Causal versus local $GW+\mathrm{EDMFT}$ scheme and application to the triangular-lattice extended Hubbard model

Using the triangular-lattice extended Hubbard model as a test system, we compare $GW+\mathrm{EDMFT}$ results for the recently proposed self-consistency scheme with causal auxiliary fields to those obtained from the standard implementation which identifies the impurity Green's functions with the corresponding local lattice Green's functions. Both for short- and long-ranged interactions we find similar results, but the causal scheme yields slightly stronger correlation effects at half-filling. We use the two implementations of $GW+\mathrm{EDMFT}$ to compute spectral functions and dynamically screened interactions in the parameter regime relevant for $1T\text{−}{\mathrm{TaS}}_2$. These simulations allow us to clarify if the sample used in a recent photoemission study [M. Ligges et al., Phys. Rev. Lett. 120, 166401 (2018)] was half-filled or hole doped.

Figure 1

Comparison of the interaction $V\left(q\right)$ in the first Brillouin zone for three different potentials, with the same NN interaction $V_1=1$ and lattice spacing $a=1$. ($\mathrm{\Gamma }, M, K$ denote the center, the midpoint of the edge, and the corner of the hexagonal first Brillouin zone, respectively.)

Figure 2

Left: the extended Hubbard model on the triangular lattice, with NN hopping $t$, onsite interaction $U$, and nonlocal interactions $V$. To reduce the number of lines, only $V_1, V_2$ and $V_4$ are indicated in the figure. Right: EDMFT maps the lattice model to an impurity model defined by a hybridization function $\mathrm{\Delta }$ (or fermionic Weiss field ${\mathcal{G}}^{-1}=i{\omega }_n+\tilde{\mu }-\mathrm{\Delta }$) and a bosonic field (retarded interaction) $\mathcal{U}$.

Figure 3

Flowchart of the $GW+\mathrm{EDMFT}$ schemes. The red parts in brackets are added in the causal version of $GW+\mathrm{EDMFT}$, while in the absence of these terms, the scheme corresponds to the standard local version of $GW+\mathrm{EDMFT}$.

Figure 4

$GW +\mathrm{DMFT}$ phase diagrams of the half-filled single-band extended Hubbard model on the triangular lattice. Top: phase diagram with only local and NN interactions, for an initial metallic (triangles) and Mott insulating (stars) solution in the self-consistency loop. The black marks indicate the $U$ and $V$ parameters which will be further studied below. The black line corresponds to the fixed ratio $V/U=0.19$. Bottom: phase diagram for the model with long-range interaction treated by Ewald summation (triangles, starting from a Mott insulating solution), and for comparison the result for the NN interaction case (stars). The red and blue arrows indicate the shift of the phase boundary as a result of the longer-ranged interaction. (Local self-consistency scheme, $\beta =500, t=-0.02$.)

Figure 5

From left to right column: real part of the impurity interaction $\mathrm{Re}\mathcal{U}\left(i{\nu }_n\right)$, imaginary part of the hybridization function $\mathrm{Im}\mathrm{\Delta }\left(i{\omega }_n\right)$, and spectral function $-\text{Im}\mathrm{\Delta }\left(\omega \right)$ obtained by Padé analytical continuation. The interaction parameters (triangles in the top panel of Fig. 4) are chosen to illustrate the effect of an increase in the nonlocal interaction in the metallic phase. The data are for the model with NN nonlocal interactions, treated with the local and causal self-consistency scheme.

Figure 6

From left to right column: real part of the impurity interaction $\mathrm{Re}\mathcal{U}\left(i{\nu }_n\right)$, imaginary part of hybridization function $\mathrm{Im}\mathrm{\Delta }\left(i{\omega }_n\right)$, and spectral function $-\text{Im}\mathrm{\Delta }\left(\omega \right)$ for increasing onsite interaction $U$ and fixed $V/U$ ratio (crosses in the top panel of Fig. 4) with NN nonlocal interactions.

Figure 7

From left to right column: fully screened local interaction $\mathrm{Re}{W}_{\text{loc}}\left(i{\nu }_n\right)$, local self-energy $\mathrm{Im}{\mathrm{\Sigma }}_{\text{loc}}\left(i\omega \right)$, and local fermionic spectral function $A_{\text{loc}}\left(\omega \right)$ obtained by the local and causal $GW+\mathrm{EDMFT}$ scheme for the model with fixed $U=0.2$ and increasing NN intersite interactions.

Figure 8

From left to right column: fully screened local interaction $\mathrm{Re}{W}_{\text{loc}}\left(i{\nu }_n\right)$, local self-energy $\mathrm{Im}{\mathrm{\Sigma }}_{\text{loc}}\left(i\omega \right)$, and local fermionic spectral function $A_{\text{loc}}\left(\omega \right)$ obtained for increasing onsite interaction $U$ and fixed $V/U$ ratio. In the right top panel, the inset shows the comparison between the local lattice and impurity Greens' function in the causal scheme. The impurity Green's function is slightly more metallic than the local lattice Green's function. The downturn at low Matsubara frequencies in the self-energy of the Mott insulating solution (bottom panels) is explained by the particle-hole asymmetry.

Figure 9

The C-CDW pattern in the low-temperature phase of $1T\text{−}{\mathrm{TaS}}_2$, sketched based on the experimental results in Refs. [17, 31, 35]. Purple dots are the Ta atoms, distorted in the direction of the purple arrows, and the blue lines indicate the David-star-like supercells, each including 13 lattice sites. The blue disk shows the area with high charge density seen by STM. $R=\sqrt{13}$ is the distance between two neighboring supercells and $r$ is the size of the disk.

Figure 10

Local self-energy ${\mathrm{\Sigma }}_{\text{loc}}$, fermionic spectral function $A_{\text{loc}}\left(\omega \right)$, and distribution function $A_{\text{loc}}\left(\omega \right)F\left(\omega \right)$ (from left to right). The doping is increased row-wise from the undoped configuration to $5\%$ and $10\%$ hole concentration. The interaction parameters are $U=0.42$ and $V=0.08$.

Figure 11

The local fully screened interaction $W_{\text{loc}}$ in the Matsubara-frequency domain and in the real-frequency domain after Padé analytical continuation. In the top panels, the model is half-filled ($n=1$). The middle panels are for $5\%$ hole doping ($n=0.95$), and the bottom panels for $10\%$ hole doping ($n=0.9$). The interaction parameters are $U=0.42, V=0.08$.

Figure 12

Spectral function $A_{\text{loc}}\left(\omega \right)$ and occupation function $A_{\text{loc}}\left(\omega \right)F\left(\omega \right)$ for the half-filled, 5% hole-doped, and 10% hole-doped system, respectively, for the indicated values of temperature ($U=0.42, V=0.08$).

Figure 13

Occupation function $A_{\text{loc}}\left(\omega \right)F\left(\omega \right)$ for the half-filled, 2.5%, and 5% hole-doped system (from top to bottom). The dashed red line corresponds to the equilibrium data plotted in Fig. 2(a) of Ref. [18]. To match the experiment, the $GW+\mathrm{EDMFT}$ calculations are carried out here for $T=30$ K ($U=0.42, V=0.08$). The purple line is the experimental distribution shifted to match the calculated results (the chemical potential in the experiment is not precisely known).

Figure 14

Imaginary part of the hybridization function in the complex frequency plane $-\text{Im}\mathrm{\Delta }\left(z\right)$, obtained from a fit to the $\mathrm{\Delta }\left(i{\omega }_n\right)$ calculated with the local scheme (left) and causal scheme (right), for the parameters $U=0.42, V/U=0.19$, and $n=1$. In the bottom panels, the blue dots on the vertical axis are the Matsubara frequencies $i{\omega }_n$. The cyan and pink points show the poles and zeros of the Padé rational functions, respectively. The solid blue line shows the real-frequency axis. The top panels plot the values along this cut (spectral functions).