Figure 1

Schematic depiction of the coupled-wire array with vertex and plaquette terms $U_{\mathit{r}}^V$ (purple) and $U_{\mathit{r}}^P$ (orange). Gray circles represent the wires, and crossed and dotted circles represent right and left movers, respectively. The labels $q=1,2$ arise from viewing each wire as the intersection of a vertical and horizontal plane, respectively.

Figure 2

Action of (a) the Laughlin quasiparticle operator (4), (b) the chiral electron operator (6), and (c) the composite chiral quasiparticle operator $Q_{LL,\mathit{r}}^{12}$ (7). Integer charges of the vertex and plaquette solitons are indicated in purple and orange, respectively.

Figure 3

Subdimensional excitations of the coupled-wire array include (a) immobile fractons, (b) lineons mobile only along fixed lattice directions, and (d) planons mobile only within 2D planes. Panel (c) depicts the fusion of $x$ and $y$ lineons into a $z$ lineon, while panel (d) depicts a pair of operators that can be multiplied to allow a planon to “turn a corner” between the $x$ and $z$ directions.

Figure 4

Schematic of the surface termination obtained by adding auxiliary boundary wires (gray ovals) and new boundary couplings (green, blue, and red). Dangling chiral gapless modes on the top and right surfaces are clearly visible.

Figure 5

(a) A loop of p-string excitation, confined to the $x-y$ plane (green), at the junction of two topological layers. Anyons (red) are pinned to the p-string where it pierces through a topological layer. The p-string, and attached anyons, fluctuates over the $x-y$ plane (shown by red arrows). (b) A system of topological layers in $x-z$ and $y-z$ planes with an extended p-string excitation fluctuating over an $x-y$ plane (green), as indicated by red arrows. (c) A fracton model obtained from topological layers in $x-z$ and $y-z$ planes via p-string condensation within $x-y$ layers (green).

Figure 6

(a) The red $\times$ depicts a fracton excitation in the planar p-string condensed layer model. It appears at the end of a p-string that has been condensed in an $x-y$ plane (indicated by red arrows) and so does not incur an energy penalty except at the open end point. Hence the excitation is pointlike. (b) A fracton dipole oriented along $\widehat{x}$. This is equivalent to an open p-string piercing a single $y-z$ layer, which pins a single Abelian anyon (red sphere). For bosonic (fermionic) p-strings this composite excitation is a $y-z$ planon. For p-strings consisting of Abelian anyons with a modular braiding this excitation is a $\widehat{y}$ lineon as it cannot pass through the p-string condensates on the $x-y$ planes without incurring an energy penalty.

(c) A fracton dipole oriented along $\widehat{y}$. Similar to (b) for bosonic (fermionic) p-string condensation this is an $x-z$ planon. For modular Abelian anyon p-strings this is an $\widehat{x}$ lineon.

Figure 7

(a) An anyon in an $x-z$ layer that braids nontrivially with the p-strings becomes an $\widehat{x}$ lineon after condensation. (b) Similarly an anyon in a $y-z$ layer that braids nontrivially with the p-strings becomes a $\widehat{y}$ lineon. (c) A composite of $\widehat{x}$ and $\widehat{y}$ lineons that, together, braid trivially with the p-strings is a $\widehat{z}$ lineon. (d) A dipole of $\widehat{x}$ (or $\widehat{y}$) lineons that can be created by a local string operator oriented along $\widehat{z}$ is an $x-y$ planon. (e) A dipole of $\widehat{x}$ lineons separated along $\widehat{y}$ that, together, braid trivially with the p-strings is an $x-z$ planon. (f) Similarly a dipole of $\widehat{y}$ lineons separated along $\widehat{x}$ that, together, braid trivially with the p-strings is a $y-z$ planon.

Figure 8

(a) Dipole configurations are created in the vertex terms by the Laughlin interaction operators $exp\left(2is\tilde{\theta }\right)$. (b) To avoid being projected out of the low-energy subspace, these operators must enter the effective Hamiltonian in the form of a plaquette term. In such configurations, the vertex solitons generated by different Laughlin interaction operators cancel.

Figure 9

A plot of the smallest eigenvalue at a given $k_z$ along the wire direction with $U=100$ and system size $L=15$. One can see that the gap ${\mathrm{\Delta }}_{k_z\to 0}\to 4.8380+\epsilon$ ($\epsilon \ll 1$), reflecting the fact that the model is gapped at finite system size.

Figure 10

Log-log plot of ${\mathrm{\Delta }}_{\delta k_z}\left(L\right)$ vs $L$ at $U=100$. We see that the scaling relation ${\mathrm{\Delta }}_{\delta k_z}\left(L\right)\sim L^{\alpha }$ with $\alpha \approx -1.4118$.

Figure 11

Log-log plot of ${\mathrm{\Delta }}_{\delta k_z}\left(U\right)$ vs $U$ at $L=20$. We see that the scaling relation ${\mathrm{\Delta }}_{\delta k_z}\left(U\right)\sim U^{\beta }$ with $\beta \approx 0.500023$. This points to a scaling ${\mathrm{\Delta }}_{\delta k_z}\left(U\right)\sim \sqrt{U}$.

Figure 12

Pictorial definition of the gapless surface ($\mathrm{L},\mathrm{R},\mathrm{T},\mathrm{B}$) and corner ($\mathrm{TL},\mathrm{TR},\mathrm{BL},\mathrm{BR}$) modes with open boundary conditions in the $x$ and $y$ directions. Chiral modes belonging to the same surface or corner mode are encircled, and $\pm$ indicates the relative sign with which each chiral mode appears [see, e.g., Eq. (A10)].

Figure 13

Schematic depiction of the expressions for the zero modes ${\mathrm{{\rm Y}}}_{+}$ (a) and ${\mathrm{{\rm Y}}}_{-}$ (b) in terms of pinned bulk fields ${\theta }_{\mathit{r}}^{V,P}$ at system size $L_x=L_y=4$. Integers appearing in a plaquette or vertex of the square lattice signify the coefficient with which the corresponding ${\theta }_{\mathit{r}}^P$ or ${\theta }_{\mathit{r}}^V$ field (respectively) enters the expression of ${\mathrm{{\rm Y}}}_{\pm }$ as a linear combination of these fields.

Figure 14

(a) A representation of the model with the alternative boundary conditions discussed in the main text. The gray circles correspond to the vertex terms ${\theta }_{\mathit{r}}^V$ and the squares (both complete and partially complete) correspond to the plaquette terms ${\theta }_{\mathit{r}}^P$. The blue and red ovals, which are shared between the left/right and top/bottom faces, respectively, correspond to the argument of the Laughlin interaction term, $2{\tilde{\theta }}^{1,2}$. Note that the bottom-left corner does not have a plaquette term, as discussed in the main text. (b) Here we provide examples of some useful phase-shift patterns which are employed to clean a general configuration in the ground-state manifold. Note $C^{\prime }$ in particular, which is a special case of pattern $C$ in the bottom-left corner where the plaquette term is absent. This will be key for cleaning the entire configuration of plaquettes.

Figure 15

(a) A subsystem symmetry on an $x-y$ plane (green), consisting of a product of Abelian anyon string operators (red) where the plane intersects topological layers in the $x-z$ planes. (b) The domain wall created by applying a partial subsystem symmetry is a p-string excitation. Gauging the subsystem symmetry condenses these domain walls within the plane, hence inducing planar p-string condensation. The connection between subsystem symmetries and p-string excitations depicted in panels (a) and (b) holds similarly when topological layers in $y-z$ planes are also included; the simpler case has been depicted for clarity of presentation.

Figure 16

(a) A $\widehat{y}$-oriented one-strata where $x-y$ -oriented two-strata supporting ${\mathbb{Z}}_N$ gauge theory (green) and $y-z$ -oriented two-strata supporting a general topological order that contains ${\mathbb{Z}}_N$ Abelian anyons meet. (b) A $\widehat{y}$-oriented one-strata where $x-y$ -oriented two-strata supporting ${\mathbb{Z}}_N$ gauge theory (green) and $x-z$ -oriented two-strata supporting a general topological order that contains ${\mathbb{Z}}_N$ Abelian anyons meet. (c) A $\widehat{z}$-oriented one-strata linking $x-z$ -oriented two-strata and $y-z$ -oriented two-strata, by a tensor product of identity domain walls.