Renormalization Group in Field Theories with Quantum Quenched Disorder

We study the renormalization group flow in general quantum field theories with quenched disorder, focusing on random quantum critical points. We show that in disorder-averaged correlation functions the flow mixes local and nonlocal operators. This leads to a new critical exponent related to the disorder (as in classical disorder). We show that the time coordinate is rescaled at each renormalization group step, leading to anisotropic spacetime scaling at critical points. We write a universal formula for the dynamical scaling exponent $z$ for weak disorder.

Figure 1

The diagrams contributing to the anomalous dimension of the (nonlocal in time) disorder operator $\mathrm{\Psi }$. The solid lines are propagators, and the dashed lines correspond to the nonlocal $v$ interaction. The dot in each diagram, with two dashed lines attached, represents $\mathrm{\Psi }(\mathbf{x})$.