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 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 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 NN2+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 x 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.

Automated summary

We investigate how symmetry and topological order are coupled in the (2 + 1)-dimensional Z(N) rank-2 toric code. The particles in the system show position-dependent excitations and different properties on different lattice sites, with a nontrivial realization of square-lattice translation symmetry. Despite the complexity of the system with multiple anyon types and permutations, a Chern-Simons theory with six U(1) gauge fields is used to describe all of them.