Topological phases in $3+1D$ are less well understood than their lower dimensional counterparts. A useful approach to the study of such phases is to look at toy models that we can solve exactly. In this talk I will present new results for an existing model for certain topological phases in $3+1D$ (the model was first presented in [1]). This model is based on a generalisation of lattice gauge theory known as higher lattice gauge theory, which treats parallel transport of lines as well as points. I will first provide a brief introduction to higher lattice gauge theory and the Hamiltonian model constructed from it. Then we will look at the simple excitations (both point-like and loop-like) that are present in this model and how these excitations can be constructed explicitly using so-called ribbon and membrane operators. Some of the quasi-particles are confined and we discuss how this arises from a condensation-confinement transition. We will then look at the (loop-)braiding relations of the excitations and finish by examining the conserved topological charges realised by the Higher Lattice Gauge Theory Model.

[1] A Bullivant, M. Calcada et al., Topological phases from higher gauge symmetry in 3+1D, Phys. Rev. B 95, 155118 (2017).