Multifractal Orthogonality Catastrophe in 1D Random Quantum Critical Points

Abstract: We study the response of random singlet quantum critical points to local perturbations. Despite being insulating, these systems are dramatically affected by a local cut in the system, so that the overlap $G=\left|\langle \Psi_B |\Psi_A \rangle\right|$ of the groundstate wave functions with and without a cut vanishes algebraically in the thermodynamic limit. We analyze this Anderson orthogonality catastrophe in detail using a real-space renormalization group approach. We show that both the typical value of the overlap G and the disorder average of $G^\alpha$ with $\alpha>0$ decay as power-laws of the system size. In particular, the disorder average of $G^\alpha$ shows a "multifractal" behavior, with a non-trivial limit $\alpha \to \infty$ that is dominated by rare events. We also discuss the case of more generic local perturbations and generalize these results to local quantum quenches.