Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions

TL;DR

<3-5 sentences>We address the problem of distinguishing symmetric, gapped Z2 spin liquids by classifying how symmetry acts on the topological excitations. The authors develop a symmetry-class framework in which each anyon carries a projective representation classified by elements of $H^2(G,\mathbb{Z}_2)$, with the full symmetry class of a phase specified by the pair of classes for $e$ and $m$ (and a twisted product for $\epsilon$ on space groups). They provide explicit results for square-lattice space group symmetry (time reversal and SO(3) spin rotation), showing $|H^2(G,\mathbb{Z}_2)|=2^{11}$ fractionalization classes and $2^{21}$ symmetry classes, and realize four classes in the toric code. The work also extends to general Abelian topological orders and contrasts with PSG classifications, offering a robust framework to identify and distinguish symmetric spin liquids and guiding future classifications and experiments.

Abstract

We classify distinct types of quantum number fractionalization occurring in two-dimensional topologically ordered phases, focusing in particular on phases with Z2 topological order, that is, on gapped Z2 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,Z2). This result leads us to a symmetry classification of gapped Z2 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 symmetry, braiding statistics plays no role, but can affect the classification when point group symmetries are present. For square lattice space group, time reversal and SO(3) spin rotation symmetry, we find 2,098,176 ~ 2^21 distinct symmetry classes. 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, 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 Z2 spin liquids, and are a first step toward a full classification of symmetric topologically ordered phases.

Figures (25)

• Figure 1: (a) Two $e$-string operators (solid lines) with a single crossing point commute. (b) $e$-string (solid line) and $m$-string (dashed line) operators with a single crossing point anticommute.
• Figure 2: (a) $e$-string (solid line) and $m$ (dashed line) strings can be viewed in combination as an $\epsilon$-string. The arrow points from the $m$ string toward the $e$-string. (b) Twisted $\epsilon$ string where the $e$ string passes underneath the $m$-string. (c) Twisted $\epsilon$ string where the $e$ string passes over the $m$ string. Note that the configurations in (b) and (c) are related by a minus sign.
• Figure 3: (a) State with two isolated $e$-particles, with $e$-sector regions $R^e_1$ and $R^e_2$. The $e$-string connecting the particles is shown as a solid line. The two regions can be combined together to give the $1$-sector region $R^1$. (b) State as in (a), but where the $e$-particle in $R^e_1$ has split into isolated $m$ and $\epsilon$ particles as allowed by the $e = m \times \epsilon$ fusion rule. $R^e_1$ can thus be subdivided into $R^m_1$ and $R^\epsilon_1$ as shown. Strings connecting the particles are not shown.
• Figure 4: Thick bonds depict the four edges meeting the vertex $s$ and bounding the plaquette $p$.
• Figure 5: Depiction of contours $C^e_x$ (thick solid line) and $C^m_x$ (thick dashed line) used to define the loop operators ${\cal L}^e_x$ and ${\cal L}^m_x$, respectively.