Fractonic order in infinite-component Chern-Simons gauge theories

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.