Wavefunctions of Symmetry Protected Topological Phases from Conformal Field Theories

We propose a method for analyzing two-dimensional symmetry protected topological (SPT) wavefunctions using a correspondence with conformal field theories (CFTs). This method generalizes the CFT approach for the fractional quantum Hall effect wherein the wavefunction amplitude is written as a many-operator correlator in the CFT. Adopting a bottom-up approach, we start from various known microscopic wavefunctions of SPTs with discrete Abelian symmetries and show how the CFT description emerges at large scale. This includes an identification of both the CFT and its operator content, along with the association between microscopic and CFT operators. The CFTs obtained vary substantially and may have negative, zero, or positive central charge. Based on this approach, a Laughlin-like picture for the $Z_{N>2}$ group-cohomology wavefunctions is derived analytically, and includes the flux attachment, plasma, and hidden order structures. Flux responses and bulk-edge correspondences are also discussed for several cases.