Zeller, R. C. & Pohl, R. O. Thermal conductivity and specific heat of noncrystalline solids. Phys. Rev. B 4, 2029–2041 (1971).
Alexander, S. Amorphous solids: their structure, lattice dynamics and elasticity. Phys. Rep. 296, 65–236 (1998).
Anderson, P. W. Through the glass lightly. Science 267, 1615–1616 (1995).
Ramos, M. A. Low-Temperature Thermal and Vibrational Properties of Disordered Solids: A Half-Century of Universal ‘Anomalies’ of Glasses (World Scientific, 2022).
Phillips, W. A. & Anderson, A. C. Amorphous Solids: Low-Temperature Properties (Springer, 1981).
Yu, C. C. & Carruzzo, H. M. in Low-Temperature Thermal and Vibrational Properties of Disordered Solids: A Half-Century of Universal ‘Anomalies’ of Glasses 113–139 (World Scientific, 2023).
Elliott, S. R. A unified model for the low-energy vibrational behaviour of amorphous solids. Europhys. Lett. 19, 201–206 (1992).
Leonforte, F., Tanguy, A., Wittmer, J. P. & Barrat, J. L. Inhomogeneous elastic response of silica glass. Phys. Rev. Lett. 97, 055501 (2006).
Schirmacher, W. Thermal conductivity of glassy materials and the ‘boson peak’. Europhys. Lett. 73, 892–898 (2006).
Schirmacher, W., Ruocco, G. & Scopigno, T. Acoustic attenuation in glasses and its relation with the boson peak. Phys. Rev. Lett. 98, 025501 (2007).
Marruzzo, A., Schirmacher, W., Fratalocchi, A. & Ruocco, G. Heterogeneous shear elasticity of glasses: the origin of the boson peak. Sci. Rep. 3, 1407 (2013).
Schirmacher, W., Scopigno, T. & Ruocco, G. Theory of vibrational anomalies in glasses. J. Non-Cryst. Solids 407, 133–140 (2015).
Schirmacher, W. et al. The nature of non-phononic excitations in disordered systems. Nat. Commun. 15, 3107 (2024).
Lerner, E. & Bouchbinder, E. Boson-peak vibrational modes in glasses feature hybridized phononic and quasilocalized excitations. J. Chem. Phys. 158, 194503 (2023).
Moriel, A., Lerner, E. & Bouchbinder, E. Boson peak in the vibrational spectra of glasses. Phys. Rev. Res. 6, 023053 (2024).
Mahajan, S. & Ciamarra, M. P. Unifying description of the vibrational anomalies of amorphous materials. Phys. Rev. Lett. 127, 215504 (2021).
Mahajan, S., Seow Yang Han, D., Jiang, C., Baggioli, M. & Ciamarra, M. P. Geometrical and vibrational properties of the defects driving the boson peak. Phys. Rev. E 112, 035413 (2025).
Galperin, Y. M., Karpov, V. G. & Kozub, V. I. Localized states in glasses. Adv. Phys. 38, 669–737 (1989).
Buchenau, U., Galperin, Y. M., Gurevich, V. L. & Schober, H. R. Anharmonic potentials and vibrational localization in glasses. Phys. Rev. B 43, 5039–5045 (1991).
Buchenau, U. et al. Interaction of soft modes and sound waves in glasses. Phys. Rev. B 46, 2798 (1992).
Klinger, M. I. & Kosevich, A. M. Soft-mode dynamics model of boson peak and high frequency sound in glasses: ‘inelastic’ Ioffe-Regel crossover and strong hybridization of excitations. Phys. Lett. A 295, 311–317 (2002).
Gurevich, V. L., Parshin, D. A. & Schober, H. R. Anharmonicity, vibrational instability, and the boson peak in glasses. Phys. Rev. B 67, 094203 (2003).
Parshin, D. A., Schober, H. R. & Gurevich, V. L. Vibrational instability, two-level systems, and the boson peak in glasses. Phys. Rev. B 76, 064206 (2007).
Schober, H. R. Quasi-localized vibrations and phonon damping in glasses. J. Non-Cryst. Solids 357, 501–505 (2011).
Pazmiño Betancourt, B. A., Starr, F. W. & Douglas, J. F. String-like collective motion in the α- and β-relaxation of a coarse-grained polymer melt. J. Chem. Phys. 148, 104508 (2018).
Lund, F. Normal modes and acoustic properties of an elastic solid with line defects. Phys. Rev. B 91, 094102 (2015).
Bianchi, E., Giordano, V. M. & Lund, F. Elastic anomalies in glasses: elastic string theory understanding of the cases of glycerol and silica. Phys. Rev. B 101, 174311 (2020).
Zhang, H., Wang, X., Yu, H.-B. & Douglas, J. F. Fast dynamics in a model metallic glass-forming material. J. Chem. Phys. 154, 084505 (2021).
Hu, Y.-C. & Tanaka, H. Origin of the boson peak in amorphous solids. Nat. Phys. 18, 669–677 (2022).
Hu, Y.-C. & Tanaka, H. Universality of stringlet excitations as the origin of the boson peak of glasses with isotropic interactions. Phys. Rev. Res. 5, 023055 (2023).
Jiang, C., Baggioli, M. & Douglas, J. F. Stringlet excitation model of the boson peak. J. Chem. Phys. 160, 214505 (2024).
Jiang, C. & Baggioli, M. Phonons in stringlet-land and the boson peak. J. Phys. Condens. Matter 36, 505101 (2024).
Liu, A. J. & Nagel, S. R. The jamming transition and the marginally jammed solid. Annu. Rev. Condens. Matter Phys. 1, 347–369 (2010).
DeGiuli, E., Laversanne-Finot, A., Düring, G., Lerner, E. & Wyart, M. Effects of coordination and pressure on sound attenuation, boson peak and elasticity in amorphous solids. Soft Matter 10, 5628–5644 (2014).
Mizuno, H., Shiba, H. & Ikeda, A. Continuum limit of the vibrational properties of amorphous solids. Proc. Natl Acad. Sci. USA 114, E9767–E9774 (2017).
Götze, W. & Mayr, M. R. Evolution of vibrational excitations in glassy systems. Phys. Rev. E 61, 587 (2000).
Grigera, T. S., Martín-Mayor, V., Parisi, G. & Verrocchio, P. Phonon interpretation of the ‘boson peak’ in supercooled liquids. Nature 422, 289–292 (2003).
Baggioli, M. & Zaccone, A. Universal origin of boson peak vibrational anomalies in ordered crystals and in amorphous materials. Phys. Rev. Lett. 122, 145501 (2019).
Taraskin, S. N., Loh, Y. L., Natarajan, G. & Elliott, S. R. Origin of the boson peak in systems with lattice disorder. Phys. Rev. Lett. 86, 1255–1258 (2001).
Chumakov, A. I. et al. Equivalence of the boson peak in glasses to the transverse acoustic Van Hove singularity in crystals. Phys. Rev. Lett. 106, 225501 (2011).
Wang, Y., Qian, Z., Tong, H. & Tanaka, H. Hyperuniform disordered solids with crystal-like stability. Nat. Commun. 16, 1398 (2025).
Torquato, S. Hyperuniform states of matter. Phys. Rep. 745, 1–95 (2018).
Lerner, E. & Bouchbinder, E. Frustration-induced internal stresses are responsible for quasilocalized modes in structural glasses. Phys. Rev. E 97, 032140 (2018).
Gelin, S., Tanaka, H. & Lemaître, A. Anomalous phonon scattering and elastic correlations in amorphous solids. Nat. Mater. 15, 1177–1181 (2016).
Shintani, H. & Tanaka, H. Universal link between the boson peak and transverse phonons in glass. Nat. Mater. 7, 870–877 (2008).
Nakayama, T. Boson peak and terahertz frequency dynamics of vitreous silica. Rep. Prog. Phys. 65, 1195 (2002).
Baldi, G. et al. Thermal conductivity and terahertz vibrational dynamics of vitreous silica. Phys. Rev. B 77, 214309 (2008).
Baldi, G., Giordano, V. M., Ruta, B. & Monaco, G. On the nontrivial wave-vector dependence of the elastic modulus of glasses. Phys. Rev. B 93, 144204 (2016).
Monaco, G. & Mossa, S. Anomalous properties of the acoustic excitations in glasses on the mesoscopic length scale. Proc. Natl Acad. Sci. USA 106, 16907–16912 (2009).
Ruocco, G. et al. Relaxation processes in harmonic glasses? Phys. Rev. Lett. 84, 5788 (2000).
Szamel, G. & Flenner, E. Microscopic analysis of sound attenuation in low-temperature amorphous solids reveals quantitative importance of non-affine effects. J. Chem. Phys. 156, 144502 (2022).
Booij, H. C. & Thoone, G. P. J. M. Generalization of Kramers–Kronig transforms and some approximations of relations between viscoelastic quantities. Rheol. Acta 21, 15–24 (1982).
Lerner, E. & Bouchbinder, E. Low-energy quasilocalized excitations in structural glasses. J. Chem. Phys. 155, 200901 (2021).
Douglas, J. F., Yuan, Q.-L., Zhang, J., Zhang, H. & Xu, W.-S. A dynamical system approach to relaxation in glass-forming liquids. Soft Matter 20, 9140–9160 (2024).
Lerner, E., Düring, G. & Bouchbinder, E. Statistics and properties of low-frequency vibrational modes in structural glasses. Phys. Rev. Lett. 117, 035501 (2016).
Richard, D., Kapteijns, G. & Lerner, E. Detecting low-energy quasilocalized excitations in computer glasses. Phys. Rev. E 108, 044124 (2023).
Dean, P. in Localized Excitations in Solids 109–116 (Springer, 1968).
Kapteijns, G., Bouchbinder, E. & Lerner, E. Universal nonphononic density of states in 2D, 3D, and 4D glasses. Phys. Rev. Lett. 121, 055501 (2018).
Beltukov, Y. M., Fusco, C., Tanguy, A. & Parshin, D. A. Transverse and longitudinal vibrations in amorphous silicon. J. Phys. Conf. Ser. 661, 012056 (2015).
Mizuno, H., Saitoh, K. & Silbert, L. E. Elastic moduli and vibrational modes in jammed particulate packings. Phys. Rev. E 93, 062905 (2016).
Caroli, C. & Lemaître, A. Fluctuating elasticity fails to capture anomalous sound scattering in amorphous solids. Phys. Rev. Lett. 123, 055501 (2019).
Baggioli, M. & Zaccone, A. Theory of sound attenuation in amorphous solids from nonaffine motions. J. Phys. Condens. Matter 34, 215401 (2022).
Monaco, G. & Giordano, V. M. Breakdown of the Debye approximation for the acoustic modes with nanometric wavelengths in glasses. Proc. Natl Acad. Sci. USA 106, 3659–3663 (2009).
Tanguy, A., Wittmer, J. P., Leonforte, F. & Barrat, J. L. Continuum limit of amorphous elastic bodies: a finite-size study of low-frequency harmonic vibrations. Phys. Rev. B 66, 174205 (2002).
Ding, G. et al. Unified theory of phonon in solids with phase diagram of non-Debye anomalies. Nat. Phys. 21, 1911–1919 (2025).
Etrillard, J., Lasjaunias, J. C., Biljakovic, K., Toudic, B. & Coddens, G. Excess low temperature specific heat and related phonon density of states in a modulated incommensurate dielectric. Phys. Rev. Lett. 76, 2334–2337 (1996).
Cano, A. & Levanyuk, A. P. Explanation of the glasslike anomaly in the low-temperature specific heat of incommensurate phases. Phys. Rev. Lett. 93, 245902 (2004).
Reményi, G. et al. Incommensurate systems as model compounds for disorder revealing low-temperature glasslike behavior. Phys. Rev. Lett. 114, 195502 (2015).
Jiang, C., Zaccone, A., Setty, C. & Baggioli, M. Glassy heat capacity from overdamped phasons and hypothetical phason-induced superconductivity in incommensurate structures. Phys. Rev. B 108, 054203 (2023).
Zhang, H., Wang, X., Chremos, A. & Douglas, J. F. Superionic UO2: a model anharmonic crystalline material. J. Chem. Phys. 150, 174506 (2019).
Zhang, J., Zhang, H. & Douglas, J. F. A closer examination of the nature of atomic motion in the interfacial region of crystals upon approaching melting. J. Chem. Phys. 160, 114506 (2024).
Zhang, J., Douglas, J. F. & Zhang, H. String-like collective motion mediates the martensitic α–β transition in titanium. J. Chem. Phys. 163, 044504 (2025).
Meyer, A. et al. Harmonic behavior of metallic glasses up to the metastable melt. Phys. Rev. B 53, 12107 (1996).
Bruna, P. et al. Communication: are metallic glasses different from other glasses? A closer look at their high frequency dynamics. J. Chem. Phys. 135, 101101 (2011).
Ghosh, A. et al. Density of states of colloidal glasses and supercooled liquids. Soft Matter 6, 3082–3090 (2010).
Tan, P., Xu, N., Schofield, A. B. & Xu, L. Understanding the low-frequency quasilocalized modes in disordered colloidal systems. Phys. Rev. Lett. 108, 095501 (2012).
Zhang, L. et al. Experimental studies of vibrational modes in a two-dimensional amorphous solid. Nat. Commun. 8, 67 (2017).
Wang, Y., Hong, L., Wang, Y., Schirmacher, W. & Zhang, J. Disentangling boson peaks and Van Hove singularities in a model glass. Phys. Rev. B 98, 174207 (2018).
Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117, 1–19 (1995).
Wang, L. et al. Low-frequency vibrational modes of stable glasses. Nat. Commun. 10, 26 (2019).
Omar, M. A. Elementary Solid State Physics: Principles and Applications (Pearson Education India, 1999).