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 + 1d 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 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 + 1d. 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 Lx × Ly torus is N 3 gcd(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.

Effective Fractonic Behavior in a Two-Dimensional Exactly Solvable Spin Liquid

“…Following the discussion in [37], we can derive the ground state degeneracy in the continuum theory by studying the boundary conditions of the gauge fields. Since the translation operations do not act linearly on the gauge fields, it is difficult to write their general transformation under arbitrary shifts.…”

“…These "quasi-fracton" [32][33][34][35][36][37] systems thus bridge canonical topological order with fracton topological order. Here we explore this bridge and argue that, as function of N and the coupling constants in the model, an exponentially large window of time opens in which the system behaves effectively as a fracton in d = 2.…”

“…Recently, certain lattice models have been obtained by Higgsing a rank-2 U (1) lattice gauge theory, yielding to Z N quantum spin liquids [32][33][34][35][36][37]. These models display features resembling those of fracton systems, but the elementary excitations, in contrast to fractons, can still hop, not in lattice spacing steps but in steps whose size scales linearly with N .…”

“…Refs. [32][33][34][35][36][37]. The Chern-Simons-like theory captures many of the features of the lattice model, such as dipole conservation, the fact that the system is gapped, and the mutual statistics of quasi-fractons and quadrupoles.…”

“…We use such data to write down a [U (1)] 4 continuum mutual Chern-Simons theory and use this effective description to explicitly compute the ground state degeneracy. In this regime, the UV/IR aspects of the theory are implemented through the boundary conditions on the gauge fields on a compact space [37].…”