We investigate how symmetry and topological order are coupled in the ($2+1$)–dimensional ${\mathbb{Z}}_N$ rank-2 toric code for general $N$, which is an exactly solvable point in the Higgs phase of a symmetric rank-2 $\text{U}\left(1\right)$ gauge theory. The symmetry-enriched topological order present has a nontrivial realization of square-lattice translation (and rotation and reflection) symmetry, where anyons on different lattice sites have different types and belong to different superselection sectors. We call such particles “position-dependent excitations.” As a result, in the rank-2 toric code anyons can hop by one lattice site in some directions while only by $N$ lattice sites in others, reminiscent of fracton topological order in $3+1$ dimensions. We find that while there are $N^2$ flavors of $e$ charges and $2N$ flavors of $m$ fluxes, there are not $N^{N^2+2N}$ anyon types. Instead, there are $N^6$ anyon types, and we can use Chern-Simons theory with six $\text{U}\left(1\right)$ gauge fields to describe all of them. While the lattice translations permute anyon types, we find that such permutations cannot be expressed as transformations on the six $\text{U}\left(1\right)$ gauge fields. Thus, the realization of translation symmetry in the ${\text{U}}^6\left(1\right)$ Chern-Simons theory is not known. Despite this, we find a way to calculate the translation-dependent properties of the theory. In particular, we find that the ground-state degeneracy on an $L_x\times L_y$ torus is $N^{3}gcd(L_x,N)gcd(L_y,N)gcd(L_x,L_y,N)$, where $gcd$ stands for “greatest common divisor.” We argue that this is a manifestation of UV/IR mixing which arises from the interplay between lattice symmetries and topological order.

• Received 15 April 2022
• Accepted 21 July 2022

DOI:https://doi.org/10.1103/PhysRevB.106.045145