Alternative names: String-membrane-net code.
Description
A commuting-projector code family obtained by coupling layers of two-dimensional topological orders and condensing extended one-dimensional flux strings [1]. A modern lattice realization starts from isotropic stacks of \(G\)-graded string-net models [3]. The family includes stabilizer examples such as the \(\mathbb{Z}_2\) string-membrane-net realization of the X-cube model [2] as well as non-stabilizer examples with non-Abelian restricted-mobility excitations [1]. String-membrane-net and cage-net constructions can realize the same fracton phases; for isotropic stacks, this equivalence can be understood via generalized local unitaries [3]. The cage-net construction can be used to realize various fracton phases, stabilizer and otherwise.Cousins
- Fracton stabilizer code— The cage-net construction can be used to realize various fracton phases, stabilizer and otherwise.
- String-net code— Cage-net codes are obtained by coupling layers of \(G\)-graded string-net models [3].
Primary Hierarchy
Parents
Cage-net codewords form ground-state subspaces of frustration-free commuting projector Hamiltonians.
Cage-net codewords form ground-state subspaces of frustration-free commuting projector Hamiltonians.
Cage-net code
Children
A field-theoretic description of the qudit X-cube model can be obtained by coupling layers of 2D \(\mathbb{Z}_q\) gauge theory [3]. For three orthogonal foliations with \(\mathbb{Z}_q\) layers, the string-membrane-net model is equivalent to the \(\mathbb{Z}_q\) X-cube model [2]. String-membrane-net models are phase-equivalent to cage-net models under generalized local unitaries [3].
References
- [1]
- A. Prem, S.-J. Huang, H. Song, and M. Hermele, “Cage-Net Fracton Models”, Physical Review X 9, (2019) arXiv:1806.04687 DOI
- [2]
- K. Slagle, D. Aasen, and D. Williamson, “Foliated field theory and string-membrane-net condensation picture of fracton order”, SciPost Physics 6, (2019) arXiv:1812.01613 DOI
- [3]
- P. Gorantla, A. Prem, N. Tantivasadakarn, and D. J. Williamson, “String membrane nets from higher-form gauging: An alternate route to p -string condensation”, Physical Review B 112, (2025) arXiv:2505.13604 DOI
Page edit log
- Victor V. Albert (2025-06-13) — most recent
Cite as:
“Cage-net code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/cage_net
Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/categories/cage_net.yml.