A systematic construction of gapped non-liquid states

Gapped non-liquid state (also known as fracton state) is a very special gapped quantum state of matter that is characterized by a microscopic cellular structure. Such microscopic cellular structure has a macroscopic effect at arbitrary long distances and cannot be removed by renormalization group flow, which makes gapped non-liquid state beyond the description of topological quantum field theory with a finite number of fields. Using Abelian and non-Abelian topological orders in 2-dimensional (2d) space and the different ways to glue them together via their gapped boundaries, we propose a systematic way to construct 3d gapped states (and in other dimensions). The resulting states are called cellular topological states, which include gapped non-liquid states, as well as gapped liquid states in some special cases. Some new fracton states with fractal excitations are constructed even using 2d $\mathbb{Z}_2$ topological order. More general cellular topological states can be constructed by connecting 2d domain walls between different 3d topological orders. The constructed cellular topological states can be viewed as fixed-point states for a reverse renormalization of gapped non-liquid states.