Landaus Fermi liquid theory is a cornerstone of quantum many body physics. At its heart is the adiabatic connection between the elementary excitations of an interacting fermion system and those of the same system with the interactions turned off. Recently, this tenet has been challenged with the finding of a non-Landau Fermi liquid, that is a strongly interacting Fermi liquid that cannot be adiabatically connected to a non-interacting system. In particular, a spin-1 two-channel Kondo impurity with single-ion magnetic anisotropy $D$ has a topological quantum phase transition at a critical value $D_c$: for $D < D_c$ the system behaves as an ordinary Fermi liquid with a large Fermi level spectral weight, while above $D_c$ the system is a non-Landau Fermi liquid with a pseudogap at the Fermi level, topologically characterized by a non-trivial Friedel sum rule with non-zero Luttinger integrals. Here, we develop a non-trivial extension of this new Fermi liquid theory to general multi-orbital problems with finite magnetic field and we reinterpret in a unified and consistent fashion several experimental studies of iron phthalocyanine molecules on Au(111) metal substrate that were previously described in disconnected and conflicting ways. The differential conductance measured using a scanning tunneling microscope (STM) shows a zero-bias dip that widens when the molecule is lifted from the surface and is transformed continuously into a peak under an applied magnetic field. Numerically solving a spin-1 impurity model with single-ion anisotropy for realistic parameter values, we robustly reproduce all these central features, allowing us to conclude that iron phthalocyanine molecules on Au(111) constitute the first confirmed experimental realization of a non-Landau Fermi liquid.
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