Renormalization of Ising cage-net model and generalized foliation

Understand

A large class of type-I fracton models, including the X-cube model, have been found to be fixed points of the foliated renormalization group (RG).

• The system size of such foliated models can be changed by adding or removing decoupled layers of $2$D topological states and continuous deformation of the Hamiltonian.
• In this paper, we study a closely related model -- the Ising cage-net model -- and find that this model is not foliated in the same sense.
• In fact, we point out certain unnatural restrictions in the foliated RG, and find that removing these restrictions leads to a generalized foliated RG under which the Ising cage-net model is a fixed point, and which includes the original foliated RG as a special case.