Renormalization group for quantum critical theories with Harris-marginal disorder

We develop a renormalization group for small-amplitude Harris-marginal disorder in otherwise strongly interacting quantum critical theories, focusing on systems that have emergent conformal invariance. Using conformal perturbation theory, we argue that previously proposed random fixed points with Lifshitz scaling in fact flow towards other universal fixed points and that this flow can be captured by a "one-loop calculation". Our approach appears best controlled in theories with only a few operators of low scaling dimensions. In this regime, we compare our predictions for the flow of disorder to holographic models and find complete agreement.