  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.
