A String-Membrane Network Model of Topological Order in 3d Space

Topological order refers to a phase of matter whose description goes beyond the Landau theory of phases in terms of global symmetries. A quintessential example of topologically ordered phases in two spatial dimensions is a fractional quantum Hall phase at one of the stable quantized values for the Hall conductivity. Many experimental and theoretical studies have been done for topological order in two spatial dimensions, however understanding this class of phases in three-dimensional space is an active area of research. We propose an exactly solvable Hamiltonian model for 3d topological order as a lattice system of 2d topologically ordered membranes joined along trivalent edges by inter-membrane interactions. Our model is based on the coupled-wire construction where various topological phases of matter can be simulated by a system of strongly interacting quasi-one-dimensional subsystems (i.e. wires). In the case of topological order, such interactions are motivated from the anyon condensation of 2d topologically ordered phases as the analogue to the familiar Bose-Einstein condensation of Landau phases. We explore examples of membrane networks built from the toric code phases and the double-semion phases and we demonstrate how an exactly solvable Hamiltonian lattice model in three-dimensional space could be constructed.