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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 ${\mathrm{G}}_{1}\ifmmode\times\else\texttimes\fi{}{\mathrm{G}}_{2}$. 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 ${\mathrm{G}}_{2}$ …