Exponential Orthogonality Catastrophe at the Anderson Metal-Insulator Transition

TL;DR

This paper investigates how the orthogonality catastrophe behaves at the Anderson Metal-Insulator transition, revealing an exponential decay of ground state overlap with system size due to multifractal correlations, contrasting with metallic and insulating regimes.

Contribution

It introduces the exponential decay of fidelity at the AMIT and analyzes the distribution of Anderson integral, highlighting multifractality's role in fidelity behavior.

Findings

Fidelity decays exponentially with system size at the AMIT.

Fidelity follows a power law on the metallic side, increasing near the mobility edge.

Distribution of Anderson integral is log-normal with diverging width at the transition.

Abstract

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 $F \sim \exp (- \langle I_A\rangle /2)= \exp(-c L^{\eta})$, 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 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 $q (E_F) \sim (\frac{E_F-E_M}{E_M})^{-\nu \eta},$ where $\nu$ 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.