A fracton is an emergent topological quasiparticle excitation which is immobile when in isolation. 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.

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.

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

The Hamiltonian is given by

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 ${\displaystyle {\frac {1+B_{v,\mu }}{2}}}$.