The Ising cage-net model, first proposed in Phys. Rev. X 9, 021010 (2019), is a representative type I fracton model with nontrivial non-abelian features. In this paper, we calculate the ground state degeneracy of this model and find that, even though it follows a similar coupled layer structure as the X-cube model, the Ising cage-net model cannot be "foliated" in the same sense as X-cube as defined in Phys. Rev. X 8, 031051 (2018). A more generalized notion of "foliation'' is hence needed to understand the renormalization group transformation of the Ising cage-net model. The calculation is done using an operator algebra approach that we develop in this paper, and we demonstrate its validity through a series of examples.

Infinite-component Chern-Simons-Maxwell theories with a periodic $K$ matrix provide abundant examples of gapped and gapless, foliated and non-foliated fracton orders. In this paper, we study the ground state degeneracy of these theories. We show that the ground state degeneracy exhibit various patterns as a function of the linear system size -- the size of the $K$ matrix. It can grow exponentially or polynomially, cycle over finitely many values, or fluctuate erratically inside an exponential envelope. We relate these different patterns of the ground state degeneracy with the roots of the ``determinant polynomial'', a Laurent polynomial, associated to the periodic $K$ matrix. These roots also determine whether the theory is gapped or gapless. Based on the ground state degeneracy, we formulate a necessary condition for a gapped theory to be a foliated fracton order.

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. The Ising cage-net model thus gives a prototypical example of the generalized foliated RG, and its system size can be changed either by condensing / uncondensing bosonic planon excitations near a 2D plane or through a linear depth quantum circuit in the same plane. We show that these two apparently different RG procedures are closely related, as they lead to the same gapped boundary when implemented in part of a plane. Finally, we briefly discuss the implications for foliated fracton phases, whose universal properties will need to be reexamined in light of the generalized foliated RG.

Fracton order features point excitations that either cannot move at all or are only allowed to move in a lowerdimensional submanifold of the whole system. In this paper, we generalize the (2 + 1)-dimensional [(2 + 1)D] U(1) Chern-Simons (CS) theory, a powerful tool in the study of (2 + 1)D topological orders, to include infinite gauge field components and find that they can describe interesting types of (3 + 1)-dimensional fracton order beyond what is known from exactly solvable models and tensor gauge theories. On the one hand, they can describe foliated fractonic systems for which increasing the system size requires insertion of nontrivial (2 + 1)D topological states. The CS formulation provides an easier approach to study the phase relation among foliated models. More interestingly, we find simple examples that lie beyond the foliation framework, characterized by 2D excitations of infinite order and irrational braiding statistics. This finding extends our realm of understanding of possible fracton phenomena.

The infinite-component Chern-Simons-Maxwell (iCSM) theory is a 3+1D generalization of the 2+1D Chern-Simons-Maxwell theory by including an infinite number of coupled gauge fields. It can be used to describe interesting 3+1D systems. In Phys. Rev. B 105, 195124 (2022), it was used to construct gapped fracton models both within and beyond the foliation framework. In this paper, we study the nontrivial features of gapless iCSM theories. In particular, we find that while gapless 2+1D Maxwell theories are confined and not robust due to monopole effect, gapless iCSM theories are deconfined and robust against all local perturbation and hence represent a robust 3+1D deconfined gapless order. The gaplessness of the gapless iCSM theory can be understood as a consequence of the spontaneous breaking of an exotic one-form symmetry. Moreover, for a subclass of the gapless iCSM theories, we find interesting topological features in the correlation and response of the system. Finally, for this subclass of theories, we propose a fully continuous field theory description of the model that captures all these features.

It has been known that at high density, banana-shaped molecules are packed into a bent configuration locally, giving rise to interesting liquid-crystalline phases such as splay bend and twist bend. These spontaneous bend states can be modeled theoretically by allowing the bend elastic constant in the Frank elastic energy to become negative. Here, we extend this idea to banana-shaped and pizza-slice-shaped molecules with ferrointeractions, which can display both splayed and bent states. We model this by allowing both splay and bend elastic constants in the effective Frank elastic energy to be negative. In particular, we predict two new liquid-crystalline splay-bend phases. Furthermore, using renormalization group analysis, we showed that the phase transition belongs to a new constrained ferromagnet universality class.