Figure 1

The two different 2D architectures considered in this paper. They are composed of superconductors (SC, blue) and ferromagnets (FM, red) deposited on top of a 2D array of fractional topological insulators (FTI, grey). ${\mathbb{Z}}_{2m}$ PF zero modes, marked as black stars, arise at each SC/FM interface along the edge. We consider two possible geometries: in (a), the FTIs extend for the whole length of the system, while in (b) the FTIs have fixed size. If we enlarged on the horizontal direction the system in (a), the number of FTIs would stay constant and the edge length would increase, while the vice versa would happen in (b). In the main text, we refer to the two architectures as the stripe and tile models, respectively. As shown schematically in (c), the two models can also be distinguished by different tunneling regimes between the two edge segments gapped out by the same superconducting island. If the SC covers a single FTI (left), tunneling of fractional charge $e/m$ may take place between the two edges, while if the SC covers two different FTIs (right), only electron tunneling is allowed, since transport of a fractional charge cannot take place via a topologically trivial bulk. The tile model (b) only has SC islands of this second kind, while the stripe architecture (a) has both.Reuse & Permissions

Figure 2

A square lattice of ${\mathbb{Z}}_{2m}$ PF zero modes of dimensions $L_x=8$ and $L_y=4$. To label the PFs, we follow the convention established in the main text; first, we order the FTI edges with an index $\mathsf{a}$, and then, we order the PFs along each edge with an index $j$, starting from an arbitrary origin. Each PF zero mode is then denoted as ${\alpha }_{\{\mathsf{a},j\}}$, and all commutation rules between operators at different sites are fixed unambiguously.Reuse & Permissions

Figure 3

The transformation defined in Eqs. (30) and (31) maps the PF operators living on the sites of a square lattice into a set of clock operators $\sigma ,\tau$ defined on the links of the lattice occupied by a superconductor (marked in figure as black dots). Here, as an example, we show the positions of the clock operators in the case of a FTI in the tile model. The mapping between ${\alpha }_1\cdots ,{\alpha }_8$ and ${\sigma }_1,{\tau }_1,\cdots ,{\sigma }_4,{\tau }_4$ shown in this figure is explicitly written down in Eq. (37).Reuse & Permissions

Figure 4

Layout of the stripe and tile architecture in terms of clock operators $\sigma ,\tau$ (black dots), living on the links of the square lattice occupied by a superconductor. Since clock operators at different sites commute, it is not necessary to order the FTI nor to assign an orientation to the FTI edges. However, notice that the clock operators for the two models live on two inequivalent lattices.Reuse & Permissions

Figure 5

Sketch of the phase diagram of the two models, as outlined in the introduction to Sec. 4. We distinguish between two regimes, depending on whether Coulomb or Josephson energy dominates. The Coulomb regime shows, for both models, a nonlocal behavior, with degenerate ground states distinguished by the expectation values of string-operators. Only the tile model, however, presents a truly topological order characterized by anyonic excitations (see Sec. 4a). The stripe model is instead dual to a ${\mathbb{Z}}_{2m}$ lattice gauge theory (see Sec. 4b), which undergoes a deconfinement/confinement phase transition with increasing $J/\Delta$.[66]Reuse & Permissions

Figure 6

(a) The notation adopted for the decorated square lattice on which the tile model can be conveniently rearranged. The grey diamonds, sitting on the sites $\mathit{r}$ of a square lattice, are FTI. To each site $\mathit{r}$ there correspond four clock operators ${\sigma }_{\mathit{r},i},{\tau }_{\mathit{r},i}$, arranged counterclockwise. Blue links are SC, red links are FM. (b) The operator $B_{\mathit{r}}$ in Eq. (48) is the counterclowise product of four $\sigma {\sigma }^{†}$ operators around the same plaquette.Reuse & Permissions

Figure 7

The four non trivial loop operators that define the ground-state manifold of the tile model on a torus. The operators $H_{\tau },V_{\tau }$ are defined as the product of all $\tau$ along the path described by the two blue lines, in the order established by the arrows. Similarly, the operators $H_{\sigma },V_{\sigma }$ are defined as the product of all $\sigma {\sigma }^{†}$ operators along the path given by the red lines. Loop operators corresponding to different paths only differ by a product of stabilizer operators $Q_{〈\mathit{r},{\mathit{r}}^{\prime }〉}$ or $B_{\mathit{r}}$.Reuse & Permissions

Figure 8

Vortex and charge excitations in the tile model, appearing at the ends of open string of $\tau$ and $\sigma {\sigma }^{†}$ operators, respectively. The vortices live on the plaquettes of the lattice, while the charges on the superconducting links. They are mutual Abelian anyons, with an exchange phase ${\mathrm{e}}^{i\pi /2m}$.Reuse & Permissions

Figure 9

Notation adopted to study the stripe model. The black and white sites identify clock operators $\sigma ,\tau$, distinguished by a sublattice degree of freedom. Grey stripes are FTI, blue links are superconducors, and red links ferromagnets. Notice that half of the vertical links are missing, thus the stripe model is effectively defined on a brick-wall lattice. The flux threaded through a single plaquette of the lattice is measured by the operator $B$ in the figure [see Eq. (62)]. As in the tile model, flux excitations (the oriented blue circles) can be created by an open string of $\tau$ operators: however, the geometry constrains their movement in the horizontal direction.Reuse & Permissions

Figure 10

The dual lattice on which the $Z_{2m}$ lattice gauge theory of Eq. (67) is defined. Clock operators now live on those links of the original square lattice which are marked by a blue dot. On this new lattice, we find that in perturbation theory the physical interactions are given by the plaquette and star operators $B_{\mathsf{d}}$ and $A_{\mathsf{d}}$ defined in Eqs. (66) and (69).Reuse & Permissions

Figure 11

The generalized order parameter (string tension) in both the original and dual lattices.Reuse & Permissions