A fracton is a collective quantized
vibration
Vibration () is a mechanical phenomenon whereby oscillations occur about an equilibrium point. Vibration may be deterministic if the oscillations can be characterised precisely (e.g. the periodic motion of a pendulum), or random if the os ...
on a
substrate
Substrate may refer to:
Physical layers
*Substrate (biology), the natural environment in which an organism lives, or the surface or medium on which an organism grows or is attached
** Substrate (aquatic environment), the earthy material that exi ...
with a
fractal
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
structure.
[.]
Fractons are the fractal analog of
phonon
A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. In the context of optically trapped objects, the quantized vibration mode can be defined a ...
s. Phonons are the result of applying
translational symmetry
In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation (without rotation). Discrete translational symmetry is invariant under discrete translation.
Analogously, an operato ...
to the potential in a
Schrödinger equation
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
. Fractal
self-similarity
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar ...
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
Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
. 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.
References
Literature
*
External links
The ‘Weirdest’ Matter, Made of Partial Particles, Defies Description
Fractals
Quasiparticles
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