Wavefunctions of Symmetry Protected Topological Phases from Conformal Field Theories

Abstract

We propose a method for analyzing two-dimensional symmetry protected topological (SPT) wavefunctions using a correspondence with conformal field theories (CFTs) and integrable lattice models. 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 symmetries and show how the CFT description emerges at large scale, thereby revealing a deep connection between group cocyles and critical, sometimes integrable, models. We show that the CFT describing the bulk wavefunction is often also the one describing the entanglement spectrum, but not always. Using a plasma analogy, we also prove the existence of hidden quasi-long-range order for a large class of SPTs. Finally, we show how response to symmetry fluxes is easily described in terms of the CFT.

21 pages, 6 figures, previous results generalized to non-Abelian groups and new relations established between group cocycles and integrable models and between the CFT and the entanglement spectrum. The loop fugacity for the Z2xZ2xZ2 type 3 SPT has been corrected to a value of 2, instead of 1 previously, thereby leading to an exact mapping to an SU(2) level 1 integrable model