Conflicting symmetries in topologically ordered surface states of three-dimensional bosonic symmetry protected topological phases

We study the ${\mathbb{Z}}_2$ topologically ordered surface state of three-dimensional bosonic SPT phases with the discrete symmetries ${\mathrm{G}}_1\times {\mathrm{G}}_2$. It has been argued that the topologically ordered state cannot be realized on a purely two-dimensional lattice model. We carefully examine the statement and show that the surface state should break ${\mathrm{G}}_2$ if the symmetry ${\mathrm{G}}_1$ is gauged on the surface. This manifests the conflict of the symmetry ${\mathrm{G}}_1$ and ${\mathrm{G}}_2$ on the surface of the three-dimensional SPT phase. Given that there is no such phenomena in the purely two-dimensional model, it signals that the symmetries are encoded anomalously on the surface of the three-dimensional SPT phases and that the surface state can never be realized on the purely two-dimensional models.