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

Abstract

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 $\text{U}(1)$ 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 ${N}^{2}$ 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 $\text{U}(1)$ 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 $\text{U}(1)$ gauge fields. Thus, the realization of translation symmetry in the ${\text{U}}^{6}(1)$ 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 ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ 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.

Topics & Concepts