Generalized Modulated Symmetries in $\mathbb{Z}_2$ Topological Ordered Phases

TL;DR

This paper explores $ ext{Z}_2$ topological phases with generalized modulated symmetries in 2+1 dimensions, revealing their lattice size dependence, unique anyon mobility, and rich boundary physics including various symmetry-breaking and SPT phases.

Contribution

It introduces a new class of $ ext{Z}_2$ topological phases characterized by modulated symmetries with support sizes 3, 5, 7, etc., and analyzes their lattice size dependence and boundary phenomena.

Findings

Symmetries are sensitive to lattice sizes, leading to spontaneous or explicit breaking.

Anyons can only move in fixed steps dictated by the modulation size.

Rich boundary physics includes trivial, symmetry-breaking, and SPT phases.

Abstract

We study $\mathbb{Z}_2$ topological ordered phases in 2+1 dimensions characterized by generalized modulated symmetries. Such phases have explicit realizations in terms of fixed-point Hamiltonians involving commuting projectors with support $h=3,5,7,\ldots$ in the horizontal direction, which dictates the modulation of the generalized symmetries. These symmetries are sensitive to the lattice sizes. For certain sizes, they are spontaneously broken and the ground state is degenerated, while for the remaining ones, the symmetries are explicitly broken and the ground state is unique. The ground state dependence on the lattice sizes is a manifestation of the ultraviolet/infrared (UV/IR) mixing. The structure of the modulated symmetries implies that the anyons can move only in rigid steps of size $h$, leading to the notion of position-dependent anyons. The phases exhibit rich boundary physics with a variety of gapped phases, including trivial, partial and total symmetry-breaking, and SPT phases. Effective field theory descriptions are discussed, making transparent the relation between the generalized modulated symmetries and the restrictions on anyon mobility, incorporating the boundary physics in a natural way, and showing how the short-distance details can be incorporated into the continuum by means of twisted boundary conditions.