Dynamical mean-field theory using Wannier functions: A flexible route to electronic structure calculations of strongly correlated materials

A versatile method for combining density functional theory in the local density approximation with dynamical mean-field theory (DMFT) is presented. Starting from a general basis-independent formulation, we use Wannier functions as an interface between the two theories. These functions are used for the physical purpose of identifying the correlated orbitals in a specific material, and also for the more technical purpose of interfacing DMFT with different kinds of band-structure methods (with three different techniques being used in the present work). We explore and compare two distinct Wannier schemes, namely the maximally localized Wannier function and the $N\text{th}$ order muffin-tin-orbital methods. Two correlated materials with different degrees of structural and electronic complexity, $\mathrm{Sr}\mathrm{V}{\mathrm{O}}_3$ and $\mathrm{Ba}\mathrm{V}{\mathrm{S}}_3$, are investigated as case studies. $\mathrm{Sr}\mathrm{V}{\mathrm{O}}_3$ belongs to the canonical class of correlated transition-metal oxides, and is chosen here as a test case in view of its simple structure and physical properties. In contrast, the sulfide $\mathrm{Ba}\mathrm{V}{\mathrm{S}}_3$ is known for its rich and complex physics, associated with strong correlation effects and low-dimensional characteristics. Insights into the physics associated with the metal-insulator transition of this compound are provided, particularly regarding correlation-induced modifications of its Fermi surface. Additionally, the necessary formalism for implementing self-consistency over the electronic charge density in a Wannier basis is discussed.

Figure 1

(Color online) Complete self-consistency loop for $\mathrm{LDA}+\mathrm{DMFT}$. The charge density $\rho$ determines the KS potential $V_{\mathrm{KS}}$, from which KS eigenvalues ${\epsilon }_{\mathbf{k}\nu }$ and eigenfunctions ${\psi }_{\mathbf{k}\nu }$ follow. The KS Green’s function is then constructed and passed on to the DMFT cycle (in practice, the KS Hamiltonian ${\mathbf{H}}_{\mathrm{KS}}$ is constructed in the basis set used to implement the method, and transferred to the DMFT cycle). The DMFT loop consists in (i) solving the effective impurity problem for the impurity Green’s function, hence obtaining an impurity self-energy, (ii) combining the self-energy correction with the KS Green’s function in order to obtain the local Green’s function ${\mathbf{G}}_{\mathrm{loc}}$ projected in the correlated subset, and (iii) obtaining an updated Weiss mean field. An initial guess for the Weiss dynamical mean field must be made at the beginning of the DMFT loop, e.g., by choosing ${\mathcal{G}}_0^{\mathrm{init}}={\widehat{P}}^{\left(\mathcal{C}\right)}{\widehat{G}}_{\mathrm{KS}}{\widehat{P}}^{\left(\mathcal{C}\right)}$. Is the DMFT loop converged, the chemical potential is updated in order to ensure global charge neutrality, and the new charge density (including many-body effects) is constructed (described in Appendix. ). This new density determines a new KS potential. Note that in addition one may want to update the set $\left\{\mid {\chi }_m⟩\right\}$ when preparing for the next DMFT loop (cf. Appendix ). The whole process must be iterated until the charge density, the impurity self-energy, and the chemical potential are converged. In practice, good convergence of the DMFT loop is reached before a new $\rho$ is calculated. Note that in the present paper using a Wannier implementation, the global self-consistency on the charge density is not implemented in practice. Thus the self-consistent LDA Hamiltonian ${\mathbf{H}}_{\mathrm{KS}}$ enters the DMFT loop, which is iterated until convergence of the self-energy.

Figure 2

(Color online) (a) LDA data for $\mathrm{Sr}\mathrm{V}{\mathrm{O}}_3$ calculated with the MBPP code. (a) Band structure. (b) DOS. For the local $\mathrm{V}\left(3d\right)∕\mathrm{O}\left(2p\right)\text{−}\mathrm{DOS}$ the cutoff radius was one-half the nearest-neighbor distance, respectively.

Figure 3

(Color online) $t_{2g}$ bands for $\mathrm{Sr}\mathrm{V}{\mathrm{O}}_3$ using different schemes to compute the $t_{2g}$ Wannier functions (and the underlying LDA band structure or potential). Dark, MLWF(MBPP); dashed-red (dashed-gray), MLWF(FLAPW); and green (light gray), NMTO(LMTO-ASA). The $t_{2g}$ bandwidth is marginally larger in FLAPW, leading to small differences.

Figure 4

(Color online) $t_{2g}$-like MLWF $w_{xy}$ for $\mathrm{Sr}\mathrm{V}{\mathrm{O}}_3$ derived from the MBPP code. First row: $\mathrm{Sr}\mathrm{V}{\mathrm{O}}_3$ structure with Sr (large blue/dark), V (red/gray), and O (small yellow/light gray) and perspective view on $w_{xy}$. Second row: $w_{xy}$ viewed along the $c$ axis and along the $a$ axis. The contour value $w_{xy}^{\left(0\right)}$ was chosen as $0.05{\left(\mathrm{a.u.}\right)}^{-3∕2}$.

Figure 5

(Color online) Contour-lines plot of the $t_{2g}$-like MLWF $w_{xy}$ for $\mathrm{Sr}\mathrm{V}{\mathrm{O}}_3$. Distinct contour values [in ${\left(\mathrm{a.u.}\right)}^{-3∕2}$] are given in the plot.

Figure 6

(Color online) (a) $t_{2g}$-like $w_{xy}$ WF. (a) From V to V along [110], and (b) from O to O along [010].

Figure 7

(Color online) Distinct WFs for $\mathrm{Sr}\mathrm{V}{\mathrm{O}}_3$ obtained from the MLWF construction using the MBPP code. First row: $\mathrm{O}\left(p_x\right)$, $\mathrm{O}\left(p_y\right)$, and $\mathrm{O}\left(p_z\right)$ for a chosen oxygen site. Second row: $\mathrm{V}(t_{2g},xy)$ as well as $\mathrm{V}(e_g,3z^2-r^2)$ and $\mathrm{V}(e_g,x^2-y^2)$. The contour value for each of the MLWFs was chosen as $0.05{\left(\mathrm{a.u.}\right)}^{-3∕2}$.

Figure 8

(Color online) Spectral function for $\mathrm{Sr}\mathrm{V}{\mathrm{O}}_3$ for different values of $U$ resulting from $\mathrm{LDA}+\mathrm{DMFT}$ in the MBPP implementation. Inset: comparison of Green’s functions from MBPP, FLAPW, and NMTO implementations of $\mathrm{LDA}+\mathrm{DMFT}$ for $U=4\mathrm{eV}$. In all calculations $J=0.65\mathrm{eV}$ was used.

Figure 9

$\mathrm{Sr}\mathrm{V}{\mathrm{O}}_3$ spectral function convoluted with a Gaussian experimental resolution (assumed to be $0.15\mathrm{eV}$) and with the Fermi function at $\beta =30{\mathrm{eV}}^{-1}$.

Figure 10

(Color online) $\mathrm{Ba}\mathrm{V}{\mathrm{S}}_3$ in the $Cmc{2}_1$ structure. The V ions are shown as smaller (red/gray) spheres, the Ba ions as larger (blue/dark) spheres.

Figure 11

(Color online) LDA data for $\mathrm{Ba}\mathrm{V}{\mathrm{S}}_3$ after the MBPP method. (a) Band structure and (b) DOS. For the local $\mathrm{V}\left(3d\right)∕\mathrm{S}\left(3p\right)$-DOS the cutoff radius was one-half the minimum nearest-neighbor distance, respectively. The inset in (b) shows the symmetry-adapted local $\mathrm{V}\left(3d\right)$-DOS close to the Fermi level.

Figure 12

(Color online) (a) $t_{2g}$ fat-band resolved band structure of $\mathrm{Ba}\mathrm{V}{\mathrm{S}}_3$. The color code is as follows: $A_{1g}$ (blue/dark), $E_{g1}$ (red/gray), and $E_{g2}$ (green/light gray). (b) Downfolded $t_{2g}$ Wannier bands (light blue/light gray) for $\mathrm{Ba}\mathrm{V}{\mathrm{S}}_3$ obtained from the MLWF construction using the MBPP code.

Figure 13

(Color online) $t_{2g}$ Wannier orbitals for $\mathrm{Ba}\mathrm{V}{\mathrm{S}}_3$ forming the maximally localized basis. The columns from left to right show the $A_{1g}$, $E_{g1}$, and $E_{g2}$ orbitals, while the second row displays the orbitals viewed along the $c$ axis. There the $a$ axis is vertically oriented, while the $b$ axis horizontally. The contour value for each of the orbitals was chosen as $0.045{\left(\mathrm{a.u.}\right)}^{-3∕2}$.

Figure 14

(Color online) $t_{2g}$ Wannier orbitals for $\mathrm{Ba}\mathrm{V}{\mathrm{S}}_3$ forming the crystal-field basis. The order of the columns and row same as in Fig. 13. Note that the $E_{g2}$ orbital remains invariant under this transformation. The contour value for each of the orbitals was equally chosen as $0.045{\left(\mathrm{a.u.}\right)}^{-3∕2}$.

Figure 15

(Color online) $t_{2g}$ Wannier fat bands for $\mathrm{Ba}\mathrm{V}{\mathrm{S}}_3$, (a) with respect to the maximally localized basis, and (b) with respect to the crystal-field basis (see text). Color coding: $A_{1g}$-like WF (blue/dark), $E_{g1}$-like WF (red/gray).

Figure 16

(Color online) $t_{2g}$ Wannier DOS for $\mathrm{Ba}\mathrm{V}{\mathrm{S}}_3$. (a) In the maximally localized basis, and (b) in the crystal-field basis.

Figure 17

(Color online) (a) Wannier DOS for $\mathrm{Ba}\mathrm{V}{\mathrm{S}}_3$ in the crystal-field basis. (b) Corresponding $\mathbf{k}$-integrated spectral function for $\mathrm{Ba}\mathrm{V}{\mathrm{S}}_3$ within $\mathrm{LDA}+\mathrm{DMFT}$ for $\beta =30{\mathrm{eV}}^{-1}$.

Figure 18

(Color online) $\mathbf{k}$-resolved correlation effect for $\mathrm{Ba}\mathrm{V}{\mathrm{S}}_3$. QP band structure from $\mathrm{LDA}+\mathrm{DMFT}$ (dark green dashed/dark dashed) in comparision with the Wannier band structure based on LDA (cyan/light gray) close to the Fermi level.

Figure 19

(Color online) (a) BZ of the $Cmc{2}_1$ structure, including the high-symmetry lines used to plot the band structures. The $a^{*}$ axis runs towards $M$, the $b^{*}$ axis towards $Y$. (b) LDA Fermi surface, and (c) QP Fermi surface from $\mathrm{LDA}+\mathrm{DMFT}$. The second row of the columns show the BZ along $a^{*}$, the third row along $b^{*}$. In the latter column the experimental CDW nesting vector ${\mathbf{q}}_c$ is displayed.

Figure 20

(Color online) (a) LDA band structure (dark) and according Wannier bands (cyan/light gray) along a closed path in the 1. BZ including the line connecting “$M∕2$” and “$A∕2$” (see text). (b) Pure (cyan/light gray) and renormalized (dark green dashed/dark dashed) Wannier bands.