Position-Dependent Excitations and UV/IR Mixing in the ${\mathbb{Z}}_N$ Rank-2 Toric Code and its Low-Energy Effective Field Theory

Abstract

We investigate how symmetry and topological order are coupled in the $2+1$d ${\mathbb{Z}}_N$ rank-2 toric code for general $N$, which is an exactly solvable point in the Higgs phase of a symmetric rank-2 $U(1)$ gauge theory. The symmetry enriched topological order present has a non-trivial realization of square-lattice translation (and rotation/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 $3+1$d. 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 $U(1)$ 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 $U^6(1)$ 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 $L_x\times L_y$ 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.

22 pages, 5 figures

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