Hidden order and flux attachment in symmetry protected topological phases: a Laughlin-like approach

Topological phases of matter are distinct from conventional ones by their lack of a local order parameter. Still in the quantum Hall effect, hidden order parameters exist and constitute the basis for the celebrated composite-particle approach. Whether similar hidden orders exist in 2D and 3D symmetry protected topological phases (SPTs) is a largely open question. Here we introduce a new approach for generating SPT groundstates, based on a generalization of the Laughlin wavefunction. This approach gives a simple and unifying picture of some classes of SPTs in 1D and 2D, and reveals their hidden order and flux attachment structures. For the 1D case, we derive exact relations between the wavefunctions obtained in this manner and group cohomology wavefunctions, as well as matrix product state classification. For the 2D Ising SPT, strong analytical and numerical evidence is given to show that the wavefunction obtained indeed describes the desired SPT. The Ising SPT then appears as a state with quasi-long-range order in composite degrees of freedom consisting of Ising-symmetry charges attached to Ising-symmetry fluxes.