Excitations in the Higher Lattice Gauge Theory Model for Topological Phases III: the (3+1)-Dimensional Case

Abstract

In this, the third paper in our series describing the excitations of the higher lattice gauge theory model for topological phases, we will examine the 3+1d case in detail. We will explicitly construct the ribbon and membrane operators which create the topological excitations, and use these creation operators to find the pattern of condensation and confinement. We also use these operators to find the braiding relations of the excitations, and to construct charge measurement operators which project to states of definite topological charge.

Figures (145)

• Figure 1: (Copy of Figure 37 from Ref. HuxfordPaper1) An electric ribbon operator measures the value of a path and assigns a weight to each possibility, creating excitations at the two ends of the path. In order to find the group element associated to the path, we must first find the contribution of each edge to the path. In this example, the edges along the path are shown in black. Some of the edges are anti-aligned with the path and so we must invert the elements associated to these edges to find their contribution to the path. This is represented by the grey dashed lines, which are labeled with the contribution of each edge to the path.
• Figure 2: Two adjacent surfaces can be combined into one if their base-point (represented by the yellow dot) and circulation (represented by the blue arrow in the middle of each plaquette) match. The label of the combined surface is given by the product of the two individual elements in reverse order. That is, if the surfaces $A$ and $B$ have labels $e_A$ and $e_B$ respectively, then the combined surface has label $e_{AB}=e_B e_A$. If two adjacent surfaces do not have the same base-point and orientation then we can still combine them by using a set of rules that describe what happens when we change the orientation or move the base-point of a surface.
• Figure 3: Given a plaquette with label $e_p$, the label of the corresponding plaquette with the opposite orientation is $e_p^{-1}$. Note that when we reverse the orientation of a plaquette, we leave its base-point, here $v_0(p)$, in the same position.
• Figure 4: We can move the base-point of a surface, either along the boundary of the surface (resulting in the case shown in the bottom-left) or away from the surface (in which case we say that we whisker the surface, and obtain the situation shown in the top-right). When we move the base-point of plaquette $p$ along a path $t$ in this way, the surface label goes from $e_p$ to $g(t)^{-1} \rhd e_p$.
• Figure 5: When we combine two surfaces (top image) into one (bottom image), the boundary of the combined surface is the product of the two individual boundaries (here the boundary path is represented as the dashed red line). This boundary can be simplified by removing edges that appear twice consecutively in the boundary with opposite orientation. In this case the combined boundary includes $i_4^{-1}i_4$ in the top image (this is the section that dips down in the image), which can be removed to give the boundary shown in the bottom image.