Fractionalization of subsystem symmetries in two dimensions

TL;DR

This paper explores the fractionalization of subsystem symmetries in two-dimensional topological phases, revealing new mechanisms involving global relations that enable fractionalization despite previous no-go constraints.

Contribution

It introduces a novel fractionalization mechanism based on global relations among symmetry generators, expanding understanding of symmetry fractionalization in topological phases.

Findings

Fractionalization is possible via global relations among symmetry generators.

Anyons can fractionalize global relations, carrying non-trivial total charge.

Fractionalized anyons are confined to lines or points depending on symmetry type.

Abstract

The fractionalization of global symmetry charges is a striking hallmark of topological quantum order. Here, we discuss the fractionalization of subsystem symmetries in two-dimensional topological phases. In line with previous no-go arguments, we show that subsystem symmetry fractionalization is not possible in many cases due to the additional rigid geometric structure of the symmetries. However, we identify a new mechanism that allows fractionalization, involving global relations between macroscopically many symmetry generators. We find that anyons can fractionalize such relations, meaning that the total charge carried under all generators involved in the global relation is non-trivial, despite the fact that these generators multiply to the identity. We first discuss the general algebraic framework needed to characterize this new type of fractionalization, and then explore this framework using a number of exactly solvable models with $\mathbb{Z}_2$ topological order, including models having line and fractal symmetries. These models all showcase another necessary property of subsystem symmetry fractionalization: fractionalized anyons must have restricted mobility when the symmetry is enforced, such that they are confined to a single line or point in the case of line and fractal symmetries, respectively. Looking forward, we expect that our identification of the importance of global relations in fractionalization will hold significance for the classification of phases with subsystem symmetries in all dimensions.