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

TL;DR

This work analyzes $\,\mathbb{Z}_2$ topological order in 2+1D under spatially modulated generalized symmetries, realized by fixed-point commuting-projector Hamiltonians with horizontal support $h$ (odd). Focusing on the $h=3$ tripartite case, the authors show UV/IR-like ground-state degeneracies depending on lattice sizes, reveal position-dependent anyons with mobility constrained to rigid steps, and map the boundary physics to a sequence of gapped phases and SPT-like boundaries via anyon condensations. They develop two complementary effective field theories: a short-distance lattice-respecting description exposing the mobility restrictions and boundary couplings, and a long-distance $K$-matrix Chern–Simons theory with twisted boundary conditions that encode lattice sizes and reproduce the observed degeneracies. The paper further extends to general $h$ (the $h$-partite case), detailing the generalized symmetry operators, anomaly structure, particle content, and boundary phenomena, highlighting a robust framework for modulated symmetries in $\,\mathbb{Z}_2$ SET phases and illustrating a concrete continuum correspondence via twisted boundaries. Overall, the results illuminate how modulated symmetries constrain anyon dynamics, yield UV/IR mixing, and give rise to a rich tapestry of boundary and defect physics with potential applications to broader fracton-like and SET systems.

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.

Figures (12)

• Figure 1: Definition of the commuting projector $P_{\vec{r}}$.
• Figure 2: Representation of the solutions of the rule \ref{['eq:const2']}, for $L=12$.
• Figure 3: Horizontal symmetry operators with $i_y\,\in\, \text{even}$.
• Figure 4: The filled red balls indicate the initial and final positions of an excitation in a transport process. The picture shows that an excitation can move along the $x$-direction only in rigid steps of size three in units of the lattice spacing.
• Figure 5: Vertical symmetry operators in the even $i_y$ sublattice.