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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 $F \sim \exp (- \langle I_A\rangle /2)= \exp(-c L^η)$, where $I_A$ is the so called Anderson integral, $η$ 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 Anderson's 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 $q (E_F) \sim (\frac{E_F-E_M}{E_M})^{-νη},$ where $ν$ is the critical exponent of correlation length $ξ_c$. On the insulating side of the transition $F$ is constant for system sizes exceeding localization length $ξ$. 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.