${\mathrm{BaOsO}}_3$: A Hund's metal in the presence of strong spin-orbit coupling

We investigate the $5d$ transition metal oxide ${\mathrm{BaOsO}}_3$ within a combination of density functional theory and dynamical mean-field theory, using a matrix-product-state impurity solver. ${\mathrm{BaOsO}}_3$ has four electrons in the $t_{2g}$ shell akin to ruthenates but stronger spin-orbit coupling (SOC) and is thus expected to reveal an interplay of Hund's metal behavior with SOC. We explore the paramagnetic phase diagram as a function of SOC and Hubbard interaction strengths, identifying metallic, band (van Vleck) insulating, and Mott insulating regions. At the physical values of the two couplings, we find that ${\mathrm{BaOsO}}_3$ is still situated inside the metallic region and has a moderate quasiparticle renormalization $m^{*}/m\approx 2$, consistent with specific heat measurements. SOC leads to a splitting of a van Hove singularity close to the Fermi energy and a subsequent reduction of electronic correlations (found in the vanishing SOC case), but the SOC strength is insufficient to push the material into an insulating van Vleck regime. In spite of the strong effect of SOC, ${\mathrm{BaOsO}}_3$ can be best pictured as a moderately correlated Hund's metal.

Figure 1

Upper panels: Density functional theory (DFT) band structure (black dots) and Wannier fit (colored) of ${\mathrm{BaOsO}}_3$ (a) without and (b) with spin-orbit coupling (SOC) along a high-symmetry path through the Brillouin zone. Lower panels: Total DFT density of states (DOS; black) and partial contributions (colored) (c) without and (d) with SOC. The DOS of the constructed $t_{2g}$-like Wannier orbitals is shown with thicker lines. For the case with SOC, we have added a local SOC term to the Wannier Hamiltonian with a coupling strength of $\lambda =0.3\mathrm{eV}$.

Figure 2

Momentum-resolved spectral function for $U=2.55\mathrm{eV}$ and $J_H=0.27\mathrm{eV}$ along a high-symmetry path through the Brillouin zone (a) without and (c) with spin-orbit coupling (SOC) together with (b) and (d) the respective momentum-integrated spectral functions. The SOC strength is denoted as $\lambda$. The dashed lines in panels (a) and (c) correspond to the bare bands of the Wannier model. The inset in panel (b) shows a low-energy zoom of the spectral function.

Figure 3

Mass enhancement $Z^{-1}=m^{*}/m$ as a function of the absolute value of $\lambda$ for $t_{2g}$ fillings of $n=2$ (orange lines) and $n=4$ (purple lines) electrons with interaction parameters $U=2.55\mathrm{eV}$ and $J_H=0.27\mathrm{eV}$. To obtain a band-insulating state also in the case $n=2$ (at large $\lambda$), we set ${\lambda }_{n=2}=-{\lambda }_{n=4}$. The dotted and dashed lines correspond to mass enhancement of the $j=\frac{3}{2}$ and $j=\frac{1}{2}$ bands, respectively, while the thick solid line shows the mass enhancement in the cubic basis. The dashed vertical line indicates the physical value of $\lambda$.

Figure 4

(a) and (b) Diagonal elements of the mass enhancement $Z_c^{-1}=m^{*}/m$ in the cubic basis and (c) and (d) polarization $\mathrm{\Delta }n:=n_{j=3/2}-n_{j=1/2}$ for different values of $U$. The occupation numbers used for the polarization were taken directly from the ground state. Open and full circles display metallic and insulating states, respectively.

Figure 5

(a) Phase diagram in the $U\text{−}\lambda$ plane at constant $J_H=0.27\mathrm{eV}$. We find two different phases in the parameter regime considered, namely, a metallic phase (open circles) and an insulting phase (squares). The character of the insulating states (Mott-like or bandlike) is encoded in the symbol colors (see main text). The colors are scaled linearly between $\chi =0.5$ and 0eV, while for $\chi \ge 0.5\mathrm{eV}$, the color is set to the one for $\chi =0.5$. The dashed lines indicate the physical values of $\lambda$ (vertical line) and $U$ (horizontal line). (b) Real and (c) imaginary part of the self-energy for $J_H=0.27\mathrm{eV}$ and selected values of $U$ and $\lambda$ (in eV). The solid and dashed lines correspond to the $j=\frac{3}{2}$ and $j=\frac{1}{2}$ bands, respectively. To show all self-energies on a comparable scale, we shifted the real parts by the chemical potential and rescaled both the imaginary and real part by $1/U$.

Figure 6

Local spectral function of a model calculation on a semicircular density of states (DOS) with $U=2.55\mathrm{eV}, J_H=0.27\mathrm{eV}, \lambda =0$, and half bandwidth $D=1\mathrm{eV}$.

Figure 7

Noninteracting density of states (DOS) at a filling of (a) four and (b) two electrons. For the system with two electrons, we also flipped the sign of $\lambda$ (see text).