Twisted foliated fracton phases

In the study of three-dimensional gapped models, two-dimensional gapped states should be considered as a free resource. This is the basic idea underlying the notion of ``foliated fracton order'' proposed in Shirley et al. Phys. Rev. X 8, 031051 (2018). We have found that many of the known type-I fracton models, although they appear very different, have the same foliated fracton order, known as ``X-cube'' order. In this paper, we identify three-dimensional fracton models with different kinds of foliated fracton order. Whereas the X-cube order corresponds to the gauge theory of a simple paramagnet with subsystem planar symmetry, the different orders correspond to twisted versions of the gauge theory for which the system prior to gauging has nontrivial order protected by the planar subsystem symmetry. We present constructions of the twisted models and demonstrate that they possess nontrivial order by studying their fractional excitation contents.