Z N generalizations of three-dimensional stabilizer codes

The authors generalize here three-dimensional \ensuremath{\mathbb{Z}}${}₂$ stabilizer models---the X-cube, toric code, and Haah's code---to their \ensuremath{\mathbb{Z}}${}_N$ counterparts, revealing size-dependent ground-state degeneracy and novel excitations. They demonstrate that the generalized X-cube hosts quasi-fractons that remain immobile under local operations but become mobile via global ones. In the toric code, both closed-loop and open-string excitations emerge, while Haah's code features new fracton tripole and monopole excitations. These findings uncover unconventional topological orders and broaden the understanding of fracton phases beyond the conventional \ensuremath{\mathbb{Z}}${}₂$ framework.