Anomalous dynamical scaling from nematic and U(1) gauge field fluctuations in two-dimensional metals

We analyze the scaling theory of two-dimensional metallic electron systems in the presence of critical bosonic fluctuations with small wave vectors, which are either due to a U(1) gauge field or generated by an Ising nematic quantum critical point. The one-loop dynamical exponent z = 3 of these critical systems was shown previously to be robust up to three-loop order. We show that the cancellations preventing anomalous contributions to z at three-loop order have special reasons, such that anomalous dynamical scaling emerges at four-loop order.