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.
Phonons are an outcome of translational symmetry applied to the potential within a Schrödinger equation. Fractal self-similarity can be considered as a type of symmetry comparable to translational symmetry. Both involve some form of invariance, either under displacement or change of position (translational symmetry) and under changes in scale (fractal self-similarity).