Here is a list of fracton codes.
| Code | Description |
|---|---|
| Chamon model code | A foliated type-I fracton non-CSS code defined on a cubic lattice using one weight-eight stabilizer generator acting on the eight vertices of each cube in the lattice [1; Eq. (D38)]. |
| Checkerboard model code | A foliated type-I fracton code defined on a cubic lattice that admits weight-eight \(X\)- and \(Z\)-type stabilizer generators on the eight vertices of each cube in the lattice. A tetrahedral Ising model can be used to obtain the checkerboard model by gauging [2–4,4] its subsystem symmetry [5]. |
| Fibonacci fractal spin-liquid code | A fractal type-I fracton CSS code defined on a cubic lattice [1; Eq. (D23)]. |
| Four Color Cube (FCC) fracton model code | A fracton code obtained from four coupled X-cube models using p-membrane condensation. A modular-qudit generalization has been proposed [6]. |
| Fracton stabilizer code | A 3D translationally invariant modular-qudit stabilizer code whose codewords make up the ground-state space of a Hamiltonian in a fracton phase. Unlike topological phases, whose excitations can move in any direction, fracton phases are characterized by excitations whose movement is restricted. |
| Haah cubic code (CC) | A 3D lattice stabilizer code on a length-\(L\) cubic lattice with one or two qubits per site. Admits two types of stabilizer generators with support on each cube of the lattice. In the non-CSS case, these two are related by spatial inversion. For CSS codes, we require that the product of all corner operators is the identity. We lastly require that there are no non-trival string operators, meaning that single-site operators are a phase, and any period one logical operator \(l \in \mathsf{S}^{\perp}\) is just a phase. |
| Hsieh-Halasz (HH) code | Member of one of two families of fracton codes, named HH-I and HH-II, defined on a cubic lattice with two qubits per site. HH-I (HH-II) is a CSS (non-CSS) stabilizer code family, with the former identified as a foliated type-I fracton code [1]. |
| Hsieh-Halasz-Balents (HHB) code | Member of one of two families of fracton codes, named HHB model A and B, defined on a cubic lattice with two qubits per site. Both are expected to be foliated type-I fracton codes [1; Eqs. (D42-D43)]. |
| Layer code | Member of a family of 3D lattice CSS codes with stabilizer generator weights \(\leq 6\) that are obtained by coupling layers of 2D surface code according to the Tanner graph of a QLDPC code. Geometric locality is maintained because, instead of being concatenated, each pair of parallel surface-code squares are fused (or quasi-concatenated) with perpendicular surface-code squares via lattice surgery. |
| Majorana checkerboard code | A Majorana analogue of the X-cube model defined on a cubic lattice. The code admits weight-eight Majorana stabilizer generators on the eight vertices of each cube of a checkerboard sublattice. |
| Qudit X-cube model code | Generalization of the X-cube model code to modular qudits. |
| Qudit cubic code | Generalization of the Haah cubic code to modular qudits. |
| Sierpinski prism model code | A fractal type-I fracton CSS code defined on a cubic lattice [1; Eq. (D22)]. The code admits an excitation-moving operator shaped like a Sierpinski triangle [1; Fig. 2]. |
| Two-foliated fracton code | A type-I fracton code obtained by gauging [2–4,4] a topological phase with subsystem symmetry. |
| Type-II fractal spin-liquid code | A type-II fracton prime-qudit CSS code defined on a cubic lattice [1; Eqs. (D9-D10)]. |
| X-cube model code | A foliated type-I fracton code supporting a subextensive number of logical qubits. Variants include several generalized X-cube models [7]. A non-stabilizer commuting-projector code constructed by stacking layers of the double-semion string-net model, called the semionic X-cube model [8], is equivalent to the X-cube model [9] (see also Refs. [10,11]). |
References
- [1]
- A. Dua, I. H. Kim, M. Cheng, and D. J. Williamson, “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
- [2]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [3]
- L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
- [4]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [5]
- S. Vijay, J. Haah, and L. Fu, “Fracton topological order, generalized lattice gauge theory, and duality”, Physical Review B 94, (2016) arXiv:1603.04442 DOI
- [6]
- E. Wickenden, M. Qi, A. Dua, and M. Hermele, “Planon-modular fracton orders”, (2025) arXiv:2412.14320
- [7]
- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
- [8]
- H. Ma, E. Lake, X. Chen, and M. Hermele, “Fracton topological order via coupled layers”, Physical Review B 95, (2017) arXiv:1701.00747 DOI
- [9]
- W. Shirley, K. Slagle, and X. Chen, “Fractional excitations in foliated fracton phases”, Annals of Physics 410, 167922 (2019) arXiv:1806.08625 DOI
- [10]
- T. Wang, W. Shirley, and X. Chen, “Foliated fracton order in the Majorana checkerboard model”, Physical Review B 100, (2019) arXiv:1904.01111 DOI
- [11]
- S. Pai and M. Hermele, “Fracton fusion and statistics”, Physical Review B 100, (2019) arXiv:1903.11625 DOI