Classifying fractionalization: Symmetry classification of gapped ${\mathbb{Z}}_2$ spin liquids in two dimensions

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 $\mathrm{SO}\left(3\right)$ spin rotation symmetries, we find $2098176\approx 2^{21}$ 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.

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) $e$ (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 $R_1^e$ and $R_2^e$. 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 $R_1^e$ has split into isolated $m$ and $\epsilon$ particles as allowed by the $e=m\times \epsilon$ fusion rule. $R_1^e$ can thus be subdivided into $R_1^m$ and $R_1^{\epsilon }$ 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 $C_x^e$ (thick solid line) and $C_x^m$ (thick dashed line) used to define the loop operators ${\mathcal{L}}_x^e$ and ${\mathcal{L}}_x^m$, respectively.

Figure 6

Depiction of the state $|{\psi }_{\alpha }\rangle$ with two $e$ particles. The operator ${\mathcal{O}}_{\alpha }$ creating this state is supported on the two shaded circular regions and along the solid line connecting them. ${\mathcal{O}}_{\alpha }$ is an $e$-string operator along the solid line, which is referred to as $L_i^e$ outside of the regions $R_i^e$. ${\mathcal{O}}_{\alpha }$ is thus an $e$ operator on the union of the $R_i^e$.

Figure 7

Depiction of the action of a symmetry operation $S$ on a two $e$-particle state, where $S$ is either translation or an internal symmetry operation. The initial state on which $S$ acts is shown in Fig. 6. $S$ acts nontrivially in the shaded circular regions, and also as the closed $e$-string loop (solid line), as shown in (a). Outside the regions $R_i^e$, $L_i^e$ is the $e$ string of the initial state $|{\psi }_{\alpha }\rangle$, and $L_f^e$ is the $e$ string of the transformed state $S|{\psi }_{\alpha }\rangle$. In (b), the $e$-string loop is divided into a product of smaller loops, chosen so that each gives unity acting on the ground state, except possibly the two loops at the ends. Therefore the loops in the center can be eliminated (c), leaving only two loops contained entirely within the regions $R_i^e$, which can be absorbed into the definition of $S^e\left(i\right)$.

Figure 8

(a) Illustration of the initial state $|{\psi }_{\alpha }\rangle$ and the action of $P_x$ on $|{\psi }_{\alpha }\rangle$. The vertical dashed line is the reflection axis, and the regions $R_i^e$ and $R_{i^{\prime }}^e$ are defined in the main text. The $e$ strings connecting regions for operators ${\mathcal{O}}_{\alpha }$ and $P_x^e\left(i\right)$ are shown. The dashed-line ${\mathcal{O}}_{\alpha }^{\prime }$ string is the image of the ${\mathcal{O}}_{\alpha }$ string under $P_x$. (b) By acting with a closed $e$-string loop as shown, the ${\mathcal{O}}_{\alpha }$ and $P_x^e\left(i\right)$ strings can be eliminated in favor of the ${\mathcal{O}}_{\alpha }^{\prime }$ string. Following the argument depicted in Figs. 7b and 7c, the closed loop can be decomposed into smaller regions with unit eigenvalue acting on the ground state, and the effect of transforming the string can be absorbed into the definition of $P_x^e\left(i\right)$. In the $\epsilon$-particle case, depending on the orientations of the strings in (a), the loop in (b) may need to be twisted in some of the regions, to ensure the correct orientation of the ${\mathcal{O}}_{\alpha }^{\prime }$ string.

Figure 9

Computation of $W\left(i\right)$ phase factor for $P_x$ reflection symmetry. (a) The initial state $|{\psi }_{\alpha }\rangle$ is depicted on the right-hand side, while the vertical dotted line is the reflection axis, and the final state $P_x|{\psi }_{\alpha }\rangle$ lies to the left of the axis. Regions $R_i^{\epsilon }$ and their images under $P_x$ are indicated with dotted lines. All particles and strings are shown in black in the initial states and in gray (blue online) in the final states. $e$ particles are filled circles, $m$ particles are crosses, $e$ strings are solid lines, and $m$ strings are dashed lines. (b) Depiction of the state $P_x^e\left(1\right)P_x^m\left(1\right)P_x^e\left(2\right)P_x^m\left(2\right)|{\psi }_{\alpha }\rangle$. To compute $W\left(i\right)$, we simply bring the strings into the final-state configuration shown in the left-hand side of (a). We find $W\left(1\right)=-1$, where the minus sign arises from crossing the $m$ string beneath the $e$ string in $R_1^{\epsilon }$. We also find $W\left(2\right)=-1$, where in this case, the sign arises from sliding the $e$ string over the final-state $m$ particle in $R_{2^{\prime }}^{\epsilon }$.

Figure 10

Computation of ${\sigma }_{px}^t$, for the group relation $P_x^2=1$. The graphical notation used here is introduced in Fig. 9. (a) The reflection axis is the vertical dotted line. The state $|{\psi }_{\alpha }\rangle$ is depicted to the right of the axis, with $e$ and $m$ particles and (vertical) strings drawn in black. This is almost the same state considered in Fig. 9, but with a simpler arrangement of strings. The state $P_x|{\psi }_{\alpha }\rangle$ is shown to the left of the axis in gray (blue online). The horizontal strings depict the operators $P_x^e\left(i\right)$ and $P_x^m\left(i\right)$. From this figure, it can be seen that $W\left(i\right)=-1$ in the symmetry localization of $P_x|{\psi }_{\alpha }\rangle$, and the same is easily seen to hold in the symmetry localization of $P_x\left(P_x\right|{\psi }_{\alpha }\rangle )$. (b) Closed loops of string obtained from $P_x^e\left(i^{\prime }\right)P_x^e\left(i\right)$ and $P_x^m\left(i^{\prime }\right)P_x^m\left(i\right)$. Positions of $e$ and $m$ particles in $|{\psi }_{\alpha }\rangle$ are shown. Since these loops do not link, and, for instance, each $e$ loop encloses no $m$ particles, we have $\ell \left(i\right)=Z^{em}\left(i\right)=Z^{me}\left(i\right)=1$.

Figure 11

Symmetry localization of $\pi /2$-rotation $R_{\pi /2}$ on a state $|{\psi }_{\alpha }\rangle$. The center of rotation symmetry is the solid square. The initial state $|{\psi }_{\alpha }\rangle$ has $e$ and $m$ particles arranged along a line extending below the center of symmetry, and $\epsilon$-sector regions $R_i^{\epsilon }$ as shown. The particles and strings of the final state are shown in gray (blue online). The angled $e$ and $m$ strings are the strings of the $R_{\pi /2}^e\left(i\right)$ and $R_{\pi /2}^m\left(i\right)$ operators. Inspection of this figure shows that $W\left(i\right)=1$ for $i=1,2$.

Figure 12

Closed $e$ and $m$ string operators obtained upon expressing $R_{\pi /2}^4|{\psi }_{\alpha }\rangle$ in terms of products of one-particle symmetry operators acting on $|{\psi }_{\alpha }\rangle$. The locations of $e$ and $m$ particles in $|{\psi }_{\alpha }\rangle$ are shown, and the closed strings are labeled as shown.

Figure 13

The state $\left|\text{ref}\right(-1)\rangle$. Dark links carry ${\sigma }^z=-1$, others have ${\sigma }^z=1$. The square at the center of all figures identifies the origin of coordinates and of point group operations.

Figure 14

(a) Electric string. (b) Magnetic string.

Figure 15

The action of translation $T_x$ by one lattice spacing along $\widehat{\mathbf{x}}$ on the two-charge state in (a). (b) Result of moving the quasiparticles with single spin flips, and the shaded area in (c) is swept out by sliding the strings to their final position. In (b), the thicker string is doubled; this convention will be used in subsequent figures, except where noted.

Figure 16

(a) Under $P_x$, charges and fluxes in the $•$ region follow contours above the origin, those in the $\circ$ region go below. (b) The conventions for $P_{xy}$ are analogous.

Figure 17

The action of $P_x$ on a two-$e$ state.

Figure 18

The action of $P_x$ on a two-$m$ state.

Figure 19

The action of $P_{xy}$ on a two-$e$ state.

Figure 20

The action of $P_{xy}$ on a two-$m$ state.

Figure 21

(a) The action of $P_{xy}{T}_x{P}_{xy}^{-1}$ on a single charge $e$. (b) Sliding the string of (a) to obtain a contribution to the single-particle phase of $T_y$.

Figure 22

The strings for the group relation $P_{xy}^{m2}$. In (a), recall that the thicker strings are doubled; that is, they represent two strings acting in the same position.

Figure 23

The strings for the group relation ${\left(P_x{P}_{xy}\right)}^4$. In (a), the darker curves are triple strings. In (b), all segments are drawn with the same weight.

Figure 24

The strings for the group relation $T_y^m{P}_x^m{T}_y^{m-1}{P}_x^{m-1}$.

Figure 25

Illustration of states used for symmetry localization to account for nonsinglet ground states. The thick solid line is the “boundary” of the periodic system—opposite edges are identified. The region $R$ is shaded. Noncontractible strings of the loop algebra act along the dashed lines, outside the region $R$.