Effect of nonmagnetic disorder on criticality in the "dirty" U(1) spin liquid

We investigate the effect of nonmagnetic disorder on the stability of the algebraic spin liquid ($ASL$) by deriving an effective field theory, nonlinear $\sigma$ model ($NL{\sigma}M$). We find that the anomalous critical exponent characterizing the criticality of the $ASL$ causes an anomalous gradient in the $NL{\sigma}M$. We show that the sign of the anomalous gradient exponent or the critical exponent of the $ASL$ determines the stability of the "dirty" $ASL$. A positive exponent results in an unstable fixed point separating delocalized and localized phases, which is consistent with our previous study [Phys. Rev. B {\bf 70}, 140405 (2004)]. We find power law suppression for the density of spinon states in contrast to the logarithmic correction in the free Dirac theory. On the other hand, a negative exponent destabilizes the $ASL$, causing the Anderson localization. We discuss the implication of our study in the pseudogap phase of high $T_c$ cuprates.