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 $F sim exp (- langle I_Arangle /2)= exp(-c L^{eta})$, where $I_A$ is the so called Anderson integral, $eta $ is the power of multifractal intensity correlations and $langle ... rangle$ denotes the ensemble average. Thus, strong disorder typically increases the sensitivity of a system to an additional impurity exponentially. We recover on the metallic side of the transition Andersons result that fidelity $F$ decays with a power law $F sim L^{-q (E_F)}$ with system size $L$. This power increases as Fermi energy $E_F$ approaches mobility edge $E_M$ as $q (E_F) sim (frac{E_F-E_M}{E_M})^{- u eta},$ where $ u$ is the critical exponent of correlation length $xi_c$. On the insulating side of the transition $F$ is constant for system sizes exceeding localization length $xi$. While these results are obtained from the mean value of $I_A,$ giving the typical fidelity $F$, we find that $I_A$ is widely, log normally, distributed with a width diverging at the AMIT. As a consequence, the mean value of 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.