Topological Orders with Global Gauge Anomalies

الأصل · EN

By definition, the physics of the d-dimensional (dim) boundary of a (d+1)-dim symmetry protected topological (SPT) state cannot be realized as itself on a d-dim lattice. If the symmetry of the system is unitary, then a formal way to determine whether a d-dim theory must be a boundary or not, is to couple this theory to a gauge field (or to "gauge" its symmetry), and check if there is a gauge anomaly. In this paper we discuss the following question: can the boundary of a SPT state be driven into a fully gapped topological order which preserves all the symmetries? We argue that if the gauge anomaly of the boundary is "perturbative", then the boundary must remain gapless; while if the boundary only has global gauge anomaly but no perturbative anomaly, then it is possible to gap out the boundary by driving it into a topological state, when d ≥ 2. We will demonstrate this conclusion with two examples: (1) the 3d spin-1/2 chiral fermion with the well-known Witten's global anomaly, which is the boundary of a 4d topological superconductor with SU(2) or U(1) Z₂ symmetry; and (2) the 4d boundary of a 5d topological superconductor with the same symmetry. We show that these boundary systems can be driven into a fully gapped Z₂ₙ topological order with topological degeneracy, but this Z₂ₙ topological order cannot be future driven into a trivial confined phase that preserves all the symmetries due to some special properties of its topological defects.