A fracton is an emergent topological quasiparticle excitation which is immobile when in isolation.^[1][2] Many theoretical systems have been proposed in which fractons exist as elementary excitations. Such systems are known as fracton models. Fractons have been identified in various CSS codes as well as in symmetric tensor gauge theories.

Gapped fracton models often feature a topological ground state degeneracy that grows exponentially and sub-extensively with system size. Among the gapped phases of fracton models, there is a non-rigorous phenomenological classification into "type I" and "type II". Type I fracton models generally have fracton excitations that are completely immobile, as well as other excitations, including bound states, with restricted mobility. Type II fracton models generally have fracton excitations and no mobile particles of any form. Furthermore, isolated fracton particles in type II models are associated with nonlocal operators with intricate fractal structure.^[3]

Models

Type I

The paradigmatic example of a type I fracton model is the X-cube model. Other examples of type I fracton models include the semionic X-cube model, the checkerboard model, the Majorana checkerboard model, the stacked Kagome X-cube model, the hyperkagome X-cube model, and more.

X-cube model

The X-cube model is constructed on a cubic lattice, with qubits on each edge of the lattice.

The Hamiltonian is given by

${\displaystyle H=-\sum _{c}A_{c}-\sum _{{\textrm {}}v}(B_{v,x}+B_{v,y}+B_{v,z})}$

Here, the sums run over cubic unit cells and over vertices. For any cubic unit cell ${\displaystyle c}$, the operator ${\displaystyle A_{c}}$ is equal to the product of the Pauli ${\displaystyle X}$ operator on all 12 edges of that unit cube. For any vertex of the lattice ${\displaystyle v}$, operator ${\displaystyle B_{v,\mu }}$ is equal to the product of the Pauli ${\displaystyle Z}$ operator on all four edges adjacent to vertex ${\displaystyle v}$ and perpendicular to the ${\displaystyle \mu }$ axis. Other notation conventions in the literature may interchange ${\displaystyle X}$ and ${\displaystyle Z}$.

In addition to obeying an overall ${\displaystyle Z_{2}\times Z_{2}}$ symmetry defined by global symmetry generators ${\displaystyle \prod _{\ell }X_{\ell }}$ and ${\displaystyle \prod _{\ell }Z_{\ell }}$ where the product runs over all edges in the lattice, this Hamiltonian obeys subsystem symmetries acting on individual planes.

All of the terms in this Hamiltonian commute and belong to the Pauli algebra. This makes the Hamiltonian exactly solvable. One can simultaneously diagonalise all the terms in the Hamiltonian, and the simultaneous eigenstates are the Hamiltonian's energy eigenstates. A ground state of this Hamiltonian is a state ${\displaystyle |\mathrm {GS} \rangle }$ that satisfies ${\displaystyle A_{c}|\mathrm {GS} \rangle =1}$ and ${\displaystyle B_{v,\mu }|\mathrm {GS} \rangle =1}$ for all ${\displaystyle c,v,\mu }$. One can explicitly write down a ground state using projection operators ${\displaystyle {\frac {1+A_{c}}{2}}}$ and ${\displaystyle {\frac {1+B_{v,\mu }}{2}}}$.

It is important to note that the constraints posed by ${\displaystyle A_{c}=1}$ and ${\displaystyle B_{v,\mu }=1}$ are not all linearly independent when the X cube model is embedded on a compact manifold. This leads to a large ground state degeneracy that increases with system size. On a torus with dimensions ${\displaystyle L_{x},L_{y},L_{z}}$, the ground state degeneracy is exactly ${\displaystyle 2^{2L_{x}+2L_{y}+2L_{z}-3}}$ .^[4] A similar degeneracy scaling, ${\displaystyle \log \mathrm {GSD} \propto L}$, is seen on other manifolds as well as in the thermodynamic limit.

Restricted mobility excitations

The X cube model hosts two types of elementary excitations, the fracton and lineon (also known as the one-dimensional particle).

If a quantum state is such that the eigenvalue of ${\displaystyle A_{c}=-1}$ for some unit cube ${\displaystyle c}$, then we say that, in this quantum state, there is a fracton located at the position ${\displaystyle c}$. For example, if ${\displaystyle |\mathrm {GS} \rangle }$ is a ground state of the Hamiltonian, then for any edge ${\displaystyle \ell }$, the state ${\displaystyle X_{\ell }|\mathrm {GS} \rangle }$ features four fractons, one each on the cubes adjacent to ${\displaystyle \ell }$.

Given a rectangle ${\displaystyle R}$ in a plane, one can define a "membrane" operator as ${\displaystyle Z_{R}\equiv \prod _{\ell \in R}Z_{\ell }}$ where the product runs over all edges ${\displaystyle \ell }$ perpendicular to the rectangle that pass through this rectangle. Then the state ${\displaystyle Z_{R}|\mathrm {GS} \rangle }$ features four fractons each located at the cubes next to the corners of the rectangle. Thus, an isolated fracton can appear in the limit of taking the length and width of the rectangle ${\displaystyle R}$ to infinity. The fact that a nonlocal membrane operator acts on the ground state to produce an isolated fracton is analogous to how, in smaller dimensional systems, nonlocal string operators can produce isolated ${\displaystyle \pi }$ flux particles and domain walls.

This construction shows that an isolated fracton cannot be mobile in any direction. In other words, there is no local operator that can be acted on an isolated fracton to move it to a different location. In order to move an individual isolated fracton, one would need to apply a highly nonlocal operator to move the entire membrane associated with it.

If a quantum state is such that the eigenvalue of ${\displaystyle B_{v,x}}$Zdroj:https://en.wikipedia.org?pojem=Fracton_(subdimensional_particle)
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