| We discuss a short-range entangled topological phase in 3+1 dimensions that is protected by time-reversal symmetry. Two models are compared that realize this phase: The first is a construction developed by Chen, Gu, Liu and Wen, which encodes the system's topological properties in the representation of the symmetry group. The second theory uses a non-linear sigma model in which the distinct topological phases differ by the way the symmetry acts on the order parameter. Both theories have in common that the modeled phases are in one to one correspondence with the elements of the co-homology group Hd+1(Z2 T, UT(1)). In this work, we extend the Chen-Gu construction to 3+1 dimensional systems. Furthermore, we show that both models coincide with respect to their topological properties. This is proved by comparing spin-flip processes and their associated topological phase factors. We derive spin-flip operators on the surface of the (3+1)-dimensional Chen-Gu construction that commute with time-reversal symmetry. To implement spin-flip processes in the non-linear sigma model, we interpolate spin-configurations from a discrete, triangular lattice into the continuum. We proceed by analyzing the phases, generated by the theta-term, for spacetime configurations of the O(4) order parameter that correspond to these spin-flip processes. |
