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.

Excitations in the Higher Lattice Gauge Theory Model for Topological Phases III: the 3+1d Case

“…(Copy of Figure 41 from Ref. [9].) 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.…”

“…As described in Ref. [9], we can reverse the orientation of a plaquette (while keeping its base-point fixed) by inverting its group label, as shown in Figure 3. If we want to move the base-point along a path t, as shown in Figure 4, then we must act on that plaquette element with g(t) −1 , so that the plaquette label goes from e p to g(t) −1 e p .…”

“…Furthermore, there are inconsistencies with regards changing the branching structure of the lattice (reversing the orientation of edges or plaquettes, or moving the base-points of plaquettes around), as we showed in the Appendix of Ref. [9]. We therefore consider the higher lattice gauge theory model in various special cases that remove these inconsistencies, or at least make them more manageable.…”

“…As we have already seen in Refs. [9,10], the Hamiltonian model for topological phases based on higher lattice gauge theory [4] hosts rich physics, including non-trivial loop braiding (in the 3+1d case), condensation and confinement. In Ref.…”

“…In Ref. [9], we gave a brief qualitative description of these features, while in Ref. [10] we examined the 2+1d model.…”