Self-consistent ladder dynamical vertex approximation

We present and implement a self-consistent $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ approach for multiorbital models and ab initio materials calculations. It is applied to the one-band Hubbard model at various interaction strengths with and without doping, to the two-band Hubbard model with two largely different bandwidths, and to ${\mathrm{SrVO}}_3$. The self-energy feedback reduces critical temperatures compared to dynamical mean-field theory, even to zero temperature in two dimensions. Compared to a one-shot, non-self-consistent calculation the nonlocal correlations are significantly reduced when they are strong. In case nonlocal correlations are weak to moderate as for ${\mathrm{SrVO}}_3$, one-shot calculations are sufficient.

Figure 1

Schematic explanation of DMFT and $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ loops.

Figure 2

Step-by-step illustration of a self-consistent ladder $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ calculation. The first box (orange color) shows preliminary calculations to set up the model. The second box (blue color) concerns the DMFT calculations, and the third one (green color) shows the $\mathrm{sc}\text{−}\mathrm{D}\mathrm{\Gamma }\mathrm{A}$cycle. Quantities written in red are kept constant throughout the whole calculation; those in blue are self-consistently determined in $\mathrm{sc}\text{−}\mathrm{D}\mathrm{\Gamma }\mathrm{A}$.

Figure 3

Phase diagram of the square-lattice Hubbard model at half filling. Blue crosses denote points at which the sc-$\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ could be converged. The DMFT-Néel temperature is shown in gray (from Ref. [73]). The magenta line indicates the DMFT metal-insulator transition. We also show the first two vertex divergence lines (from Ref. [56]).

Figure 4

Static magnetic susceptibility of the square-lattice Hubbard model with $U=2$ and $n=1$ at momentum $\mathbf{q}=(\pi ,\pi )$ as a function of inverse temperature. Different colors and symbols denote different methods. The gray vertical line marks the DMFT Néel temperature.

Figure 5

The imaginary part of sc-$\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ self-energy (left) and the corresponding spectral function (right) at $U=2$ and half filling for two momenta on the Fermi surface: nodal point, ${\mathbf{k}}_{\mathrm{N}}=(\pi /2,\pi /2)$, and antinodal point, ${\mathbf{k}}_{\mathrm{AN}}=(\pi ,0)$. Different colors and symbols denote different temperatures. The spectral functions were obtained by analytic continuation with the maximum entropy method [74, 75].

Figure 6

Inverse temperature dependence of the imaginary part of self-energy at $U=2$ and half filling for the first three Matsubara frequencies ${\omega }_n=\{\pi T,3\pi T,5\pi T\}$ for two momenta on the Fermi surface: ${\mathbf{k}}_{\mathrm{N}}=(\pi /2,\pi /2)$ and ${\mathbf{k}}_{\mathrm{AN}}=(\pi ,0)$. Different colors and symbols denote different methods. The $\lambda \text{−}\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ and DiagMC data in this figure were kindly provided by the authors of Ref. [63].

Figure 7

Static magnetic susceptibility for $U=4$ on a path through the Brillouin zone for $T=0.25$ and $T=0.1$. The value of $\lambda \text{−}\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ susceptibility at the $M$ point is ${\chi }_m(\omega =0,\mathbf{q}=(\pi ,\pi ))=415$ (beyond the $y$ range of the plot). A smaller momentum window is shown in the insets.

Figure 8

Imaginary part of self-energy for the nodal (N) and antinodal (AN) points as a function of Matsubara frequencies for $U=4$, $n=1$ and two temperatures: $T=0.25$ and $T=0.1$. Different methods are distinguished by different symbols and colors. The 1FF and 9FF p-$\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ data are reproduced from Ref. [37].

Figure 9

Static magnetic susceptibility in sc-$\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ along the $X\text{−}M$ path in the Brillouin zone ($q_y=\pi$) for different temperatures $T$, $U=8t$, and $n=0.85$.

Figure 10

Left panel: logarithmic plot of the dynamic magnetic structure factor $\mathrm{Im}{\chi }_m(\mathbf{q},\omega )/(1-e^{-\omega /T})$ at $U=8$, $n=0.85$, and $T=0.05$, obtained by analytic continuation. The analytic continuation was done with the maximum entropy method [74, 75]. Right panel: $Y$-shaped spin-excitation dispersion obtained from the dynamic magnetic susceptibility at $\mathbf{q}=(q_x,\pi )$ for different temperatures $T$.

Figure 11

Fermi surface (FS) and the corresponding real and imaginary part of self-energy as a function of Matsubara frequency as obtained in DMFT and sc-$\mathrm{D}\mathrm{\Gamma }\mathrm{A}$. For sc-$\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ we show two different momenta on the FS, as indicated with green and blue stars on the FS plot. The FS was obtained by plotting $\overline{A_{\mathbf{k}}\left(0\right)}\approx G(\mathbf{k},\tau =1/\left(2T\right))$ (which avoids the analytical continuation and averages the spectral function over an interval $\sim T$ around the FS). The noninteracting tight-binding FS is plotted with a thin cyan line in both FS plots. The parameters are $T=0.05$, $U=8$, $n=0.85$.

Figure 12

Two-band model: band structure (left) and Fermi surfaces at $T=0.1$ (right). The filling is $n=1.7$, such that the chemical potential is slightly temperature dependent.

Figure 13

Real (top) and imaginary part (bottom) of the self energy at the lowest Matsubara frequency for the two band Hubbard model at $T=0.1$ along a high-symmetry path through the Brillouin zone.

Figure 14

Real (top) and imaginary (bottom) part of the self-energy vs Matsubara frequency at $T=0.1$ for the $\mathrm{\Gamma }$, $X$, and $M$ point, comparing DMFT, 1-$\mathrm{D}\mathrm{\Gamma }\mathrm{A}$, and $\mathrm{sc}\text{−}\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ for a two-band Hubbard model. The different colors refer to the two bands, as plotted in Fig. 12; the different line types to the different methods.

Figure 15

Momentum-resolved spectral function of the two-orbital Hubbard model at $T=0.1$ along a high-symmetry path through the Brillouin zone. The first column shows only the component corresponding to the wide band, the second column only the component corresponding to the narrow band. In the third row the full spectral function is shown.

Figure 16

Low-frequency zoom of momentum-resolved spectral function for the two-orbital Hubbard model at temperatures $T=0.2,0.1,0.05$ on a path through the Brillouin zone. In the 1-$\mathrm{D}\mathrm{\Gamma }\mathrm{A}$spectrum at $T=0.1$ it is clearly visible that the analytic continuation does not work well in the vicinity of the $\mathrm{\Gamma }$ point. This is a typical issue for 1-$\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ calculations where nonlocal self-energy corrections become large.

Figure 17

Spectral weight at the Fermi level $\overline{A(\mathbf{k},0)}\approx G(\mathbf{k},\tau =1/2T)$ of the two-band Hubbard model for $T=0.1$ (top) and $T=0.05$ (bottom), comparing DMFT (left) and $\mathrm{sc}\text{−}\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ (right) and the tight-binding model without interaction (red and blue lines).

Figure 18

Momentum dependence of the self-energy of strontium vanadate at the lowest positive Matsubara frequency and $k_z=0$. Upper row: one-shot ab initio $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$, lower row: self-consistent $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$. The DMFT value is $(1.861–0.104i)$ eV.

Figure 19

Momentum-dependent spectral function of ${\mathrm{SrVO}}_3$, comparing DMFT, one-shot $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$, and self-consistent $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$.

Figure 20

Extrapolation of antiferromagnetic susceptibility of the square-lattice Hubbard model with $U=2$ at $T=0.05$. The largest $\mathbf{q}$ grid is $80\times 80$.

Figure 21

Extrapolation of antiferromagnetic susceptibility of the square-lattice Hubbard model with $U=2$ at $T=0.04$. The largest $\mathbf{q}$ grid is $80\times 80$.