Classifying fractionalization: Symmetry classification of gappedZ2spin liquids in two dimensions

Abstract

We classify distinct types of quantum number fractionalization occurring in two-dimensional topologically ordered phases, focusing in particular on phases with ${\mathbb{Z}}_{2}$ topological order, that is, on gapped ${\mathbb{Z}}_{2}$ spin liquids. We find that the fractionalization class of each anyon is an equivalence class of projective representations of the symmetry group, corresponding to elements of the cohomology group ${H}^{2}(G,{\mathbb{Z}}_{2})$. This result leads us to a symmetry classification of gapped ${\mathbb{Z}}_{2}$ spin liquids, such that two phases in different symmetry classes cannot be connected without breaking symmetry or crossing a phase transition. Symmetry classes are defined by specifying a fractionalization class for each type of anyon. The fusion rules of anyons play a crucial role in determining the symmetry classes. For translation and internal symmetries, braiding statistics plays no role, but can affect the classification when point group symmetries are present. For square lattice space group, time-reversal, and $\mathrm{SO}(3)$ spin rotation symmetries, we find $2\phantom{\rule{0.16em}{0ex}}098\phantom{\rule{0.16em}{0ex}}176\ensuremath{\approx}{2}^{21}$ distinct symmetry classes. Our symmetry classification is not complete, as we exclude, by assumption, permutation of the different types of anyons by symmetry operations. We give an explicit construction of symmetry classes for square lattice space group symmetry in the toric code model. Via simple examples, we illustrate how information about fractionalization classes can, in principle, be obtained from the spectrum and quantum numbers of excited states. Moreover, the symmetry class can be partially determined from the quantum numbers of the four degenerate ground states on the torus. We also extend our results to arbitrary Abelian topological orders (limited, though, to translations and internal symmetries), and compare our classification with the related projective symmetry group classification of parton mean-field theories. Our results provide a framework for understanding and probing the sharp distinctions among symmetric ${\mathbb{Z}}_{2}$ spin liquids and are a first step toward a full classification of symmetric topologically ordered phases.