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

Abstract

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 $(\sigma\bar{\sigma}, 1)$-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.