Quantum phase transition of fracton topological orders

Fracton topological order (FTO) is a new classification of correlated phases in three spatial dimensions with ground-state degeneracy (GSD) scaling up system size and fractional excitations which are immobile or have restricted mobility. With the origin GSD, FTO immune to local perturbations, whereas strong enough global external perturbation expected break order. The critical point transition either characterized by broken GSD appearance topologically distinct states lower energy. In this work, we propose characterize quantum phase type-I FTOs induced terms, when can be breaking down develop theory study analytically points. particular, for term creating lineon-type excitations, predict generic formula transition, GSD. This applies board class FTOs, including X-cube model, more models under perturbations two-dimensional (2D) 3D upper limits point. Our work makes step characterizing fracton orders.