# Twisted foliated fracton phases

## TL;DR

This paper classifies various three-dimensional fracton models based on their foliated fracton order, revealing new twisted phases distinguished by their fractional excitations and subsystem symmetries.

## Contribution

It identifies and constructs new twisted foliated fracton phases, expanding the understanding of fracton order beyond the well-known X-cube model.

## Key findings

- Many known type I fracton models share the same X-cube foliated order.
- New twisted fracton models are constructed with nontrivial order.
- Twisted models exhibit distinct fractional excitations and subsystem symmetry properties.

## Abstract

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 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 novel 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.