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 lower-dimensional submanifold of the whole system. In this paper, we generalize the $(2+1)$-dimensional $\left[\right(2+1\left)\mathrm{D}\right]$ U(1) Chern-Simons (CS) theory, a powerful tool in the study of $(2+1)\mathrm{D}$ 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 $(2+1)\mathrm{D}$ 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.

Figure 1

Foliated fracton order and its interpretation in terms of $K$ matrix. In (a) (first line), we start with a system $H\left(L\right)$ of size $L$ in the $z$ direction. A finite-depth local unitary circuit $U$ is applied to the green region $\{(x,y,z):z_1\le z\le z_2\}$. The result is the same system $H(L-1)$ of size $L-1$ in the $z$ direction and a decoupled ($2+1)\mathrm{D}$ gapped system (red layer). In (b) (second line), we start with a quasidiagonal $K\left(N\right)$ of size $N\propto L$ with periodic boundary condition. Only entries in the blue region can be nonzero. We apply the transformation $K\left(N\right)\mapsto WK\left(N\right)W^T$, where $W\in \text{GL}(N,\mathbb{Z})$ shown in the dashed box is equal to the identity except in the green block, so the action of $W$ on $K\left(N\right)$ is within the green cross in the second figure. The result is the direct sum of the same system $K(N-a)$ of size $N-a$ and a decoupled block $K^{\prime }$ of size $a=O\left(1\right)$ (red block).

Figure 2

Lattice model realizing $K=\left(2\right)$. The matter content of the system is two IQH layers ${\mathrm{\Omega }}^1$ and ${\mathrm{\Omega }}^2$ (blue lines) with Chern number $C^l=1$. The layers are coupled each with unit charge to a dynamical $\text{U}\left(1\right)$ gauge field $A$.

Figure 3

The flux and Gauss's law terms in Eq. (14). Reversing the direction of the edges changes the signs in front of $A$ and $E$. In the context of our first example $K=\left(2\right)$, the index $i$ should be ignored and $q^{il}=1$ for $l=1,2$.

Figure 4

Lattice model realizing $K_{\text{nF}}$. The matter content of the system is infinitely many IQH layers ${\mathrm{\Omega }}^l$ (blue lines) with Chern number $C^l=1$. The layers are coupled with unit charge to infinitely many dynamical $\text{U}\left(1\right)$ gauge fields $A^i$ in the way indicated by the curly brackets.

Figure 5

The string operator for the lattice model realizing $K=\left(2\right)$. Fermions live in the blue layers ${\mathrm{\Omega }}^1$ and ${\mathrm{\Omega }}^2$, and the gauge field in the middle, green layer. The operators ${\mathcal{O}}_l$ are generated by hopping operators $c_{l,{\mathbf{r}}^{\prime }}^{†}{e}^{i{A}_{\mathbf{r}{\mathbf{r}}^{\prime }}}{c}_{l,\mathbf{r}}$. The action of ${\mathcal{O}}_l$ is nontrivial only near the path (gray region), and is exponentially close to the identity away from the path. The string operator $\mathcal{W}$ consists of $e^{-iA}$ acting on the dashed red line, $e^{-i\pi E}$ acting on the solid red segments, and ${\mathcal{O}}_1, {\mathcal{O}}_2$ acting near the path.