Twisted gauge theories in three-dimensional Walker-Wang models

Three-dimensional gauge theories with a discrete gauge group can emerge from spin models as a gapped topological phase with fractional point excitations (gauge charge) and loop excitations (gauge flux). It is known that 3D gauge theories can be “twisted,” in the sense that the gauge flux loops can have nontrivial braiding statistics among themselves and such twisted gauge theories are realized in models discovered by Dijkgraaf and Witten. A different framework to systematically construct three-dimensional topological phases was proposed by Walker and Wang and a series of examples have been studied. Can the Walker-Wang construction be used to realize the topological order in twisted gauge theories? This is not immediately clear because the Walker-Wang construction is based on a loop condensation picture while the Dijkgraaf-Witten theory is based on a membrane condensation picture. In this paper, we show that the answer to this question is Yes, by presenting an explicit construction of the Walker-Wang models which realize both the twisted and untwisted gauge theories with gauge group ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$. We identify the topological order of the models by performing modular transformations on the ground-state wave functions and show that the modular matrices exactly match those for the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ gauge theories. By relating the Walker-Wang construction to the Dijkgraaf-Witten construction, our result opens up a way to study twisted gauge theories with fermonic charges, and correspondingly strongly interacting fermionic symmetry protected topological phases and their surface states, through exactly solvable models.

Figure 1

A three-loop braiding process. The process involves a flux loop $\alpha$ sweeping out a torus enclosing a flux loop $\beta$ while both linked with a “base” flux loop $\gamma$. The blue curves indicate the trajectory of two points on loop $\alpha$. If $\alpha$ and $\beta$ are identical, we can similarly define the process where $\alpha$ and $\beta$ are exchanged while both linked with $\gamma$.

Figure 2

An example of the ground-state wave function of a Walker-Wang model. $a$ and $b$ here label the quasiparticle types.

Figure 3

Graphical definition of $F$ and $R$ symbols.

Figure 4

Planar projection of a trivalent resolution of the cubic lattice.

Figure 5

Plaquette term in the Walker-Wang Hamiltonian.

Figure 6

Sketch of surface excitations in a gauged SPT model. The bulk is below the plane. $q$ represents a gauge charge that can appear in the bulk and on the surface. $\chi$ represents an anyon that can appear only on the surface. $\alpha$ and $\beta$ represent the flux lines in the bulk that end on the surface.

Figure 7

Data for Rep($Q_8$) and Rep($D_4$). The (simple) objects are the irreducible representations (charges) of either $Q_8$ or $D_4$, defined in Eqs. (6) and (7). They are all self-dual. Here, $a,b=0,1,2,3$ and $x,y=1,2,3$, $x\ne y$. $\chi$ is defined by $\chi (0,a)=\chi (a,0)=1$, $\chi (x,x)=1$, and $\chi (x,y)=-1$. The quantum dimension of a charge is defined to be the product of its Frobenius-Schur indicator and dimension. The $F$ symbols with $n=-1$ (respectively, $n=1$) are the $6j$ symbols of $Q_8$ (respectively, $D_4$). The $R$ symbols are solutions to the hexagon equations (given the $F$ symbols) with the constraint that all charges are bosonic and have trivial mutual statistics. For $Q_8$, there exists a unique solution, $n=-1$, $s_0=1$, and $s_1=s_2=s_3=-1$. For $D_4$, there exists three solutions, $n=1$, $s_0=1$, $s_1=-1$, $s_2=1$, $s_3=1$, and also the ones resulting from the permutations $1\leftrightarrow 2$ and $1\leftrightarrow 3$, respectively.

Figure 8

$S$ and $T$ transformations on a three-torus.

Figure 9

$T$ transformation as the Dehn twist of a hollow cylinder.

Figure 10

A “minimal” trivalent lattice on the three-torus, which consists of four vertices, six edges, and three plaquettes.

Figure 11

Decomposition of a three-loop braiding process into two separate braiding processes in the dimensionally reduced 2D system.