Wire constructions of Abelian topological phases in three or more dimensions

Coupled-wire constructions have proven to be useful tools to characterize Abelian and non-Abelian topological states of matter in two spatial dimensions. In many cases, their success has been complemented by the vast arsenal of other theoretical tools available to study such systems. In three dimensions, however, much less is known about topological phases. Since the theoretical arsenal in this case is smaller, it stands to reason that wire constructions, which are based on one-dimensional physics, could play a useful role in developing a greater microscopic understanding of three-dimensional topological phases. In this paper, we provide a comprehensive strategy, based on the geometric arrangement of commuting projectors in the toric code, to generate and characterize coupled-wire realizations of strongly interacting three-dimensional topological phases. We show how this method can be used to construct pointlike and linelike excitations, and to determine the topological degeneracy. We also point out how, with minor modifications, the machinery already developed in two dimensions can be naturally applied to study the surface states of these systems, a fact that has implications for the study of surface topological order. Finally, we show that the strategy developed for the construction of three-dimensional topological phases generalizes readily to arbitrary dimensions, vastly expanding the existing landscape of coupled-wire theories. Throughout the paper, we discuss ${\mathbb{Z}}_m^{}$ topological order in three and four dimensions as a concrete example of this approach, but the approach itself is not limited to this type of topological order.

Figure 1

A single unit cell of the square array of wires, consisting of a single star $s$ and plaquette $p$. The dashed nearest-neighbor links belong to neighboring unit cells. The midpoint of each link hosts a quantum wire, represented by the symbol $\times$, aligned along the $z$ direction (out of the page). Any plaquette $p$ is surrounded by four quantum wires located at the four cardinal points $p_N,p_W,p_S$, and $p_E$, respectively. Similarly, any star $s$ is surrounded by four quantum wires located at the four cardinal points $s_N,s_W,s_S$, and $s_E$.

Figure 2

Pictorial representation of the tunneling vectors (2.13). The $2M$-dimensional integer-valued vectors $v_{1,2}^{\left(\mathtt{j}\right)}$ and $w_{1,2}^{\left(\mathtt{j}\right)}$ determine the linear combinations of bosonic fields in each wire that enter the cosine term associated with each star or plaquette, respectively.

Figure 3

Pictorial representation of star and plaquette excitations created by the operators (2.31). The filled blue circle represents an application of the vertex operator (2.31a) in the corresponding wire, while the purple and orange crosses represent defective stars hosting solitons of opposite signs. Similarly, the filled green square represents an application of the vertex operator (2.31b), and the purple and orange squares represent defective plaquettes hosting solitons of opposite signs.

Figure 4

Deconfinement of star defects within a plane. (a) A single application of the vertex operator (2.31a) (blue circle) creates a star defect (purple cross) and an antistar defect (orange cross). (b) Applying a string of vertex operators (2.31a) moves a star defect by healing one while creating another. Consequently, it costs no extra energy to separate the two star defects. (c) To turn a corner, it may be necessary to heal a defect with an application of the inverse of the vertex operator (2.31a) in the appropriate wire (white circle). (d) When two star defects meet, they annihilate one another. A completely analogous description of plaquette defects also holds starting from the operators Eq. (2.31b).

Figure 5

(a) A pair of linelike defects along the $z$ direction created by the operators (2.37). (b) Propagating one linelike defect away from the other using the operators (2.37) creates a membrane in the $x\text{−}z$ plane. (c) Growing a star membrane in the $x\text{−}y$ plane from the product of the operators (2.31a). (d) Membrane in the $x\text{−}y$ plane with defect lines on its boundaries. Darker purple or orange crosses indicate defective stars with a “double-strength” soliton, where two kinks of the same sign coexist in the same star.

Figure 6

Pictorial representation for braiding a pointlike plaquette defect [defined in Eq. (2.31b)] around a linelike star defect along the $z$ direction [defined in Eq. (2.37a)]. The green colored loop represents the world line during adiabatic braiding of one end of an open plaquette string, say the plaquette counterpart to the open string depicted in Fig. 4. By choice, the world line encloses the rightmost boundary of the star membrane in Fig. 5.

Figure 7

Pictorial representation of braiding a star line and a plaquette line. Because periodic boundary conditions are imposed in all directions and because we choose to represent the world sheet induced by the adiabatic evolution of plaquette linelike defects by the green membrane that extends across the width of the lattice in the $y$ direction, this world sheet forms a cylinder with its symmetry axis parallel to the $z$ direction. By choice, this cylinder encircles the star-type line that defines the rightmost boundary of the blue membrane. The algebra encoded by Eq. (2.42) implies that such a braiding yields no overall statistical phase.

Figure 8

Pictorial representation of the plaquette-type string operators defined in Eqs. (2.45). The star-type string operators are defined similarly.

Figure 9

Pictorial representations of the star-type membrane operators defined in Eqs. (2.46). In (a), the membrane is parallel to the $x\text{−}z$ plane (there is also a similarly defined membrane parallel to the $y\text{−}z$ plane). In (b), the membrane is parallel to the $x\text{−}y$ plane. Plaquette-type membrane operators are represented similarly.

Figure 10

Pictorial representations of the tunneling vectors (a) ${\tilde{\mathcal{T}}}_s$ and (b) ${\tilde{\mathcal{T}}}_p$ built using the vectors $\tilde{v}$ and $\tilde{w}$ defined in Eqs. (2.62). The signs $\pm$ indicate whether a $\pm 1$ appears in the tunneling vector associated with that link.

Figure 11

Example of an array of quantum wires with open boundary conditions in the $y$ direction and periodic boundary conditions along all other directions. The dashed links indicate the presence of periodic boundary conditions in the $x$ direction. The crosses represent quantum wires on the links of the square lattice that are inequivalent modulo the periodic boundary conditions. In this example, $N_x=3$ and $N_y=2$. Consequently, there are $2N_x{N}_y+N_x=15$ wires in the array, and $2N_x{N}_y-N_x=9$ wires are gapped by the allowed tunneling vectors. Consequently, there are six wires in the array that remain gapless when these tunneling vectors are included (three on the top face and three on the bottom face, represented by the purple crosses).

Figure 12

Examples of hypercubic stars and plaquettes for arrays of quantum wires of types (a) (2,1), (b) (3,1), and (c) (3,2). Black crosses represent wires extending perpendicular to all principal directions of the respective hypercubic lattices. The numbers label generalized cardinal directions $C_{\mathrm{s}}$ and $C_{\mathrm{p}}$ defined in Eqs. (3.3b) and (3.4b).

Figure 13

Pictorial representations of the tunneling vectors (a) ${\tilde{\mathcal{T}}}_s$ and (b) ${\tilde{\mathcal{T}}}_p$, defined in Eqs. (3.6), for the array of quantum wires of type $(3,1)$. As in Fig. 10, they are built using the vectors $\tilde{v}$ and $\tilde{w}$ defined in Eqs. (2.62).

Figure 14

Pictorial representations of the tunneling vectors (a) ${\tilde{\mathcal{T}}}_s$ and (b) ${\tilde{\mathcal{T}}}_p$, defined in Eqs. (3.6), for the array of quantum wires of type $(3,2)$. As in Figs. 10 and 13, they are built using the vectors $\tilde{v}$ and $\tilde{w}$ defined in Eqs. (2.62).

Figure 15

Pictorial representation of the six plaquette-centered tunneling vectors ${\mathcal{T}}_p$ surrounding a cubic cell of the array of quantum wires labeled by (3,1). (Folding sides $2,\cdots ,5$ upwards out of the page and placing side 6 on top constructs the cubic cell.) The numbers $1,\cdots ,6$ label these tunneling vectors in the order in which they appear in the Gram matrix $G$ in Eq. (3.10). The signs $\pm$ indicate whether a $\pm 1$ appears in the tunneling vector associated with that link.