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

Abstract

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 ${\mathrm{G}}_{2}$ 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.

Locations

• arXiv (Cornell University) - View - PDF
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• Physical Review B - View