Volume 54 Issue 1
Mar.  2024
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Geng L L, Yuan J B, Cheng W, Hu G K, Zhou X M. Fundamental principles and research progress of non-Hermitian mechanical systems. Advances in Mechanics, 2024, 54(1): 1-60 doi: 10.6052/1000-0992-23-034
Citation: Geng L L, Yuan J B, Cheng W, Hu G K, Zhou X M. Fundamental principles and research progress of non-Hermitian mechanical systems. Advances in Mechanics, 2024, 54(1): 1-60 doi: 10.6052/1000-0992-23-034

Fundamental principles and research progress of non-Hermitian mechanical systems

doi: 10.6052/1000-0992-23-034
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  • Corresponding author: zhxming@bit.edu.cn
  • Received Date: 2023-09-11
  • Accepted Date: 2023-12-27
  • Available Online: 2024-01-09
  • Publish Date: 2024-03-24
  • Non-Hermitian theory, originated from quantum mechanics, is a theoretical framework for investigating the dynamics of open systems. New phenomena can be revealed with this theory, including exceptional point, chiral mode switching, and topological skin effect, which provide novel concepts for unusual wave and vibration control. This review will provide a comprehensive introduction to basic concepts of non-Hermitian theory in terms of classical mechanical systems, clarify the relationship between classical and non-Hermitian systems, and summarize the cutting-edge research progress in this field. Exceptional points and parity-time symmetry in non-Hermitian systems are firstly introduced. Then, the perturbation theory near exceptional points and its application to enhanced sensitivity are presented. Subsequently, the eigenvalue topological structure near exceptional points and eigenmode evolution in the process of dynamical encircling of exceptional points are discussed. Finally, the topological phase property of non-Hermitian mechanical systems is introduced.

     

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  • [1]
    Ahmed W W, Farhat M, Staliunas K, et al. 2023. Machine learning for knowledge acquisition and accelerated inverse-design for non-Hermitian systems. Communications Physics, 6: 2. doi: 10.1038/s42005-022-01121-9
    [2]
    Ashida Y, Gong Z, Ueda M. 2020. Non-Hermitian physics. Advances in Physics, 69 : 249-435.
    [3]
    Bender C M. 2007. Making sense of non-Hermitian Hamiltonians. Reports on Progress in Physics, 70 : 947-1018.
    [4]
    Bender C M, Boettcher S. 1998. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Physical Review Letters, 80 : 5243-5246.
    [5]
    Bender C M, Boettcher S, Meisinger P N. 1999. PT-symmetric quantum mechanics. Journal of Mathematical Physics, 40 : 2201-2229.
    [6]
    Berry M V. 1984. Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 392 : 45-57.
    [7]
    Berry M V. 2004. Physics of nonhermitian degeneracies. Czechoslovak Journal of Physics, 54 : 1039-1047.
    [8]
    Cai R, Jin Y, Li Y, et al. 2022. Exceptional points and skin Modes in non-Hermitian metabeams. Physical Review Applied, 18: 014067. doi: 10.1103/PhysRevApplied.18.014067
    [9]
    Cai R, Jin Y, Li Y, et al. 2023. Absorption-lasing effects and exceptional points in parity-time symmetric non-Hermitian metaplates. Journal of Sound and Vibration, 555: 117710. doi: 10.1016/j.jsv.2023.117710
    [10]
    Callaway J. 2013. Quantum Theory of the Solid State. Academic Press; London.
    [11]
    Chen W, Kaya Özdemir Ş, Zhao G, et al. 2017. Exceptional points enhance sensing in an optical microcavity. Nature, 548: 192-196. doi: 10.1038/nature23281
    [12]
    Chen Y, Li X, Scheibner C, et al. 2021. Realization of active metamaterials with odd micropolar elasticity. Nature communications, 12: 5935. doi: 10.1038/s41467-021-26034-z
    [13]
    Cheng W, Hu G. 2021. Odd elasticity realized by piezoelectric material with linear feedback. Science China Physics, Mechanics & Astronomy, 64 : 2.
    [14]
    Cheng W, Hu G. 2022. Acoustic skin effect with non-reciprocal Willis materials. Applied Physics Letters, 121 : 041701.
    [15]
    Cheng Z, Yu Z. 2021. Supervised machine mearning topological states of one-dimensional non-Hermitian systems. Chinese Physics Letters, 38 : 070302.
    [16]
    Choi Y, Hahn C, Yoon J W, et al. 2018. Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators. Nature Communications, 9: 2182. doi: 10.1038/s41467-018-04690-y
    [17]
    Choi Y, Hahn C, Yoon J W, et al. 2017. Extremely broadband, on-chip optical nonreciprocity enabled by mimicking nonlinear anti-adiabatic quantum jumps near exceptional points. Nature Communications, 8: 14154. doi: 10.1038/ncomms14154
    [18]
    Christensen J, Willatzen M, Velasco V R, et al. 2016. Parity-time synthetic phononic media. Physical Review Letters, 116: 207601. doi: 10.1103/PhysRevLett.116.207601
    [19]
    De Carlo M, De Leonardis F, Soref R A, et al. 2022. Non-Hermitian sensing in photonics and electronics: A review. Sensors, 22: 3977. doi: 10.3390/s22113977
    [20]
    Dembowski C, Dietz B, Gräf H D, et al. 2004. Encircling an exceptional point. Physical Review E, 69: 056216. doi: 10.1103/PhysRevE.69.056216
    [21]
    Ding K, Fang C, Ma G. 2022. Non-Hermitian topology and exceptional-point geometries. Nature Reviews Physics, 4 : 745-760.
    [22]
    Ding K, Ma G, Xiao M, et al. 2016. Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization. Physical Review X, 6: 021007.
    [23]
    Ding K, Ma G, Zhang Z Q, et al. 2018. Experimental demonstration of an anisotropic exceptional point. Physical Review Letters, 121: 085702. doi: 10.1103/PhysRevLett.121.085702
    [24]
    Dong Z, Li Z, Yang F, et al. 2019. Sensitive readout of implantable microsensors using a wireless system locked to an exceptional point. Nature Electronics, 2: 335-342. doi: 10.1038/s41928-019-0284-4
    [25]
    Doppler J, Mailybaev A A, Böhm J, et al. 2016. Dynamically encircling an exceptional point for asymmetric mode switching. Nature, 537: 76-79. doi: 10.1038/nature18605
    [26]
    Duan Y, Geng L, Guo Q, et al. 2023. Acoustic chiral mode switching by dynamic encircling of exceptional points. Applied Physics Letters, 123:101701
    [27]
    El-Ganainy R, Makris K G, Khajavikhan M, et al. 2018. Non-Hermitian physics and PT symmetry. Nature Physics, 14: 11-19. doi: 10.1038/nphys4323
    [28]
    Elbaz G, Pick A, Moiseyev N, et al. 2022. Encircling exceptional points of Bloch waves: mode conversion and anomalous scattering. Journal of Physics D:Applied Physics, 55: 235301. doi: 10.1088/1361-6463/ac5859
    [29]
    Fan H, Chen J, Zhao Z, et al. 2020. Antiparity-time symmetry in passive nanophotonics. ACS Photonics, 7: 3035-3041. doi: 10.1021/acsphotonics.0c01053
    [30]
    Fleury R, Sounas D, Alù A. 2015. An invisible acoustic sensor based on parity-time symmetry. Nature Communications, 6 : 5905.
    [31]
    Fruchart M, Hanai R, Littlewood P B, et al. 2019. Non-reciprocal robotic metamaterials. Nature communications, 10: 4608. doi: 10.1038/s41467-019-12599-3
    [32]
    Geib N, Sasmal A, Wang Z, et al. 2021. Tunable nonlocal purely active nonreciprocal acoustic media. Physical Review B, 103: 165427. doi: 10.1103/PhysRevB.103.165427
    [33]
    Geng L, Zhang W, Zhang X, et al. 2021a. Chiral mode transfer of symmetry-broken states in anti-parity-time-symmetric mechanical system. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 477 : 20210641.
    [34]
    Geng L, Zhang W, Zhang X, et al. 2021b. Topological mode switching in modulated structures with dynamic encircling of an exceptional point. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 477 : 20200766.
    [35]
    Ghosh S N, Chong Y D. 2016. Exceptional points and asymmetric mode conversion in quasi-guided dual-mode optical waveguides. Scientific Reports, 6 : 19837.
    [36]
    Gilary I, Mailybaev A A, Moiseyev N. 2013. Time-asymmetric quantum-state-exchange mechanism. Physical Review A, 88 : 010102.
    [37]
    Gong Z, Ashida Y, Kawabata K, et al. 2018. Topological phases of non-hermitian systems. Physical Review X, 8: 031079.
    [38]
    Graefe E M, Mailybaev A A, Moiseyev N. 2013. Breakdown of adiabatic transfer of light in waveguides in the presence of absorption. Physical Review A, 88 : 033842.
    [39]
    Gu Z, Gao H, Cao P C, et al. 2021. Controlling sound in non-hermitian acoustic systems. Physical Review Applied, 16: 057001. doi: 10.1103/PhysRevApplied.16.057001
    [40]
    Guo A, Salamo G J, Duchesne D, et al. 2009. Observation of PT-symmetry breaking in complex optical potentials. Physical Review Letters, 103: 093902. doi: 10.1103/PhysRevLett.103.093902
    [41]
    Haken H. 1983. Synergetics—an Introduction. Springer-Verlag Berlin Heidelberg; Berlin.
    [42]
    Hassan A U, Zhen B, Soljačić M, et al. 2017. Dynamically encircling exceptional points: Exact evolution and polarization state conversion. Physical Review Letters, 118: 093002. doi: 10.1103/PhysRevLett.118.093002
    [43]
    Haus H A, Huang W. 1991. Coupled-mode theory. Proceedings of the IEEE, 79 : 1505-1518.
    [44]
    Heiss W D. 1999. Phases of wave functions and level repulsion. The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics, 7 : 1-4.
    [45]
    Hodaei H, Hassan A U, Wittek S, et al. 2017. Enhanced sensitivity at higher-order exceptional points. Nature, 548: 187-191. doi: 10.1038/nature23280
    [46]
    Hou Z, Assouar B. 2018. Tunable elastic parity-time symmetric structure based on the shunted piezoelectric materials. Journal of Applied Physics, 123 : 085101.
    [47]
    Huang H, Chen J, Huo S. 2021. Recent advances in topological elastic metamaterials. Journal of Physics: Condensed Matter, 33 : 503002.
    [48]
    Huang J, Zhou X. 2019. A time-varying mass metamaterial for non-reciprocal wave propagation. International Journal of Solids and Structures, 164 : 25-36.
    [49]
    Huang J, Zhou X. 2020. Non-reciprocal metamaterials with simultaneously time-varying stiffness and mass. Journal of Applied Mechanics, 87 : 071003.
    [50]
    Kato T. 1966. Perturbation Theory for Linear Operators. Springer; Verlag Berlin Heidelberg.
    [51]
    Kazemi H, Hajiaghajani A, Nada M Y, et al. 2021. Ultra-sensitive radio frequency biosensor at an exceptional point of degeneracy induced by time modulation. IEEE Sensors Journal, 21: 7250-7259.
    [52]
    Kazemi H, Nada M Y, Nikzamir A, et al. 2022. Experimental demonstration of exceptional points of degeneracy in linear time periodic systems and exceptional sensitivity. Journal of Applied Physics, 131: 144502. doi: 10.1063/5.0084849
    [53]
    Kittel C A M, Paul and McEuen, Paul 1996. Introduction to Solid State Physics. Wiley; New York.
    [54]
    Klitzing K V, Dorda G, Pepper M. 1980. New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Physical Review Letters, 45 : 494-497.
    [55]
    Kononchuk R, Cai J, Ellis F, et al. 2022. Exceptional-point-based accelerometers with enhanced signal-to-noise ratio. Nature, 607: 697-702. doi: 10.1038/s41586-022-04904-w
    [56]
    Lai Y H, Lu Y K, Suh M G, et al. 2019. Observation of the exceptional-point-enhanced Sagnac effect. Nature, 576: 65-69. doi: 10.1038/s41586-019-1777-z
    [57]
    Li A, Chen W, Wei H, et al. 2022. Riemann-encircling exceptional points for efficient asymmetric polarization-locked devices. Physical Review Letters, 129: 127401. doi: 10.1103/PhysRevLett.129.127401
    [58]
    Li D, Huang S, Cheng Y, et al. 2021. Compact asymmetric sound absorber at the exceptional point. Science China Physics, Mechanics & Astronomy, 64 : 244303.
    [59]
    Li H X, Rosendo-López M, Zhu Y F, et al. 2019. Ultrathin acoustic parity-time symmetric metasurface cloak. Research, 2019: 1-7.
    [60]
    Li H, Mekawy A, Krasnok A, et al. 2020. Virtual parity-time symmetry. Physical Review Letters, 124: 193901. doi: 10.1103/PhysRevLett.124.193901
    [61]
    Li P H, Wang Y Z. 2023. Negative refraction and exceptional point with Parity-Time symmetry in a piezoelectric mechanical metamaterial. Mechanics of Materials, 181 : 104647.
    [62]
    Liu X, Huang G, Hu G. 2012. Chiral effect in plane isotropic micropolar elasticity and its application to chiral lattices. Journal of the Mechanics and Physics of Solids, 60 : 1907-1921.
    [63]
    Liu Z P, Zhang J, Özdemir Ş K, et al. 2016. Metrology with PT-Symmetric Cavities: Enhanced Sensitivity near the PT-Phase Transition. Physical Review Letters, 117: 110802. doi: 10.1103/PhysRevLett.117.110802
    [64]
    Lu J, Deng W, Huang X, et al. 2023. Non-hermitian topological phononic metamaterials. Advanced Materials: 2307998.
    [65]
    Luigi Lugiato F P, Massimo Brambilla. 2015. Nonlinear Optical Systems. Cambridge University Press; Cambridge.
    [66]
    Ma F, Imam A, Morzfeld M. 2009. The decoupling of damped linear systems in oscillatory free vibration. Journal of Sound and Vibration, 324 : 408-428.
    [67]
    Milburn T J, Doppler J, Holmes C A, et al. 2015. General description of quasiadiabatic dynamical phenomena near exceptional points. Physical Review A, 92: 052124. doi: 10.1103/PhysRevA.92.052124
    [68]
    Miri M A, Alù A. 2019. Exceptional points in optics and photonics. Science, 363 : eaar7709.
    [69]
    Mousavi S H, Khanikaev A B, Wang Z. 2015. Topologically protected elastic waves in phononic metamaterials. Nature Communications, 6 : 8682.
    [70]
    Muhlestein M B, Sieck C F, Alù A, et al. 2016. Reciprocity, passivity and causality in Willis materials. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472 : 20160604.
    [71]
    Özdemir Ş K, Rotter S, Nori F, et al. 2019. Parity–time symmetry and exceptional points in photonics. Nature Materials, 18: 783-798. doi: 10.1038/s41563-019-0304-9
    [72]
    Peng B, Özdemir Ş K, Lei F, et al. 2014. Parity–time-symmetric whispering-gallery microcavities. Nature Physics, 10: 394-398. doi: 10.1038/nphys2927
    [73]
    Peng Y G, Mazor Y, Alù A. 2022. Fundamentals of acoustic Willis media. Wave Motion, 112: 102930.
    [74]
    Qi Y, Qiu C, Xiao M, et al. 2020. Acoustic realization of quadrupole topological insulators. Physical Review Letters, 124: 206601. doi: 10.1103/PhysRevLett.124.206601
    [75]
    Quan L, Yves S, Peng Y, et al. 2021. Odd Willis coupling induced by broken time-reversal symmetry. Nature Communications, 12: 2615. doi: 10.1038/s41467-021-22745-5
    [76]
    Rosa M I N, Mazzotti M, Ruzzene M. 2021. Exceptional points and enhanced sensitivity in PT-symmetric continuous elastic media. Journal of the Mechanics and Physics of Solids, 149 : 104325.
    [77]
    Rüter C E, Makris K G, El-Ganainy R, et al. 2010. Observation of parity–time symmetry in optics. Nature Physics, 6: 192. doi: 10.1038/nphys1515
    [78]
    Scheibner C, Souslov A, Banerjee D, et al. 2020. Odd elasticity. Nature Physics, 16: 475-480. doi: 10.1038/s41567-020-0795-y
    [79]
    Shang C, Liu S, Shao R, et al. 2022. Experimental identification of the second‐order non‐Hermitian skin effect with physics‐graph‐informed machine learning. Advanced Science, 9: 2202922. doi: 10.1002/advs.202202922
    [80]
    Shankar S, Souslov A, Bowick M, et al. 2022. Topological active matter. Nature Reviews Physics, 4: 380-398. doi: 10.1038/s42254-022-00445-3
    [81]
    Shen H, Zhen B, Fu L. 2018. Topological Band Theory for non-Hermitian Hamiltonians. Physical Review Letters, 120 : 146402.
    [82]
    Sieck C F, Alù A, Haberman M R. 2017. Origins of Willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization. Physical Review B, 96 : 104303.
    [83]
    Stenholm S. 1984. Foundations of Laser Spectroscopy. Wiley-Interscience; New York.
    [84]
    Tang W, Ding K, Ma G. 2021. Direct measurement of topological properties of an exceptional parabola. Physical Review Letters, 127 : 034301.
    [85]
    Tang W, Jiang X, Ding K, et al. 2020. Exceptional nexus with a hybrid topological invariant. Science, 370: 1077-1080. doi: 10.1126/science.abd8872
    [86]
    Thouless D J, Kohmoto M, Nightingale M P, et al. 1982. Quantized hall conductance in a two-dimensional periodic potential. Physical Review Letters, 49: 405-408. doi: 10.1103/PhysRevLett.49.405
    [87]
    Tian Y, Ge H, Lu M H, et al. 2019. Research advances in acoustic metamaterials. Acta Physica Sinica, 68 : 194301-194301-194301-194312.
    [88]
    Tokura Y, Yasuda K, Tsukazaki A. 2019. Magnetic topological insulators. Nature Reviews Physics, 1 : 126-143.
    [89]
    Uzdin R, Mailybaev A, Moiseyev N. 2011. On the observability and asymmetry of adiabatic state flips generated by exceptional points. Journal of Physics A-Mathematical and Theoretical, 44 : 435302.
    [90]
    Uzdin R, Moiseyev N. 2012. Scattering from a waveguide by cycling a non-Hermitian degeneracy. Physical Review A, 85 : 031804.
    [91]
    Veletsos A S, Ventura C E. 1986. Modal analysis of non-classically damped linear systems. Earthquake Engineering & Structural Dynamics, 14 : 217-243.
    [92]
    Wang A, Meng Z, Chen C Q. 2023. Non-Hermitian topology in static mechanical metamaterials. Science advances, 9 : eadf7299.
    [93]
    Wang C, Sweeney William R, Stone A D, et al. 2021a. Coherent perfect absorption at an exceptional point. Science, 373: 1261-1265. doi: 10.1126/science.abj1028
    [94]
    Wang H, Zhang X, Hua J, et al. 2021b. Topological physics of non-Hermitian optics and photonics: a review. Journal of Optics, 23: 123001. doi: 10.1088/2040-8986/ac2e15
    [95]
    Wang X, Fang X, Mao D, et al. 2019. Extremely asymmetrical acoustic metasurface mirror at the exceptional point. Physical Review Letters, 123: 214302. doi: 10.1103/PhysRevLett.123.214302
    [96]
    Welters A. 2011. On explicit recursive formulas in the spectral perturbation analysis of a Jordan block. SIAM Journal on Matrix Analysis and Applications, 32 : 1-22.
    [97]
    Wiersig J. 2014. Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle detection. Physical Review Letters, 112 : 203901.
    [98]
    Willis J R. 1981. Variational principles for dynamic problems for inhomogeneous elastic media. Wave Motion, 3 : 1-11.
    [99]
    Wu Q, Chen Y, Huang G. 2019. Asymmetric scattering of flexural waves in a parity-time symmetric metamaterial beam. The Journal of the Acoustical Society of America, 146 : 850-862.
    [100]
    Wu Q, Xu X, Qian H, et al. 2023. Active metamaterials for realizing odd mass density. Proceedings of the National Academy of Sciences, 120: e2209829120. doi: 10.1073/pnas.2209829120
    [101]
    Wu Y, Zhou P, Li T, et al. 2021. High-order exceptional point based optical sensor. Optics Express, 29: 6080-6091. doi: 10.1364/OE.418644
    [102]
    Xiao Z, Li H, Kottos T, et al. 2019. Enhanced sensing and nondegraded thermal noise performance based on PT-symmetric electronic circuits with a sixth-order exceptional point. Physical Review Letters, 123: 213901. doi: 10.1103/PhysRevLett.123.213901
    [103]
    Xin L, Siyuan Y, Harry L, et al. 2020. Topological mechanical metamaterials: A brief review. Current Opinion in Solid State and Materials Science, 24: 100853. doi: 10.1016/j.cossms.2020.100853
    [104]
    Xing T, Pan Z, Tao Y, et al. 2020. Ultrahigh sensitivity stress sensing method near the exceptional point of parity-time symmetric systems. Journal of Physics D:Applied Physics, 53: 205102. doi: 10.1088/1361-6463/ab761d
    [105]
    Xu H, Mason D, Jiang L, et al. 2016. Topological energy transfer in an optomechanical system with exceptional points. Nature, 537: 80-83. doi: 10.1038/nature18604
    [106]
    Yang F, Liu Y C, You L. 2017. Anti-PT symmetry in dissipatively coupled optical systems. Physical Review A, 96 : 053845.
    [107]
    Yang X, Li J, Ding Y, et al. 2022. Observation of transient parity-time symmetry in electronic systems. Physical Review Letters, 128: 065701. doi: 10.1103/PhysRevLett.128.065701
    [108]
    Yang Z, Zhang K, Fang C, et al. 2020. Non-Hermitian Bulk-boundary correspondence and auxiliary generalized brillouin zone theory. Physical Review Letters, 125: 226402. doi: 10.1103/PhysRevLett.125.226402
    [109]
    Yoon J W, Choi Y, Hahn C, et al. 2018. Time-asymmetric loop around an exceptional point over the full optical communications band. Nature, 562: 86-90. doi: 10.1038/s41586-018-0523-2
    [110]
    Yuan J, Geng L, Huang J, et al. 2022. Exceptional points induced by time-varying mass to enhance the sensitivity of defect detection. Physical Review Applied, 18: 064055. doi: 10.1103/PhysRevApplied.18.064055
    [111]
    Zhang H, Huang R, Zhang S D, et al. 2020. Breaking anti-PT symmetry by spinning a resonator. Nano Letters, 20: 7594-7599. doi: 10.1021/acs.nanolett.0c03119
    [112]
    Zhang L, Yang Y, Ge Y, et al. 2021a. Acoustic non-Hermitian skin effect from twisted winding topology. Nature Communications, 12: 6297. doi: 10.1038/s41467-021-26619-8
    [113]
    Zhang L F, Tang L Z, Huang Z H, et al. 2021b. Machine learning topological invariants of non-Hermitian systems. Physical Review A, 103: 012419. doi: 10.1103/PhysRevA.103.012419
    [114]
    Zhang X L, Chan C T. 2019. Dynamically encircling exceptional points in a three-mode waveguide system. Communications Physics, 2 : 63.
    [115]
    Zhang X L, Jiang T, Chan C T. 2019. Dynamically encircling an exceptional point in anti-parity-time symmetric systems: asymmetric mode switching for symmetry-broken modes. Light-Science & Applications, 8 : 88.
    [116]
    Zhang X L, Wang S, Hou B, et al. 2018. Dynamically encircling exceptional points: In situ control of encircling loops and the role of the starting point. Physical Review X, 8: 021066.
    [117]
    Zhong J, Wang K, Park Y, et al. 2021. Nontrivial point-gap topology and non-Hermitian skin effect in photonic crystals. Physical Review B, 104: 125416. doi: 10.1103/PhysRevB.104.125416
    [118]
    Zhou Z, Jia B, Wang N, et al. 2023. Observation of perfectly-chiral exceptional point via bound state in the continuum. Physical Review Letters, 130: 116101. doi: 10.1103/PhysRevLett.130.116101
    [119]
    Zhu X, Ramezani H, Shi C, et al. 2014. PT-symmetric acoustics. Physical Review X, 4: 031042.
    [120]
    Zhu Y, Long H, Liu C, et al. 2022. An ultra-thin ventilated metasurface with extreme asymmetric absorption. Applied Physics Letters, 120: 141701. doi: 10.1063/5.0086859
    [121]
    陈阿丽, 汪越胜, 王艳锋, 等. 2022. 声学/弹性相位梯度超表面设计: 原理、功能基元、可调和编码. 力学进展, 52 : 948-1011. (Chen A L, Wang Y S, Wang Y F, et al. 2022. Design of acoustic/elastic phase gradient metasurfaces: Principles, functional elements, tunability, and coding. Advances in Mechanics, 52 : 948-1011).

    Chen A L, Wang Y S, Wang Y F, et al. 2022. Design of acoustic/elastic phase gradient metasurfaces: Principles, functional elements, tunability, and coding. Advances in Mechanics, 52: 948-1011
    [122]
    陈毅, 刘晓宁, 向平, 等. 2016. 五模材料及其水声调控研究. 力学进展, 46: 201609. (Chen Y, Liu X N, Xiang P, et al. 2016. Pentamode material for underwater acoustic wave control. Advances in Mechanics, 46: 201609). doi: 10.6052/1000-0992-16-010

    Chen Y, Liu X N, Xiang P, et al. 2016. Pentamode material for underwater acoustic wave control. Advances in Mechanics, 46: 201609. doi: 10.6052/1000-0992-16-010
    [123]
    陈毅, 张泉, 张亚飞, 等. 2021. 弹性拓扑材料研究进展. 力学进展, 51: 189-256. (Chen Y, Zhang Q, Zhang Y F, et al. 2021. Research progress of elastic topological materials. Advances in Mechanics, 51: 189-256). doi: 10.6052/1000-0992-21-015

    Chen Y, Zhang Q, Zhang Y F, et al. 2021. Research progress of elastic topological materials. Advances in Mechanics, 51: 189-256. doi: 10.6052/1000-0992-21-015
    [124]
    尹剑飞, 蔡力, 方鑫, 等. 2022. 力学超材料研究进展与减振降噪应用. 力学进展, 52: 508-586. (Yin J F, Cai L, Fang X, et al. 2022. Review on research progress of mechanical metamaterials and their applications in vibration and noise control. Advances in Mechanics, 52: 508-586). doi: 10.6052/1000-0992-22-005

    Yin J F, Cai L, Fang X, et al. 2022. Review on research progress of mechanical metamaterials and their applications in vibration and noise control. Advances in Mechanics, 52: 508-586. doi: 10.6052/1000-0992-22-005
    [125]
    祝雪丰, 彭玉桂, 沈亚西. 2017. 宇称时间对称性声学. 物理, 46: 740-748. (Zhu X F, Peng Y G, Shen Y X. 2017. Parity-time symmetric acoustics. Physics, 46: 740-748).

    Zhu X F, Peng Y G, Shen Y X. 2017. Parity-time symmetric acoustics. Physics, 46: 740-748
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