Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview

Abstract

In this series of papers, we study a Hamiltonian model for 3+1d topological phases, based on a generalisation of lattice gauge theory known as "higher lattice gauge theory". Higher lattice gauge theory has so called "2-gauge fields" describing the parallel transport of lines, just as ordinary gauge fields describe the parallel transport of points. In the Hamiltonian model this is represented by having labels on the plaquettes of the lattice, as well as the edges. In this paper we summarize our findings in an accessible manner, with more detailed results and proofs to be presented in the other papers in the series. The Hamiltonian model supports both point-like and loop-like excitations, with non-trivial braiding between these excitations. We explicitly construct operators to produce and move these excitations, and use these to find the loop-loop and point-loop braiding relations. These creation operators also reveal that some of the excitations are confined, costing energy to separate. This is discussed in the context of condensation/confinement transitions between different cases of this model. We also discuss the topological charges of the model and use explicit measurement operators to re-derive a relationship between the number of charges measured by a 2-torus and the ground-state degeneracy of the model on the 3-torus. From these measurement operators, we can see that the ground state degeneracy on the 3-torus is related to the number of types of linked loop-like excitations.

Figures (51)

• Figure 1: Composition of paths is described by group multiplication
• Figure 2: The gauge transform on a vertex is equivalent to adding an imaginary edge at that vertex and then combining this edge into the diagram, or equivalently transporting the vertex along that edge.
• Figure 3: We consider the effect of a vertex transform $A_v^x$ on a closed loop starting at the vertex $v$. The path label in this case goes from $g_1g_2$ to $xg_1g_2x^{-1}$. That is, the path label of the closed loop is conjugated by $x$ under the action of the transform.
• Figure 4: The plaquette term $B_p$ gives 1 if the closed path forming the boundary of the plaquette $p$ is $1_G$ (i.e., if it is flat) and 0 otherwise. For the example plaquette shown in this figure, where the edges are in states labelled by $g_1$ and $g_2$, that means that acting with the plaquette term gives a non-zero result only if $g_1g_2^{-1}=1_G$.
• Figure 5: Just as a path is associated to parallel transport of points, so is a surface associated to parallel transport of a path. The initial position of the path is called the source and the final position is called the target. The parallel transport of a path over a surface labelled by $e \in E$ results in the path element $g$ gaining a factor of $\partial(e)$, where $\partial$ is a group homomorphism from $E$ to $G$.