Dynamical mean-field theory within the full-potential methods: Electronic structure of ${\text{CeIrIn}}_5$, ${\text{CeCoIn}}_5$, and ${\text{CeRhIn}}_5$

We implemented the charge self-consistent combination of density-functional theory and dynamical mean-field theory (DMFT) in two full-potential methods, the augmented plane-wave and the linear muffin-tin orbital methods. We categorize the commonly used projection methods in terms of the causality of the resulting DMFT equations and the amount of partial spectral weight retained. The detailed flow of the dynamical mean-field algorithm is described, including the computation of response functions such as transport coefficients. We discuss the implementation of the impurity solvers based on hybridization expansion and an analytic continuation method for self-energy. We also derive the formalism for the bold continuous time quantum Monte Carlo method. We test our method on a classic problem in strongly correlated physics, the isostructural transition in Ce metal. We apply our method to the class of heavy-fermion materials ${\text{CeIrIn}}_5$, ${\text{CeCoIn}}_5$, and ${\text{CeRhIn}}_5$ and show that the $\text{Ce}4f$ electrons are more localized in ${\text{CeRhIn}}_5$ than in the other two, a result corroborated by experiment. We show that ${\text{CeIrIn}}_5$ is the most itinerant and has a very anisotropic hybridization, pointing mostly toward the out-of-plane In atoms. In ${\text{CeRhIn}}_5$ we stabilized the antiferromagnetic DMFT solution below 3 K, in close agreement with the experimental Neel temperature.

Figure 1

(Color online) Schematic of the projection-embedding step in the $\text{DFT}+\text{DMFT}$ algorithm. The full Green’s function of the solid $G\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)$ is truncated to its local counterpart $\widehat{P}G={\mathcal{G}}_{L{L}^{\prime }}$. The impurity solution delivers an effective local potential, which is embedded $\left(\widehat{E}\right)$ into the Dyson equation of the solid. The DMFT-SCC connects the two.

Figure 2

(Color online) All diagrams of the second, the third, and the fourth order in hybridization strength which contribute to the Luttinger-Ward functional. The pseudoparticle propagators run across the ring while the crossing lines stand for the hybridization $\Delta$. The full line represents spin-up and the dashed lines the spin-down hybridization. The black diagrams (both diagrams in ${\Phi }^{\left(2\right)}$, first four in ${\Phi }^{\left(3\right)}$ and first 24 in ${\Phi }^{\left(4\right)}$) give positive contribution to $\Phi$, and the red give negative contribution. Some diagrams seem to appear multiple times. This is because different pseudoparticles appear in the ring. Since we do not use different line for each pseudoparticle, some diagrams seem equivalent. However, it is very straightforward to deduce the pseudoparticle propagators knowing the type and the direction of the conduction electron propagators.

Figure 3

(Color online) (a) Comparison of the finite-order perturbation theory result with the exact impurity Green’s function for $1/T=100$ in units of $D$, the half bandwidth of the Bethe lattice. (b) The contributions to the impurity Green’s function up to the fifth order, plotted separately order by order. Both $\tau$ and $G$ are in units of $1/D$.

Figure 4

(Color online) The same as in Fig. 3 but we zoom-in the short-time behavior.

Figure 5

(Color online) The three panels show the contribution to the impurity Green’s function at third, fourth, and fifth order in perturbation theory. We show separately the contribution from the terms with positive $\Phi$ and the terms with negative $\Phi$.

Figure 6

The NCA Luttinger-Ward functional and the self-energies within NCA.

Figure 7

The Luttinger-Ward functional for the one-crossing approximation (OCA).

Figure 8

(Color online) Total and partial density of states of elemental cerium metal in both phases, $\alpha$ and $\gamma$ phase. We used OCA impurity solver.

Figure 9

(Color online) The same data as in Fig. 8, but obtained by continuous time quantum Monte Carlo solver, and analytical continuation method.

Figure 10

(Color online) The hybridization function (in units of electron volt) of the $j_z=5/2$ subshell within LDA and within DMFT in both phases.

Figure 11

(Color online) Optical conductivity of $\alpha \text{-Ce}$ and $\gamma \text{-Ce}$ within $\text{LDA}\left(\text{Wien}2\text{K}\right)+\text{DMFT}$ method. Note the shoulder in $\alpha \text{-Ce}$ conductivity, which is due to excitations across the two quasiparticle peaks ($4f:5/2$ and $4f:7/2$) clearly visible in Fig. 8, and also measured by experiment of Ref. 72.

Figure 12

(Color online) Crystal structure of $\text{Ce}X{\text{In}}_5$. Red, yellow, and gray spheres correspond to Ce, $X$, and In atoms, respectively.

Figure 13

(Color online) Total density of states (full lines) and partial $\text{Ce}4f$ density of states (dashed lines) for ${\text{CeCoIn}}_5$, ${\text{CeRhIn}}_5$, and ${\text{CeIrIn}}_5$ materials, in units of states per electron volt. The lower panel shows the low-energy part of the $\text{Ce}4f$ density of states for all three compounds. We used the OCA impurity solver. The frequency is in units of electron volt.

Figure 14

(Color online) The sketch of the itinerancy/localization of the three 115 compounds. In our view, the Ir compound is most itinerant while the Rh compound is most localized. The Co compound is not localized enough to develop magnetic order at low temperature while it is nor a good metal. It is thus conceivable that it would show tendency toward superconductivity. This phenomena is however beyond our current theoretical method—the single site DMFT calculation. We also show the bond distances between Ce and In atoms, and the angle between Ce and out-of-plane In atom. None of these parameters can explain the actual order of the compounds, hence the structure itself cannot explain the trend of localized to itinerant transition in these compounds.

Figure 15

(Color online) The $\text{Ce}4f$ Weiss field hybridization function $\left|{\Delta }^{″}\right|/\pi$ decomposed into crystal-field components of tetragonal field. All quantities are in units of electron volt. The upper plot shows the two-dimensional projection of the three relevant orbitals while the lower panel shows their 3D shapes (blue dots mark the position of the in-plane In atoms while the red dots the position of the out-of-plane In atoms). The radial extend of the orbitals in the polar plot of the upper panel is proportional to the value $\left|{\Delta }^{″}\right|/\pi$ at zero frequency. The full/dashed/dotted lines correspond to ${\text{CeIrIn}}_5/{\text{CeCoIn}}_5/{\text{CeRhIn}}_5$. While all three components of hybridization ${\Gamma }_7^{-}$, ${\Gamma }_7^{+}$, and ${\Gamma }_6$, are largest (smallest) in ${\text{CeIrIn}}_5$ $\left({\text{CeRhIn}}_5\right)$ compound, ${\Gamma }_7^{+}$ takes the largest value and also changes more than the other two components.

Figure 16

(Color online) The frequency dependence of the three most important hybridization functions ${\Delta }^{″}\left(\omega \right)/\pi$ (in electron volt) in all three 115 compounds: ${\text{CeCoIn}}_5$, ${\text{CeRhIn}}_5$, and ${\text{CeIrIn}}_5$. The frequency is in units of electron volt.

Figure 17

(Color online) Temperature dependence of the magnetic moment of the commensurate AFM Neel state in ${\text{CeRhIn}}_5$.

Figure 18

(Color online) Total and partial $\text{Ce}4f$ density of states for Rh-115 below and above the AFM transition. Above the Neel temperature, there is a signature of Kondo effect, which partially screens magnetic moment at elevated temperatures, even though the system develops the long-range order below 3 K. The quasiparticle peak is however much smaller than the same peak in ${\text{CeIrIn}}_5$ material. Once in the ordered state, the quasiparticle peak disappears.