The higher lattice gauge theory model for topological phases of matter

Topological phases of matter are characterised by long-range entanglement between their constituent degrees of freedom, which allows them to host excitations with non-trivial exchange statistics. While topological phases in 2+1d are relatively well studied, less is known about their 3+1d counterparts. In this thesis, we examine features of these higher-dimensional phases in a tractable example model, based on a generalisation of lattice gauge theory called higher lattice gauge theory. As well as hosting point-like excitations, the model supports loop-like excitations with non-trivial loop-loop and point-loop braiding statistics. 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, while others are condensed and can be produced locally. This is discussed in the context of condensation-confinement transitions between different cases of this model. We also explain a method for measuring topological charge 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 higher lattice gauge theory 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