Boundary states of three dimensional topological order and the deconfined quantum critical point

We study the boundary states of the archetypal three dimensional topological order, i.e. the three dimensional Z2 toric code. There are three distinct elementary types of bound-ary states that we will consider in this work. In the phase diagram that includes the three elementary boundaries there may exist a multi-critical point, which is captured by the so-called deconfined quantum critical point (DQCP) with an "easy-axis" anisotropy. Moreover, there is an emergent Z2,d symmetry that swaps two of the boundary types, and it becomes part of the global symmetry of the DQCP. The emergent Z2,d (where d represents "defect") symmetry on the boundary is originated from a type of surface de-fect in the bulk. We further find a gapped boundary with a surface topological order that is invariant under the emergent symmetry.