We study the fractionalization of space group symmetries in two-dimensional topologically ordered phases. Specifically, we focus on
Z2-fractionalized phases in two dimensions whose deconfined topological excitations transform trivially under translational symmetries but projectively under glide reflections, whose quantum numbers are hence fractionalized. We accomplish this by generalizing the dichotomy between even and odd gauge theories to incorporate additional symmetries inherent to nonsymmorphic crystals. We show that the resulting fractionalization of point group quantum numbers can be detected in numerical studies of ground state wave functions. We illustrate these ideas using a microscopic model of a system of bosons at integer unit cell filling on a lattice with space group
p4g that can be mapped to a half-magnetization plateau for an
S=1/2 spin system on the Shastry-Sutherland lattice.
Symmetry Fractionalization in Three-Dimensional
Z2 Topological Order and Fermionic Symmetry-Protected Phases
In two-dimensional topological phases, quasiparticle excitations can carry fractional symmetry quantum numbers. We generalize this notion of symmetry fractionalization to three-dimensional topological phases, in particular to loop excitations, and propose a partial classification for symmetry-enriched
Z2 toric code phase. We apply the results to the classification of fermionic symmetry-protected topological phases in three dimensions.
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 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 nontrivial, despite the fact that these generators multiply to the identity. We first discuss the general algebraic framework needed to characterize this type of fractionalization, and then explore this framework using a number of exactly solvable models with
Z2 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.