Fracton (subdimensional particle)

The immobility of fractons in symmetric tensor gauge theory can be understood as a generalization of electric charge conservation resulting from a modified Gauss's law. For example, in the U(1) scalar charge model, the fracton charge density (${\displaystyle \rho }$) is related to a symmetric electric field tensor (${\displaystyle E_{ij}}$, a theoretical generalization of the usual electric vector field) via ${\displaystyle \rho =\partial _{i}\partial _{j}E_{ij}}$, where the repeated spatial indices ${\displaystyle i,j=1,2,3}$ are implicitly summed over. Both the fracton charge (${\displaystyle q}$) and dipole moment (${\displaystyle p_{i}}$) can be shown to be conserved:

${\displaystyle {\begin{aligned}q&=\int \rho \;d^{3}x=\int \partial _{i}(\partial _{j}E_{ij})\;d^{3}x=0\\p_{i}&=\int x_{i}\rho \;d^{3}x=\int x_{i}\partial _{j}\partial _{k}E_{jk}\;d^{3}x=-\int \partial _{k}E_{ik}\;d^{3}x=0\end{aligned}}}$

When integrating by parts, we have assumed that there is no electric field at spatial infinity. Since the total fracton charge and dipole moment is zero under this assumption, this implies that the charge and dipole moment is conserved. Because moving an isolated charge changes the total dipole moment, this implies that isolated charges are immobile in this theory. However, two oppositely charged fractons, which forms a fracton dipole, can move freely since this does not change the dipole moment. [3]

Fractons were originally studied as an analytically tractable realization of quantum glassiness where the immobility of isolated fractons results in a slow relaxation rate [4] .[5] This immobility has also been shown to be capable of producing a partially self-correcting quantum memory, which could be useful for making an analog of a hard drive for a quantum computer [6] .[7] Fractons have also been shown to appear in quantum linearized gravity models [8] and (via a duality) as disclination crystal defects .[9] However, aside from the duality to crystal defects, and although it has been shown to be possible in principle [10] ,[11] other experimental realizations of fractons have not yet been realized.

It has been conjectured [14] that many type-I models are examples of foliated fracton phases; however, it remains unclear whether non-Abelian fracton models[15][16][17] can be understood within the foliated framework.