Fractionalizing glide reflections in two-dimensional $Z_2$ topologically ordered phases

We study the fractionalization of space group symmetries in two-dimensional topologically ordered phases. Specifically, we focus on $Z_2$-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.

Figure 1

SSL structure and $Z_4$ electric field configuration at $\nu =1$ on the SSL. Lattice vectors ${\mathit{a}}_1$ and ${\mathit{a}}_2$. The Gauss law constraint ${\prod }_{{\mathit{r}}^{\prime }\in \langle \mathit{r}{\mathit{r}}^{\prime }\rangle }{e}_{\mathit{r}{\mathit{r}}^{\prime }}=i$ must be satisfied at each site. Solid (black) and dashed (magenta) links indicate electric field $e_{\mathit{r}{\mathit{r}}^{\prime }}=1$ and $-1$ respectively. Blue arrows on the diagonal links represent $e_{\mathit{r}{\mathit{r}}^{\prime }}=i$ when the link is traversed in the direction of the arrow. The dotted square shows a single unit cell that includes four sites; note that the $e$-field pattern is the same in every unit cell but breaks point-group symmetries.

Figure 2

$Z_4$ electric field configuration at $\nu =2$ on the SSL. Here, the Gauss law requires ${\prod }_{{\mathit{r}}^{\prime }\in \langle \mathit{r}{\mathit{r}}^{\prime }\rangle }{e}_{\mathit{r}{\mathit{r}}^{\prime }}=-1$ on each site. Solid (black) and dashed (magenta) links indicate electric field $e_{\mathit{r}{\mathit{r}}^{\prime }}=1$ and $-1$, respectively. Note that all space-group symmetries are preserved by this pattern.

Figure 3

Vison creation operators at a square plaquette $\otimes$: products of electric field operator $e_{\mathit{r}{\mathit{r}}^{\prime }}$ and $e_{\mathit{r}{\mathit{r}}^{\prime }}^{†}$ along arrow lines.

Figure 4

Configuration of bond signs ${\eta }_{\overline{\mathit{r}}{\overline{\mathit{r}}}^{\prime }}$ on the dual pentagonal lattice for $\nu =1$. At this filling, ${\prod }_{\overline{\mathit{r}}{\overline{\mathit{r}}}^{\prime }\in ⬠}{\eta }_{\overline{\mathit{r}}{\overline{\mathit{r}}}^{\prime }}=i$ (link products are taken in the anticlockwise sense). Solid (black) and dashed (magenta) links indicate electric field ${\eta }_{\overline{\mathit{r}}{\overline{\mathit{r}}}^{\prime }}=1$ and $-1$, respectively. Blue arrows on each diagonal link represent ${\eta }_{\overline{\mathit{r}}{\overline{\mathit{r}}}^{\prime }}=i$. The dotted square shows a single six-site unit cell. Note that the pattern breaks point group symmetry but not translations, as expected.

Figure 5

Configuration of bond signs ${\eta }_{\overline{\mathit{r}}{\overline{\mathit{r}}}^{\prime }}$ on the dual pentagonal lattice for $\nu =2$. At this filling, ${\prod }_{\overline{\mathit{r}}{\overline{\mathit{r}}}^{\prime }\in ⬠}{\eta }_{\overline{\mathit{r}}{\overline{\mathit{r}}}^{\prime }}=-1$. Solid (black) and dashed (magenta) links indicate electric field ${\eta }_{\overline{\mathit{r}}{\overline{\mathit{r}}}^{\prime }}=1$ and $-1$, respectively. Note that the pattern breaks no symmetries.

Figure 6

Schematic picture of $d$-orbital ordering pattern between $d_{{(x+y)}^2}$ and $d_{{(x-y)}^2}$ on the $x-y$ plane. Dashed line indicates the enlarged unit cell in the presence of orbital ordering. Solid (black colored) and dashed (magenta colored) links have electric field configuration $e_{\mathit{r}{\mathit{r}}^{\prime }}=1$ and $-1$, respectively, satisfying ${\prod }_{{\mathit{r}}^{\prime }\in \langle \mathit{r}{\mathit{r}}^{\prime }\rangle }{e}_{\mathit{r}{\mathit{r}}^{\prime }}=-1$. (See the main text for details.)