Fractons are the fractal analog of phonons. Phonons are the result of applying translational symmetry to the potential in a Schrödinger equation. Fractal self-similarity can be thought of as a symmetry somewhat comparable to translational symmetry. Translational symmetry is symmetry under displacement or change of position, and fractal self-similarity is symmetry under change of scale. The quantum mechanical solutions to such a problem in general lead to a continuum of states with different frequencies. In other words, a fracton band is comparable to a phonon band. The vibrational modes are restricted to part of the substrate and are thus not fully delocalized, unlike phonon vibrational modes. Instead, there is a hierarchy of vibrational modes that encompass smaller and smaller parts of the substrate.
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- Srivastava, G. P. (1990), The Physics of Phonons, CRC Press, pp. 328–329, ISBN 9780852741535.