The different low energy effective Hamiltonians for cuprate high-Tc superconductors  can be considered as steps in a Wilsonian renormalization procedure from large- to small energy windows around the Fermi level. High energy models include bare interactions and all spin-, orbital-, and lattice degrees of freedom explicitly. However, at low energies many researchers consider the single band Hubbard model with a screened interaction to be adequate for capturing ground state and “near-ground state properties” like the magnetic susceptibility. Occasionally, this viewpoint is still challenged. Recent measurements  of the doping dependence in the NMR Knight shift were claimed to be in contradiction with an effective single band description calling for explicit inclusion of oxygen degrees of freedom.
In our ongoing work we revisit the single- and three band Hubbard models for cuprates on the one- and two particle level within dynamical mean-field theory. We study the temperature/doping phase diagram using material-realistic parameters  and account also for possibly incommensurate order by solving the Bethe-Salpeter equation. At lower doping levels our preliminary results seem to support the idea of a single band description for single and two particle observables in small energy windows at low temperatures (at least qualitatively). At the same time, however, excitations beyond a couple of 100meV as well as the high doping regime seem to be problematic for a static single band picture.
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