Type-I Fractons -- Foliation in Non-Abelian Models

In this thesis, we present recent contributions to the study of Type-I non-abelian fracton models, which led us to propose the notion of generalized foliated fracton orders that captures the universal properties of both abelian and non-abelian Type-I fracton models. Fracton models are known for their exotic properties such as point-like excitations with restricted mobilities and robust topological ground state degeneracy that grows sub-extensively with the system size. A multitude of Type-I fracton models whose excitations obey either abelian or non-abelian fusion rules have recently been constructed. Among them, a large number of the abelian fracton models have been shown to possess foliation structures, where models of different system sizes can be related through the addition / removal of an entire piece of topologically ordered system on a sub-dimensional manifold via the action of a finite-depth local unitary circuit. In this thesis, this is referred to as the original foliation renormalization group (RG) scheme, which leads us to the notion of original foliation fracton orders. The Ising cage-net model and other similar non-abelian models are closely related to these abelian models in terms of their excitation structures and coupled layers construction etc. However, it was not known whether their fracton orders can also be understood within the original foliation framework. We address this problem in this thesis. In Chapter 2, we show that the Ising cage-net model does not fit into the original definition of foliated fracton orders, by calculating its ground state degeneracy. We realize that there exists naturally a more general way to define foliation -- the generalized foliation scheme (Chapter 3). The Ising cage-net and other similar non-abelian fracton models are foliated according to this generalized scheme. In the generalized foliation scheme, the RG transformation is defined by, from the excitation perspective, the condensation of planons / gauging subsystem symmetries. In terms of quantum circuits, this RG transformation is equivalent to a sequential linear-depth circuit that acts near a sub-dimensional manifold. With this definition, we can study phase relation of the Ising cage-net with other known fracton models. In Chapter 4, via gauging composite subsystem symmetries, we further show that the Ising cage-net belongs to the same generalized foliated fracton phases as the prototypical X-cube model. Furthermore, gauging composite subsystem symmetries opens up a new route to constructing non-abelian fracton models hosting exotic non-abelian fractons. An example is the tri-Ising-fracton model (Sec. 4.5).