Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order

We construct an exactly soluble Hamiltonian on the D=3 cubic lattice, whose ground state is a topological phase of bosons protected by time-reversal symmetry, i. e. , a symmetry-protected topological (SPT) phase. In this model, excitations with anyonic statistics are shown to exist at the surface but not in the bulk. The statistics of these surface anyons is explicitly computed and shown to be identical to the three-fermion Z₂ model, a variant of Z₂ topological order which cannot be realized in a purely D=2 system with time-reversal symmetry. Thus the model realizes a novel surface termination for three-dimensional (3D) SPT phases, that of a fully symmetric gapped surface with topological order. The 3D phase found here was previously proposed from a field theoretic analysis but is outside the group cohomology classification that appears to capture all SPT phases in lower dimensions. Such phases may potentially be realized in spin-orbit-coupled magnetic insulators, which evade magnetic ordering. Our construction utilizes the Walker-Wang prescription to create a 3D confined phase with surface anyons, which can be extended to other topological phases.