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Low-energy physics for an iron phthalocyanine molecule on Au(111)

The system of an iron phthalocyanine molecule on the Au(111) surface, has been studied recently due to its peculiar properties. In particular, several surprising results of scanning tunneling spectroscopy changing the position of the molecule and applying magnetic field can be explained by the {\it non-Landau} Fermi liquid state of a 2-channel spin-1 Kondo model with anisotropy. The localized orbitals near the Fermi level are three, one of symmetry $z^2$ and two (nearly) degenerate $π$ orbitals of symmetry $xz$ and $yz$. Previous studies using the numerical renormalization group neglected one of these orbitals to render the problem tractable. Here we investigate, using a slave-boson mean-field approximation, if the splitting $S$ between $π$ orbitals caused by spin-orbit coupling (SOC) justifies this approximation. We obtain an abrupt transition from a 3-band regime to a 2-band one at a value of $S$ which is about 1/3 of the atomic SOC for Fe, justifying the 2-band model for the system.