Position-dependent excitations and UV/IR mixing in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:math> rank-2 toric code and its low-energy effective field theory

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