Approximation schemes for the study of multi-band Gutzwiller wave functions

The minimum of the Gutzwiller energy functional depends on the number of parameters considered in the variational state. For a three-orbital Hubbard model we find that the frequently used diagonal Ansatz is very accurate in high-symmetry situations. For lower symmetry, induced by a crystal-field splitting or the spin-orbit coupling, the discrepancies in energy between the most general and a diagonal Gutzwiller Ansatz can be quite significant. We discuss approximate schemes that may be employed in multi-band cases where a minimization of the general Gutzwiller energy functional is too demanding numerically.