Self-consistency over the charge density in dynamical mean-field theory: A linear muffin-tin implementation and some physical implications

We present a simple implementation of the dynamical mean-field-theory approach to the electronic structure of strongly correlated materials. This implementation achieves full self-consistency over the charge density, taking into account correlation-induced changes to the total charge density and effective Kohn-Sham Hamiltonian. A linear muffin-tin orbital basis set is used, and the charge density is computed from moments of the many-body momentum-distribution matrix. The calculation of the total energy is also considered, with a proper treatment of high-frequency tails of the Green’s function and self-energy. The method is illustrated on two materials with well-localized $4f$ electrons, insulating cerium sesquioxide ${\mathrm{Ce}}_2{\mathrm{O}}_3$ and the $\gamma$ phase of metallic cerium, using the Hubbard-I approximation to the dynamical mean-field self-energy. The momentum-integrated spectral function and momentum-resolved dispersion of the Hubbard bands are calculated, as well as the volume dependence of the total energy. We show that full self-consistency over the charge density, taking into account its modification by strong correlations, can be important for the computation of both thermodynamical and spectral properties, particularly in the case of the oxide material.

Figure 1

(Color online) DMFT combined with electronic structure calculations. Starting from a local electronic density $\rho \left(\mathbf{r}\right)$, the associated Kohn-Sham potential is calculated and the Kohn-Sham Hamiltonian is constructed. The Kohn-Sham Hamiltonian $H_{L{L}^{\prime }}^{\mathit{KS}}\left(\mathbf{k}\right)$ is expressed in a localized basis set (e.g., LMTOs). A double-counting term is subtracted to obtain the effective one-electron Hamiltonian $H_0\equiv H^{\mathit{KS}}-H^{\mathit{DC}}$. The local self-energy matrix for the subset of correlated orbitals is obtained through the iteration of the DMFT loop: a multiorbital impurity model for the correlated subset is solved [red (lighter gray) arrow], containing as an input the dynamical mean-field (or Weiss field ${\mathcal{G}}_0$). The self-energy ${\Sigma }_{ab}$ is combined with $H_0$ into the self-consistency condition Eqs. (13, 14) in order to update the Weiss field [blue (darker gray) arrow]. At the end of the DMFT loop, the components of the full, $\mathbf{k}$-dependent Green’s function in the local basis set can be calculated, yielding the momentum-distribution matrix $N{\left(\mathbf{k}\right)}_{L{L}^{\prime }}$ and the updated charge density $\rho \left(\mathbf{r}\right)$ as described in Sec. 2C. This updated charge density is used to compute the new Kohn-Sham potential. At the same time, the chemical potential must also be adjusted self-consistently with the total number of electrons. The full $\mathrm{LDA}+\mathrm{DMFT}$ cycle is carried out until convergence in the charge density, chemical potential, and total energy is reached.

Figure 2

(Color online) The LMTO ${\mathrm{Ce}}_2{\mathrm{O}}_3$ DOS calculated within DFT-LDA, the $\mathrm{LDA}+\mathrm{DMFT}$ with the fixed LDA charge density (DMFT-non-SC), and the $\mathrm{LDA}+\mathrm{DMFT}$ fully self-consistent over the charge density (DMFT-SC) schemes. The black (solid), green (dashed), blue (dotted), and red (dash-dotted) curves are the total, O, $\mathrm{Ce}\text{−}d$, and $\mathrm{Ce}\text{−}f$ DOS, respectively. The vertical dashed line indicates the position of the chemical potential, while the vertical dotted lines indicate positions of the lower Hubbard bands in DMFT-non-SC and DMFT-SC, respectively.

Figure 3

(Color online) The LMTO fully self-consistent $\mathrm{LDA}+\mathrm{DMFT}$ $\mathbf{k}$-resolved spectral function of ${\mathrm{Ce}}_2{\mathrm{O}}_3$ (Ref. [63]).

Figure 4

(Color online) The total energy vs lattice parameter dependence calculated within the LDA (solid line), $\mathrm{LDA}+U$ (dashed line), DMFT-non-SC (dotted line), and DMFT-SC (dash-dotted line) approaches with $U=6\mathrm{eV}$. The vertical dashed line indicates the experimental lattice parameter of ${\mathrm{Ce}}_2{\mathrm{O}}_3$ (Ref. 45).

Figure 5

(Color online) Spectral function for $\gamma$ cerium within $\mathrm{LDA}+\mathrm{DMFT}$ (Hubbard I) at $T=0\mathrm{K}$ with the self-consistent (SC) and non-self-consistent (non-SC) schemes. In the first case, two ways of treating the double-counting corrections are used: LDA number of electron and DMFT one ($N=1$ for the Hubbard-I solver). Experimental spectrum (photoemission and Bermsstrahlung isochromat spectroscopy) of $\gamma$-cerium[49, 50] are also represented.

Figure 6

(Color online) Total energy vs lattice parameter curves in LDA, $\mathrm{LDA}+U$, and $\mathrm{LDA}+\mathrm{DMFT}$ (Hubbard I) [self-consistent (SC) and non-self-consistent (non-SC)] for cerium, with $U=6\mathrm{eV}$. Shifts in energy between different curves are arbitrary. The experimental value for the lattice constant[61] is indicated by the arrow.