat. Phys., 2020, 16: 747

[63] L. Xie, L. Jin, Zhi Song. Antihelical edge states in two-dimensional photonic topological metals. Sci. Bull., 2023, 68: 255

[64] K. Kawabata, K. Shiozaki, M. Ueda, M. Sato. Symmetry and topology in non-Hermitian physics. Phys. Rev. X, 2019, 9: 041015

[65] Z. P. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, M. Ueda. Topological phases of non-Hermitian systems. Phys. Rev. X, 2018, 8: 031079

[66] K. Takata, M. Notomi. Photonic topological insulating phase induced solely by gain and loss. Phy. Rev. Lett., 2018, 121: 213902

[67] H. C. Wu, L. Jin, Z. Song. Topology of an anti-parity−time symmetric non-Hermitian Su−Schrieffer−Heeger model. Phys. Rev. B, 2021, 103: 235110

[68] T. S. Deng, W. Yi. Non-Bloch topological invariants in a non-Hermitian domain wall system. Phys. Rev. B, 2019, 100: 035102

[69] C. X. Guo, C. H. Liu, X. M. Zhao, Y. Liu, S. Chen. Exact solution of non-Hermitian systems with generalized boundary conditions: Size-dependent boundary effect and fragility of the skin effect. Phys. Rev. Lett., 2021, 127: 116801

[70] J. R. Li, C. Luo, L. L. Zhang, S. F. Zhang, P. P. Zhu, W. J. Gong. Band structures and skin effects of coupled nonreciprocal Su−Schrieffer−Heeger lattices. Phys. Rev. A, 2023, 107: 022222

[71] Y. X. Liu, Y. C. Wang, X. J. Liu, Q. Zhou, S. Chen. Exact mobility edges, PT-symmetry breaking, and skin effect in one-dimensional non-Hermitian quasicrystals. Phys. Rev. B, 2021, 103: 014203

[72] L. J. Zhai, G. Y. Huang, S. Yin. Cascade of the delocalization transition in a non-Hermitian interpolating Aubry−André−Fibonacci chain. Phys. Rev. B, 2021, 104: 014202

[73] C. Mejia-Cortes, M. I. Molina. Interplay of disorder and PT symmetry in one-dimensional optical lattices. Phys. Rev. A, 2015, 91: 033815

[74] L. Jin, Z. Song. Bulk−boundary correspondence in a non-Hermitian system in one dimension with chiral inversion symmetry. Phys. Rev. B, 2019, 99: 081103(R)

[75] F. Evers, A. D. Mirlin. Fluctuations of the inverse participation ratio at the Anderson transition. Phys. Rev. Lett., 2000, 84: 3690

[76] V. M. M. Alvarez, M. D. Coutinho-Filho. Edge states in trimer lattices. Phys. Rev. A, 2019, 99: 013833

[77] T. E. Lee. Anomalous edge state in a non-Hermitian lattice. Phys. Rev. Lett., 2016, 116: 133903

Declarations

The authors declare that they have no competing interests and there are no conflicts.

Acknowledgements

This work was financially supported by the LiaoNing Revitalization Talents Program (Grant No. XLYC1907033), the National Natural Science Foundation of China (Grant No. 11905027), the Natural Science Foundation of Liaoning province (Grant No. 2023-MS-072), and Fundamental Research Funds for the Central Universities of Ministry of Education of China (Grant Nos. N2209005 and N2205015).

RIGHTS & PERMISSIONS

2024 Higher Education Press