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We investigate the electronic structure of (Sr$_{1-x}$La$_x$)$_2$RhO$_4$using a combination of the density functional and dynamical mean-fieldtheories. Unlike the earlier local density approximation plus Hubbard $U$(LDA+U) studies, we find no sizable enhancement of the spin-orbit splitting dueto electronic correlations and show that such an enhancement is a spuriouseffect of the static mean-field approximation of the LDA+U method. The electrondoping suppresses the importance of electronic correlations, which is reflectedin quasi-particle bandwidth increasing with $x$. (Sr$_{1-x}$La$_x$)$_2$RhO$_4$can be classified as weakly correlated metal, which becomes an itinerantin-plane ferromagnet (but possibly A-type antiferromagnet) due to Stonerinstability around $x=0.2$.

Doping-dependent bandwidth renormalization and spin-orbit coupling in (Sr$_{1-x}$La$_x$)$_2$RhO$_4$