Numerical renormalization group calculations of the magnetization of Kondo impurities with and without uniaxial anisotropy

We study a Kondo impurity model with additional uniaxial anisotropy D in a nonzero magnetic field B using the numerical renormalization group (NRG). The ratio g(e)/g(S) of electron and impurity g factor is regarded as a free parameter and, in particular, the special cases of a "local" (g(e) = 0) and "bulk" (g(e) = g(S)) field are considered. For a bulk field, the relationship between the impurity magnetization M and the impurity contribution to the magnetization M-imp is investigated and it is shown that M and M-imp are proportional to each other for fixed coupling strength. Furthermore, we find that the g-factor ratio effectively rescales the magnetic field argument of the zero-temperature impurity magnetization. In case of an impurity with D = 0 and g(e) = g(S), it is demonstrated that at zero temperature M(B), unlike M-imp(B), does not display universal behavior. With additional "easy-axis" anisotropy, the impurity magnetization is "stabilized" at a D-dependent value for k(B)T << gs beta B-B << vertical bar D vertical bar and, for nonzero temperature, is well described by a shifted and rescaled Brillouin function on energy scales that are small compared to vertical bar D vertical bar. In the case of "hard-axis" anisotropy, the magnetization curves can feature steps which are due to field-induced pseudo-spin-1/2 Kondo effects. For large hard-axis anisotropy and a local field, these screening effects are described by an exchange-anisotropic spin-1/2 Kondo model with an additional scattering term that is spin dependent (in contrast to ordinary potential scattering). In accordance with the observed step widths, this effective model predicts a decrease of the Kondo temperature with every further step that occurs upon increasing the field. Our study is motivated by the question as to how the magnetic properties of a deposited magnetic molecule are modified by the interaction with a nonmagnetic metallic surface.