A Critical Analysis of the Mean-Field Approximation for the Calculation of the Magnetic Moment in the Friedel-Anderson Impurity Model

It is shown that the calculation of the magnetic moment of a Friedel-Anderson impurity in mean-field theory is unreliable. A class of approximate solutions, which contains the mean-field solution as an element, is expressed in rotated Hilbert space and optimized. The optimal state has considerably lower energy than the mean field solution and requires almost twice the Coulomb exchange U to become magnetic. Since most moment calculations of magnetic impurities, for example the spin-density-functional theory, use the mean-field approximation the resulting magnetic moments have to be critically reexamened. PACS: 75.20.Hr, The properties of magnetic impurities in a metal is one of the most intensively studied problems in solid state physics. Although some of the experimental anomalies were already discovered in the 1930’s, it is still a subject 1 of great interest. The work of Friedel [1] and Anderson [2] laid the foundation to understand why some transition metal impurities form a magnetic local moment, while others don’t. They considered a host with an s-band in which a transition metal atom is dissolved. The s-electrons can hop onto the d-impurity via the hopping matrix element Vsd. The ten-fold degeneracy of a real d-impurity is simplified and reduced to a two-fold degeneracy for spin up and spin down. If both states are occupied they repel each other due to the Coulomb exchange energy. This yields the Friedel-Anderson Hamiltonian HFA = ∑