Loop braiding statistics in exactly soluble three-dimensional lattice models

We construct two exactly soluble lattice spin models that demonstrate the importance of three-loop braiding statistics for the classification of three-dimensional gapped quantum phases. The two models are superficially similar: both are gapped and both support particlelike and looplike excitations similar to those of charges and vortex lines in a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ gauge theory. Furthermore, in both models the particle excitations are bosons, and in both models the particle and loop excitations have the same mutual braiding statistics. The difference between the two models is only apparent when one considers the recently proposed three-loop braiding process in which one loop is braided around another while both are linked to a third loop. We find that the statistical phase associated with this process is different in the two models, thus proving that they belong to two distinct phases. An important feature of this work is that we derive our results using a concrete approach: we construct string and membrane operators that create and move the particle and loop excitations and then we extract the braiding statistics from the commutation algebra of these operators.

Figure 1

(a) Two-loop braiding process. (b) Three-loop braiding process. The gray curves show the paths of two points on the moving loop.

Figure 2

Both models are built out of two species of spins. The blue spins ${\sigma }_p$ live on the plaquettes $p$ of the cubic lattice, while the red spins ${\sigma }_{\widehat{p}}$ live on the plaquettes $\widehat{p}$ of the dual cubic lattice.

Figure 3

(a) The sheared square lattice is formed by shifting the corners of the square plaquettes (the blue dots) according to $(i,j)\to (i+\epsilon j,j)$ with $\epsilon >0$. (b) Top view of the sheared cubic lattice. The solid/dashed squares denote cubes on two neighboring layers. The sheared cubic lattice is formed by shifting the corners of the cubes according to $(i,j,k)\to (i+\epsilon j+{\epsilon }^{\prime }k,j+{\epsilon }^{\prime }k,k)$ with ${\epsilon }^{\prime }>{\epsilon }^{\prime }>0$.

Figure 4

(a) A blue cube intersects a red membrane; their intersection consists of a single red loop. (b) A blue cube intersects/overlaps with a blue membrane; their intersection consists of a single blue region. (c) A blue cube intersects with both a red membrane and a blue membrane; their intersection consists of one red loop and one blue region. (d) For the membrane configuration shown in (c), the integer $m_c=1$ because there is one red loop while $n_c=2$ because there are two intersections (denoted by green dots) between the red loop and the boundary of the blue region (thick blue line).

Figure 5

(a) The integer $m_c$ is a function of the 12 spins ${\sigma }_{\widehat{p}}^z$ that are closest to the cube $c$. (b) The integer $n_c$ is a function of the 6 blue spins ${\sigma }_p^z$ and 12 red spins ${\sigma }_{\widehat{p}}^z$ around the cube $c$.

Figure 6

(a) A blue cylinder $S$ intersects with a red membrane and two blue membranes (only part of $S$ is shown for clarity). One of the blue membranes is incident from above $S$ and the other from below. (b) We represent the intersections as a picture drawn on the cylinder $S$. In this case, the picture consists of a red loop, and two blue regions, one with a solid boundary and one with a dotted boundary.

Figure 7

A typical picture representing the intersections of an (unlinked) blue cylinder with red and blue membranes. Here, we draw the cylinder as a rectangle with top and bottom identified and the left and right being the two ends of the cylinder.

Figure 8

Two typical pictures representing the intersections between (linked) blue cylinders and red and blue membranes. Here, we draw the cylinder as a rectangle with top and bottom identified. Panel (a) shows a typical picture for a blue cylinder linked with a red base loop. Panel (b) shows a typical picture for a blue cylinder linked with a blue base loop.

Figure 9

A blue torus operator $M_b\left(S\right)$ acts on two slightly different membrane configurations (a) $(X_b,X_r)$ and (b) $(X_b^{\prime },X_r^{\prime })$. Here, $X_b=X_b^{\prime }$ is a blue sphere, while $X_r$ consists of two red spheres and $X_r^{\prime }$ consists of a bigger red sphere obtained by merging the two red spheres in $X_r$.

Figure 10

The exchange statistics of the charge excitations can be computed from the commutation algebra (47) of three string operators acting on three paths $P_1,P_2,P_3$ that share a common end point.

Figure 11

(a) The statistical phase associated with braiding a charge around a loop can be computed from the commutation algebra (48) of a string operator acting along a path $P$, and a membrane operator acting along a cylinder $S$. (b) The phase associated with braiding two loops around one another can be computed from the commutation algebra (49) of two membrane operators, one acting on a cylinder $S$ and the other acting on a torus $S^{\prime }$.

Figure 12

The statistical phase associated with a three-loop braiding process can be computed from the commutation algebra of two membrane operators, one acting along the cylinder $S$ and the other acting along the torus $S^{\prime }$. Here, both $S$ and $S^{\prime }$ are linked with a base loop which lies along the boundary of the disk $D$.

Figure 13

An example of the 2D identity (A8) for the case of loops living on the honeycomb lattice. Panel (a) shows a loop configuration $X$, while panel (b) shows the corresponding configuration $X+p$. In this example, $N_{\text{loop}}\left(X\right)=2$ and $N_{\text{loop}}(X+p)=1$. Also, $n_p\left(X\right)=4$ since four of the six legs adjacent to the plaquette $p$ are occupied by strings. The identity (A8) holds for this example since ${(-1)}^{1-2}=(-1)i^4$.