Robustness of critical U(1) spin liquids and emergent symmetries in tensor networks

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 nonsymmetric perturbations. Using a field theory description, we find that in the RVB, doping constitutes a relevant perturbation that 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 it also justifies the use of such states as a variational ansatz for critical systems. Finally, we give an example of a projected entangled pair state where nonsymmetric perturbations do not open up a gap and the U⁡(1) symmetry reemerges.