Fracton
Encyclopedia
A fracton is a collective quantized vibration
on a substrate with a fractal
structure.
Fractons are the fractal analog of phonon
s. 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 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 which encompass smaller and smaller parts of the substrate.
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
on a substrate with a fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
structure.
Fractons are the fractal analog of phonon
Phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...
s. Phonons are the result of applying translational symmetry
Translational symmetry
In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation...
to the potential in a Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
. Fractal self-similarity
Self-similarity
In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...
can be thought of as a symmetry somewhat comparable to translational symmetry. Translational symmetry is symmetry under displacement or change of position, and 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 which encompass smaller and smaller parts of the substrate.