  A few years ago, a topological quantum phase transition (TQPT) has been found
in Anderson and Kondo 2-channel spin-1 impurity models that include a hard-axis
anisotropy term $DS_z^2$ with $D > 0$. The most remarkable manifestation of the
TQPT is a jump in the spectral density of localized electrons, at the Fermi
level, from very high to very low values as $D$ is increased. If the two
conduction channels are equivalent, the transition takes place at the critical
anisotropy $D_c \sim 2.5\; T_K$, where $T_K$ is the Kondo temperature for
$D=0$. This jump might be important to develop a molecular transistor. The jump
is due to a corresponding one in the Luttinger integral, which has a
topological non-trivial value $\pi/2$ for $D > D_c$. Here, we review the main
results for the spectral density and highlight the significance of the theory
for the interpretation of measurements conducted on magnetic atoms or molecules
on metallic surfaces. In these experiments, where $D$ is held constant, the
energy scale $T_K$ is manipulated by some parameters. The resulting variation
gives rise to a differential conductance $dI/dV$, measured by
scanning-tunneling spectroscopy, which is consistent with a TQPT at an
intermediate value of $T_K$. We also show that the theory can be extended to
integer spin $S>1$ and two-impurity systems. This is also probably true for
half-integer spin and non-equivalent channels in some cases.
