  The Anderson model for a magnetic impurity in a one-dimensional quasicrystal
is studied using the numerical renormalization group (NRG). The main focus is
elucidating the physics at the critical point of the Aubry-Andre (AA)
Hamiltonian, which exhibits a fractal spectrum with multifractal wave
functions, leading to an AA Anderson (AAA) impurity model with an
energy-dependent hybridization function defined through the multifractal local
density of states at the impurity site. We first study a class of Anderson
impurity models with uniform fractal hybridization functions that the NRG can
solve to arbitrarily low temperatures. Below a Kondo scale $T_K$, these models
approach a fractal strong-coupling fixed point where impurity thermodynamic
properties oscillate with $\log_b T$ about negative average values determined
by the fractal dimension of the spectrum. The fractal dimension also enters
into a power-law dependence of $T_K$ on the Kondo exchange coupling $J_K$. To
treat the AAA model, we combine the NRG with the kernel polynomial method (KPM)
to form an efficient approach that can treat hosts without translational
symmetry down to a temperature scale set by the KPM expansion order. The
aforementioned fractal strong-coupling fixed point is reached by the critical
AAA model in a simplified treatment that neglects the wave-function
contribution to the hybridization. The temperature-averaged properties are
those expected for the numerically determined fractal dimension of $0.5$. At
the AA critical point, impurity thermodynamic properties become negative and
oscillatory. Under sample-averaging, the mean and median Kondo temperatures
exhibit power-law dependences on $J_K$ with exponents characteristic of
different fractal dimensions. We attribute these signatures to the impurity
probing a distribution of fractal strong-coupling fixed points with decreasing
temperature.
