Exponential Orthogonality Catastrophe at the Anderson Metal-Insulator Transition

We consider the orthogonality catastrophe at the Anderson metal-insulator transition (AMIT). The typical overlap F between the ground state of a Fermi liquid and the one of the same system with an added potential impurity is found to decay at the AMIT exponentially with system size L as F∼exp(-cL^{η}), where η is the power of multifractal intensity correlations. Thus, strong disorder typically increases the sensitivity of a system to an added impurity exponentially. We recover, on the metallic side of the transition, Anderson's result that the fidelity F decays with a power law F∼L^{-q(E_{F})} with system size L. Its power increases as the Fermi energy E_{F} approaches the mobility edge E_{M} as q(E_{F})∼[(E_{F}-E_{M})/E_{M}]^{-νη}, where ν is the critical exponent of the correlation length ξ_{c}. On the insulating side of the transition, F is constant for system sizes exceeding the localization length ξ. While these results are obtained for the typical fidelity F, we find that logF is widely, log normally, distributed with a width diverging at the AMIT. As a consequence, the mean value of the fidelity F converges to one at the AMIT, in strong contrast to its typical value which converges to zero exponentially fast with system size L. This counterintuitive behavior is explained as a manifestation of multifractality at the AMIT.