Density matrix renormalization group study of a three-orbital Hubbard model with spin-orbit coupling in one dimension

Using the Density Matrix Renormalization Group technique we study the effect of spin-orbit coupling on a three-orbital Hubbard model in the $(t_{2g}{)}^4$ sector and in one dimension. Fixing the Hund coupling to a robust value compatible with some multiorbital materials, we present the phase diagram varying the Hubbard $U$ and spin-orbit coupling $\lambda$, at zero temperature. Our results are shown to be qualitatively similar to those recently reported using the Dynamical Mean Field Theory in higher dimensions, providing a robust basis to approximate many-body techniques. Among many results, we observe an interesting transition from an orbital-selective Mott phase to an excitonic insulator with increasing $\lambda$ at intermediate $U$. In the strong $U$ coupling limit, we find a non-magnetic insulator with an effective angular momentum $⟨({\mathbf{\text{J}}}^{eff}{)}^2⟩\mathrm{\ne }0$ near the excitonic phase, smoothly connected to the $⟨({\mathbf{\text{J}}}^{eff}{)}^2⟩=0$ regime. We also provide a list of quasi-one dimensional materials where the physics discussed in this publication could be realized.