Charge self-consistency in density functional theory combined with dynamical mean field theory: $k$-space reoccupation and orbital order

We study the effects of charge self-consistency within the combination of density functional theory (DFT; wien2k) with dynamical mean field theory (DMFT; w2dynamics) in a basis of maximally localized Wannier orbitals. Using the example of two cuprates, we demonstrate that even if there is only a single Wannier orbital with fixed filling, a noteworthy charge redistribution can occur. This effect stems from a reoccupation of the Wannier orbital in k-space when going from the single, metallic DFT band to the split, insulating Hubbard bands of DMFT. We analyze another charge self-consistency effect beyond moving charge from one site to another: the correlation-enhanced orbital polarization in a freestanding layer of ${\mathrm{SrVO}}_3$.

Figure 1

Schematic representation of the DFT+DMFT approach. In a non-CSC or “one-shot” DFT+DMFT calculation, the step labeled “density update” is not performed; both the DFT and DMFT cycles are closed separately. By contrast, in CSC DFT+DMFT, neither DFT nor DMFT is iterated individually; instead, both of them are closed together. In terms of the schematic, a one-shot calculation follows the orange arrows, which close the individual cycles. A CSC calculation follows the blue arrows instead, which close the CSC cycle. (The green arrows are followed in either case.)

Figure 2

(a) Crystal structure of ${\mathrm{Sr}}_2{\mathrm{CuTeO}}_6$ with green, golden, and red balls representing Sr, Te, and O, respectively; the Cu sites are in the center of the blue octahedra that are elongates along the $c$-axis. (b) Spectral function, $A(\mathbf{k},w)$, as calculated by DFT+DMFT along a high-symmetry path through the Brillouin zone. The white curve depicts a well-separated Wannier band; the Fermi energy is set at zero.

Figure 3

Isosurface plot of the correlation-induced charge-density difference, $\mathrm{\Delta }\rho \left(\mathbf{r}\right)={\rho }_{\mathrm{DMFT}}\left(\mathbf{r}\right)-{\rho }_{\mathrm{DFT}}\left(\mathbf{r}\right)$, in ${\mathrm{Sr}}_2{\mathrm{CuTeO}}_6$. Yellow and cyan correspond to positive and negative $\mathrm{\Delta }\rho \left(\mathbf{r}\right)$'s at an isovalue of $4\times {10}^{-3}$ electrons/${\mathrm{bohr}}^3$; the dashed and dotted lines represent the unit cell and the (rotated) four O atoms around each Cu site, respectively.

Figure 4

(a) Crystal structure of ${\mathrm{HgBa}}_2{\mathrm{CuO}}_4$. Green, pink, blue, and red balls represent Ba, Hg, Cu, and O atoms, respectively. (b) k-dependence of $\mathrm{\Delta }{N}^{\mathcal{W}}$, i.e., the change of occupation of the Wannier orbitals in k-space for the $k_z=0$ plane in the Brillouin zone. The high-symmetry points $\mathrm{\Gamma }(0,0), X(\pi ,0)$, and $M(\pi ,\pi )$ are marked. The cyan curve depicts the DFT Fermi surface, which separates positive (yellow) and negative (blue) $\mathrm{\Delta }{N}^{\mathcal{W}}$ contributions.

Figure 5

(a) k-resolved DMFT spectral function $A(\mathbf{k},\omega )$ (color) in comparison with the DFT band structure (white line) for ${\mathrm{HgBa}}_2{\mathrm{CuO}}_4$. The dashed horizontal line is the Fermi energy. (b) Isosurface plot of the DMFT charge redistribution $\mathrm{\Delta }\rho \left(\mathbf{r}\right)$. Yellow and cyan correspond to positive and negative values of $\mathrm{\Delta }\rho \left(\mathbf{r}\right)$, at an isovalue of $1.5\times {10}^{-3}$ electrons/${\mathrm{bohr}}^3$).

Figure 6

DMFT (upper panel) and DFT (lower panel) density of states projected onto the V $t_{2g}$ orbitals. Dashed lines correspond to calculations without CSC; solid lines are with CSC. Inset: Top view of the isosurface plot of charge redistribution, $\mathrm{\Delta }\rho \left(\mathbf{r}\right)$. Yellow and cyan correspond to positive and negative contributions at an isovalue of $6\times {10}^{-3}$ electrons/${\mathrm{bohr}}^3$.

Figure 7

Top view of the isosurface plot of the difference in charge redistribution, $\mathrm{\Delta }{\rho }^{*}\left(\mathbf{r}\right)$, over the iterations. Yellow and cyan correspond to positive and negative contributions at an isovalue of $6\times {10}^{-3}$ electrons/${\mathrm{bohr}}^3$.