Quantum Fidelity of the Aubry-Andr\'e Model and the Exponential Orthogonality Catastrophe

We consider the orthogonality catastrophe in the (extended) Aubry-Andr\'e (AA)-Model, by calculating the overlap $F$ between the ground state of the Fermi liquid in that quasi-crystalline model and the one of the same system with an added potential impurity, as function of the size of that impurity. Recently, the typical fidelity $F_{\rm typ}$ was found in quantum critical phases to decay exponentially with system size $L$ as $F \sim \exp(-c L^{z \eta})$\cite{Kettemann2016} as found in an analytical derivation due to critical correlations. For the critical AA model $\eta = 1/2$ is the power of multifractal intensity correlations, and $z$ the dynamical exponent due to the fractal structure of the density of states which is numerically found to be $z \gg 1$. Surprisingly, however, we find for a weak single site impurity that the fidelity decays with a power law, in the critical phase. Even though it is found to be smaller and decays faster than in the metallic phase, it does not decay exponentially as predicted. We find an exponential AOC however in the insulator phase for which we give a statistical explanation, a mechanism which is profoundly different from the AOC in metals, where it is the coupling to a continuum of states which yields there the power law suppression of the fidelity. By reexamination of the analytical derivation we identify nonperturbative corrections due to the impurity potential and multipoint correlations among wave functions as possible causes for the absence of the exponential AOC in the critical phase. For an extended impurity, however, we find indications of an exponential AOC at the quantum critical point of the AA model and at the mobility edge of the extended AA model and suggest an explanation for this finding.