Splitting of the Kondo resonance in anisotropic magnetic impurities on surfaces

Using the numerical renormalization group method, we study the splitting of the Kondo resonance by a magnetic field applied in different directions in the Kondo model for anisotropic magnetic impurities. Several types of magnetic anisotropy are considered: the XXZ exchange coupling anisotropy J_perp J_z, the longitudinal magnetic anisotropy DS_z^2, and the transverse magnetic anisotropy E(S_x^2-S_y^2). In the spin-1/2 model with the XXZ exchange coupling anisotropy we find very small direction dependence in the magnitude of the splitting. In the spin-3/2 model with the easy-plane (D>0) anisotropy, we observe very unequal magnitudes with further differences between x and y directions in the presence of an additional transverse anisotropy. A simple and rather intuitive interpretation is that the splitting is larger in magnetically soft directions. The magnitude of the splitting is directly related to the energy differences between spin states and it is only weakly modified by some multiplicative factor due to Kondo screening. The results for the S=3/2 model are in good agreement with recent scanning tunneling spectroscopy studies of Co impurities adsorbed on CuN islands on Cu(100) surfaces [A. F. Otte et al., Nature Physics 4, 847 (2008)]