Gapped fracton models often feature a topological ground state degeneracy that grows exponentially and sub-extensively with system size.
Furthermore, isolated fracton particles in type II models are associated with nonlocal operators with intricate fractal structure.
where the product runs over all edges in the lattice, this Hamiltonian obeys subsystem symmetries acting on individual planes.
The X cube model hosts two types of elementary excitations, the fracton and lineon (also known as the one-dimensional particle).
In other words, there is no local operator that can be acted on an isolated fracton to move it to a different location.
Here, as usual, "moving" refers to the repeated action of local unitary operators that translate the particles.
string operator attached to the lineon exactly one time, then at the end of the rotation of the planeon the wave function will pick up a factor of
[5] It is possible to construct the X cube model by taking three stacks of toric code sheets, on along each of the three axes, superimposing them, and adding couplings to the edges where they intersect.
Thus the Haah's code fracton model also in some sense exhibits the property that the logarithm of the ground state degeneracy tends to scale in direct proportion to the linear dimension of the system.
Just like in type I models and in topologically ordered systems, different ground states of Haah's code cannot be distinguished by local operators.
Thus we see that an infinitely large fractal-shaped operator is required to generate an isolated fracton out of the ground state in the Haah's code model.
Unlike in type I models, there are no stable bound states of a finite number of fractons that are mobile.
by applying a finite depth local unitary map and arbitrarily adding and/or removing two-dimensional gapped systems, then
It is important in this definition that the local unitary map remains at finite depth as the sizes of systems 1 and 2 are taken to the thermodynamic limit.
To state the definition more precisely, suppose one can find two (possibly empty or infinite) collections of two-dimensional gapped phases (with arbitrary topological order),
This would make it impossible to study the phases of fracton matter in the thermodynamic limit where system size
The concept of foliated fracton order resolves this issue, by allowing degenerate subsystems ( two-dimensional gapped topological phases) to be used as "free resources" that can be arbitrarily added or removed from the system to account for these differences.
In other words, the same type of model on the same manifold with the same system size may have different ground state degeneracies depending on the underlying choice of foliation.
[4] By definition, the number of superselection sectors in a fracton model is infinite (i.e. scales with system size).
This is analogous to how topological orders tend to have a finite list of ordinary superselection sectors.
Generally for fracton models in the ground state, when considering the entanglement entropy of a subregion of space with large linear size
It is possible to use a mutual information calculation to extract a contribution to entanglement entropy that is unique to the foliated fracton order.
[12][11] 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.
Various formulations and constraints of symmetric tensor gauge theory tend to result in conservation laws that imply the existence of restricted-mobility particles.
) can be shown to be conserved: When integrating by parts, we have assumed that there is no electric field at spatial infinity.
[13] One approach to constructing an explicit action for scalar fractonic matter fields and their coupling to the symmetric tensor gauge theory is the following.
A global charge conservation symmetry would imply that the action is symmetric under the transformation
A global dipole moment conservation symmetry would imply that the action is symmetric under the transformation
[16] [17] Fractons have also been shown to appear in quantum linearized gravity models [18] and (via a duality) as disclination crystal defects.
[19] However, aside from the duality to crystal defects, and although it has been shown to be possible in principle, [20] [21] other experimental realizations of gapped fracton models have not yet been realized.