Resonating-valence-bond structure of Gutzwiller-projected superconducting wave functions

Gutzwiller-projected (GP) wave functions have been widely used for describing spin-liquid physics in frustrated magnets and in high-temperature superconductors. Such wave functions are known to represent states of the resonating-valence-bond (RVB) type. In the present work I discuss the RVB structure of a GP singlet superconducting state with nodes in the spectrum. The resulting state for the undoped spin system may be described in terms of the "path integral" over loop coverings of the lattice, thus extending the known construction for RVB states. The problem of the topological order in GP states may be reformulated in terms of the statistical behavior of loops. The simple example of the projected d-wave state on the square lattice demonstrates that the statistical behavior of loops is renormalized in a nontrivial manner by the projection.