String-net condensation can give rise to non-Abelian anyons whereas loop condensation usually gives rise to Abelian anyons. It has been proposed that generalized quantum loop gases with non-orthogonal inner products can produce non-Abelian anyons. We detail an exact mapping between the string-net and the generalized loop models and explain how the non-orthogonal products arise. We also introduce a loop model of double-stranded nets where quantum loops with an orthogonal inner product and local interactions supports non-Abelian Fibonacci anyons. Finally we emphasize the origin of the sign problem in such systems and its consequences on the complexity of their ground state wave functions.String-net condensation can give rise to non-Abelian anyons whereas loop condensation usually gives rise to Abelian anyons. It has been proposed that generalized quantum loop gases with non-orthogonal inner products can produce non-Abelian anyons. We detail an exact mapping between the string-net and the generalized loop models and explain how the non-orthogonal products arise. We also introduce a loop model of double-stranded nets where quantum loops with an orthogonal inner product and local interactions supports non-Abelian Fibonacci anyons. Finally we emphasize the origin of the sign problem in such systems and its consequences on the complexity of their ground state wave functions.

arXiv:0911.1132Journal of Physics A Mathematical and TheoreticalApr 13, 20103 citations

Anyons and quantum groups

Alberto Lerda A, S. Sciuto

arXiv:hep-th/9301100Nuclear Physics BJul 1, 199389 citations

Cage-Net Fracton Models

Abhinav Prem, Sheng-Jie Huang, Hao Song +1 more

We introduce a class of gapped three-dimensional models, dubbed "cage-net fracton models," which host immobile fracton excitations in addition to non-Abelian particles with restricted mobility.Starting from layers of two-dimensional string-net models, whose spectrum includes non-Abelian anyons, we condense extended one-dimensional "flux strings" built out of pointlike excitations.Flux-string condensation generalizes the concept of anyon condensation familiar from conventional topological order and allows us to establish properties of the fracton-ordered (equivalently, flux-string-condensed) phase, such as its groundstate wave function and spectrum of excitations.Through the examples of doubled-Ising and SUð2Þ k cagenet models, we demonstrate the existence of strictly immobile Abelian fractons and of non-Abelian particles restricted to move only along one dimension.In the doubled-Ising cage-net model, we show that these restricted-mobility non-Abelian excitations are a fundamentally three-dimensional phenomenon, as they cannot be understood as bound states among two-dimensional non-Abelian anyons and Abelian particles.We further show that the ground-state wave function of such phases can be understood as a fluctuating network of extended objects-cages-and strings, which we dub a cage-net condensate.Besides having implications for topological quantum computation in three dimensions, our work may also point the way towards more general insights into quantum phases of matter with fracton order.We introduce a class of gapped three-dimensional models, dubbed "cage-net fracton models," which host immobile fracton excitations in addition to non-Abelian particles with restricted mobility.Starting from layers of two-dimensional string-net models, whose spectrum includes non-Abelian anyons, we condense extended one-dimensional "flux strings" built out of pointlike excitations.Flux-string condensation generalizes the concept of anyon condensation familiar from conventional topological order and allows us to establish properties of the fracton-ordered (equivalently, flux-string-condensed) phase, such as its groundstate wave function and spectrum of excitations.Through the examples of doubled-Ising and SUð2Þ k cagenet models, we demonstrate the existence of strictly immobile Abelian fractons and of non-Abelian particles restricted to move only along one dimension.In the doubled-Ising cage-net model, we show that these restricted-mobility non-Abelian excitations are a fundamentally three-dimensional phenomenon, as they cannot be understood as bound states among two-dimensional non-Abelian anyons and Abelian particles.We further show that the ground-state wave function of such phases can be understood as a fluctuating network of extended objects-cages-and strings, which we dub a cage-net condensate.Besides having implications for topological quantum computation in three dimensions, our work may also point the way towards more general insights into quantum phases of matter with fracton order.

arXiv:1806.04687Physical Review XApr 17, 2019113 citations

Physics And Mathematics of Anyons

Shiing-Shen Chern, C. W. Chu, C. S. Ting

Jun 1, 19915 citations

Non-Abelian Particle-Loop, Fracton, and Planon Condensation in Cage-Net Models

Yifei Wang, Yu Zhao, Yingcheng Li +2 more

We present a framework for non-Abelian p-loop, fracton, and planon condensation in 3+1 dimensions by constructing extended cage-net fracton models using decoupled layers of the Hu-Geer-Wu (HGW) string-net model. These cage-net models extend the conventional cage-net models based on the Levin-Wen (LW) string-net model in the sense that they inherit the tail degrees of freedom of the HGW models, which are essential for completely describing the internal spaces of quasiparticles. This approach allows us to explicitly derive the quasiparticle spectra of the cage-net models by projecting those of the parent 2D HGW layers. Utilizing this framework, we can condense the p-loops formed by non-Abelian anyons within a fracton phase. Specifically, we construct the condensation projector for $(σ\barσ, 1)$ -loops within the extended Ising Cage-Net (ICN) model. We demonstrate that condensing these non-Abelian loops drives a phase transition that maps the ICN model to the X-cube (XC) model defined on a truncated cubic lattice, a process that explicitly reveals the splitting of non-Abelian planons into distinct sub-dimensional excitations. Furthermore, our framework extends to the condensation of fractons and planons: we demonstrate that in the ICN model fracton condensation drives the decoupling of the 3D fracton order back into isolated 2D topological order layers, while planon condensation collapses the system entirely into a trivial phase. Our results establish a concrete Hamiltonian mechanism for phase transitions between distinct fracton orders and provide a generalizable method for analyzing the evolution of sub-dimensional excitations.We present a framework for non-Abelian p-loop, fracton, and planon condensation in 3+1 dimensions by constructing extended cage-net fracton models using decoupled layers of the Hu-Geer-Wu (HGW) string-net model. These cage-net models extend the conventional cage-net models based on the Levin-Wen (LW) string-net model in the sense that they inherit the tail degrees of freedom of the HGW models, which are essential for completely describing the internal spaces of quasiparticles. This approach allows us to explicitly derive the quasiparticle spectra of the cage-net models by projecting those of the parent 2D HGW layers. Utilizing this framework, we can condense the p-loops formed by non-Abelian anyons within a fracton phase. Specifically, we construct the condensation projector for $(σ\barσ, 1)$-loops within the extended Ising Cage-Net (ICN) model. We demonstrate that condensing these non-Abelian loops drives a phase transition that maps the ICN model to the X-cube (XC) model defined on a truncated cubic lattice, a process that explicitly reveals the splitting of non-Abelian planons into distinct sub-dimensional excitations. Furthermore, our framework extends to the condensation of fractons and planons: we demonstrate that in the ICN model fracton condensation drives the decoupling of the 3D fracton order back into isolated 2D topological order layers, while planon condensation collapses the system entirely into a trivial phase. Our results establish a concrete Hamiltonian mechanism for phase transitions between distinct fracton orders and provide a generalizable method for analyzing the evolution of sub-dimensional excitations.

arXiv:2604.24755Apr 27, 20260 citations

(σ\barσ, 1)

-loops within the extended Ising Cage-Net (ICN) model. We demonstrate that condensing these non-Abelian loops drives a phase transition that maps the ICN model to the X-cube (XC) model defined on a truncated cubic lattice, a process that explicitly reveals the splitting of non-Abelian planons into distinct sub-dimensional excitations. Furthermore, our framework extends to the condensation of fractons and planons: we demonstrate that in the ICN model fracton condensation drives the decoupling of the 3D fracton order back into isolated 2D topological order layers, while planon condensation collapses the system entirely into a trivial phase. Our results establish a concrete Hamiltonian mechanism for phase transitions between distinct fracton orders and provide a generalizable method for analyzing the evolution of sub-dimensional excitations.