  We study the response of critical Resonating Valence Bond (RVB) spin liquids
to doping with longer-range singlets, and more generally of U(1)-symmetric
tensor networks to non-symmetric perturbations. Using a field theory
description, we find that in the RVB, doping constitutes a relevant
perturbation which immediately opens up a gap, contrary to previous
observations. Our analysis predicts a very large correlation length even at
significant doping, which we verify using high-accuracy numerical simulations.
This emphasizes the need for careful analysis, but also justifies the use of
such states as a variational ansatz for critical systems. Finally, we give an
example of a PEPS where non-symmetric perturbations do not open up a gap and
the U(1) symmetry re-emerges.
