A fracton is a collective quantized
vibration
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic function, periodic, such as the motion of a pendulum ...
on a
substrate with a
fractal
In mathematics, a fractal is a 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 scales, as illu ...
structure.
[.]
Fractons are the fractal analog of
phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
s. Phonons are the result of applying
translational symmetry
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by .
In physics and mathematics, continuous translational symmetry is the invariance of a system of equatio ...
to the potential in a
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
. Fractal
self-similarity
__NOTOC__
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 se ...
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. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
. 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
eeds ref.
References
External links
The ‘Weirdest’ Matter, Made of Partial Particles, Defies Description
Fractals
Quasiparticles
{{CMP-stub