Anisotropy-driven topological quantum phase transition in magnetic impurities

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.