String flux mechanism for fractionalization in topologically ordered phases

We construct a family of exactly solvable spin models that illustrate a mechanism for fractionalization in topologically ordered phases, dubbed the string flux mechanism. The essential idea is that an anyon of a topological phase can be endowed with fractional quantum numbers when the string attached to it slides over a background pattern of flux in the ground state. The string flux models that illustrate this mechanism are ${\mathbb{Z}}_n$ quantum double models defined on specially constructed $d$-dimensional lattices, and possess ${\mathbb{Z}}_n$ topological order for $d\ge 2$. The models have a unitary, internal symmetry $G$, where $G$ is an arbitrary finite group. The simplest string flux model is a ${\mathbb{Z}}_2$ toric code defined on a bilayer square lattice, where $G={\mathbb{Z}}_2$ is layer-exchange symmetry. In general, by varying the pattern of ${\mathbb{Z}}_n$ flux in the ground state, any desired fractionalization class [element of $H^2(G,{\mathbb{Z}}_n)]$ can be realized for the ${\mathbb{Z}}_n$ charge excitations. While the string flux models are not gauge theories, they map to ${\mathbb{Z}}_n$ gauge theories in a certain limit, where they follow a magnetic route for the emergence of low-energy gauge structure. The models are analyzed by studying the action of $G$ symmetry on ${\mathbb{Z}}_n$ charge excitations, and by gauging the $G$ symmetry. The latter analysis confirms that distinct fractionalization classes give rise to distinct quantum phases, except that classes $\left[\omega \right],{\left[\omega \right]}^{-1}\in H^2(G,{\mathbb{Z}}_n)$ give rise to the same phase. We conclude with a discussion of open issues and future directions.

Figure 1

Bilayer square lattice on which the simplest model with $G={\mathbb{Z}}_2$ symmetry and ${\mathbb{Z}}_2$ topological order is defined. The degrees of freedom are spin-1/2 spins residing on links. Each pair of vertically adjacent sites is connected by two distinct links, labeled with up and down arrows.

Figure 2

Detail of the bilayer square lattice, illustrating the labeling of sites and links.

Figure 3

Two string configurations that differ by the string (thick links) sliding over a type I plaquette. The coefficients of these configurations in the ground-state wave functions differ corresponding by a phase factor $K=\pm 1$. The string feels a ${\mathbb{Z}}_2^g$ flux through the type I plaquette when $K=-1$.

Figure 4

(a) Vertices $v^{\prime }$ adjacent to $v$, and oriented edges used to form $A_v$. Here and elsewhere, orientation is denoted by arrows in the middle of an edge. (b) Edges of an oriented cycle $c$.

Figure 5

(a) Outgoing edges from a vertex $g$ in the Cayley graph of $G$, associated with $s,s^{\prime },s^{\prime \prime }\in S$. There are also incoming edges (not shown) joining each of $sg,s^{\prime }g,s^{\prime \prime }g$ back to $g$. The arrows at the end of each edge indicate Cayley graph direction. (b) Cayley graph of $G={\mathbb{Z}}_2=\{1,a\}$.

Figure 6

Depiction of plaquettes $P$ used to define the quantum double model on a Cayley graph. (a) General plaquette $p\in P$. We assume $g_1,g_2\ne 1$, otherwise $g_0,g_1,g_2\in G$ are arbitrary. The orientation is taken counterclockwise, as indicated by the arrows in the middle of each edge. (Recall that the arrows at the end of each edge indicate Cayley graph direction, and not orientation.) The plaquette in (a) reduces to that shown in (b) if $g_1{g}_2=1$ and one takes $g_2=g$ and $g_1=g^{-1}$. In this case, the orientation and Cayley graph direction of the edges agree.

Figure 7

Here, we illustrate how to decompose an arbitrary cycle in a Cayley graph into the elementary cycles. All cycles shown are to be traversed with counterclockwise orientation. In (a), we show a cycle containing five vertices and five edges. We assume the vertices are all distinct. The edges are chosen to have Cayley graph direction as shown, pointing from $g_i$ to $g_j$ if $i<j$. In (b), the cycle of (a) is decomposed into elementary cycles (each triangle). This procedure generalizes obviously to cycles of arbitrary length, but with the special choice of Cayley graph direction shown. In (c), we show a cycle with one edge of the “wrong” direction. This is “repaired” in (d) by gluing a two-edge elementary cycle as shown. We can then proceed to apply the procedure depicted in (b), looking only at the “inner” edges in (d).

Figure 8

Lattice for the $d=1$ string flux model, visualized as a stack of $\left|G\right|$ one-dimensional chains, each labeled by a group element $g\in G$. The shaded ovals represent the Cayley subgraphs that connect the chains together.

Figure 9

(a) General type II plaquette $p\in P_2$. $\mathbf{r},{\mathbf{r}}^{\prime }$ are neighboring hypercubic lattice sites, and $g_0,g\in G$, with $g\ne 1$. Cayley graph direction is indicated on the two Cayley edges, while spatial edges are undirected. (b) General type III plaquette $p\in P_3$. ${\mathbf{r}}_1,\cdots ,{\mathbf{r}}_4$ are hypercubic lattice sites forming a square face, and $g\in G$. We will not need to specify a conventional orientation for type II and type III plaquettes, so the plaquettes here are drawn without orientation.

Figure 10

(a) Spatial $g$ edge in a cycle $c$, connecting hypercubic sites $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$. (b) A type II plaquette. (c) Result of gluing together the edge in (a) and the plaquette in (b). The effect is to “move over” the segment of $c$ shown in (a) to become a spatial 1 edge, at the expense of creating additional edges within the Cayley subgraphs at $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$.

Figure 11

(a) $d=2$ model visualized as a vertical stack of square lattices. The top layer is indicated by solid lines, and the bottom layer by dashed lines (other layers in between not shown). Shaded ovals represent the Cayley graphs connecting the layers, and the shaded arrow indicated that we view the stack of square lattices from above to obtain the planar projection. (b) Shows the resulting planar projection, which is simply a square lattice, with a cut (dashed line). The cut has end points in plaquettes of the square lattice, and intersects square edges, but not vertices. $m$ and $\overline{m}$ particles lie at the end points of the cut. The dash-dot line shows a closed cut $x\left(\mathbf{r}\right)$, encircling the single square lattice site $\mathbf{r}$, which is used in Sec. 8.

Figure 12

(a) Depiction of $e$ strings used to construct a state $|{\psi }_e\rangle$, for $n=3$. The $w_i$ string $\left(i=1,2,3\right)$ is oriented away from the central vertex $v_c$, toward each vertex $v_i$ where an $e$ particle resides. (b) Detail of the path $w_i$. A path $w_i^s$ of spatial 1 edges joins $v_c$ to $v({\mathbf{r}}_i,1)$, which is then joined to $v_i$ by the Cayley $g_i$ edge $({\mathbf{r}}_i;1,g_i)$.

Figure 13

Solid lines show the $e$ string $a_{({\mathbf{r}}_i,1,g)}^{-1}{a}_{({\mathbf{r}}_i,1,g_i)}{a}_{({\mathbf{r}}_i,g_i,g{g}_i)}$, with site labels ${\mathbf{r}}_i$ suppressed. This string is moved through the three plaquettes $p_1,p_2,p_3$, to coincide with the dashed-line string $a_{({\mathbf{r}}_i;g,g{g}_i)}$ joining the same initial and final vertices. The phase factors accumulated as the string passes through $p_1,p_2,p_3$ are ${\omega }^{-1}(g^{-1},g),\omega (g,g_i)$, and $\omega (g{g}_i,g^{-1})$, respectively.