Position-dependent excitations and UV/IR mixing in the ZN rank-2 toric code and its low-energy effective field theory

We investigate how symmetry and topological order are coupled in the ($2+1$)--dimensional ZN rank-2 toric code for general N, which is an exactly solvable point in the Higgs phase of a symmetric rank-2 U(1) 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 N² flavors of e charges and $2N$ flavors of m fluxes, there are not N^N²+2N anyon types. Instead, there are N⁶ anyon types, and we can use Chern-Simons theory with six U(1) 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 U(1) gauge fields. Thus, the realization of translation symmetry in the U⁶(1) 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 Lₓ×Ly torus is N³gcd(Lₓ,N)gcd(Ly,N)gcd(Lₓ,Ly,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.