High-frequency asymptotics of the vertex function: Diagrammatic parametrization and algorithmic implementation

Vertex functions are a crucial ingredient of several forefront many-body algorithms in condensed matter physics. However, the full treatment of their frequency and momentum dependence severely restricts numerical calculations. A significant advancement requires an efficient treatment of the high-frequency asymptotic behavior of the vertex functions. In this work, we first provide a detailed diagrammatic analysis of the high-frequency structures and their physical interpretation. Based on these insights, we propose a parametrization scheme, which captures the whole high-frequency domain for arbitrary values of the Coulomb interaction and electronic density, and we discuss the details of its algorithmic implementation in many-body solvers based on parquet equations as well as functional renormalization group schemes. Finally, we assess its validity by comparing our results for a single impurity Anderson model with exact diagonalization calculations. The proposed parametrization is pivotal for the algorithmic development of all quantum many-body methods based on vertex functions arising from both local and nonlocal static microscopic interactions as well as effective dynamic interactions which uniformly approach a static value for large frequencies. In this way, our present technique can substantially improve vertex-based diagrammatic approaches including spatial correlations beyond dynamical mean-field theory.

Figure 1

Notations of the vertex functions in the three different scattering channels.

Figure 2

The upper row shows the full vertex $F_{\uparrow \downarrow }$ in $pp$ notation, and the lowest-order diagrams (excluding the bare interaction) for each of the four contributions to the parquet equation (6). The bottom row shows the numerical SIAM results for these vertices ($U=1, \beta =20, D=1$) and the corresponding decomposition through the parquet equation (6) using $pp$ notation for vanishing transfer frequency $\mathrm{\Omega }=0$. In the diagrammatic representations used throughout this paper, all external legs are to be considered as the remainder of the amputated propagators.

Figure 3

Compact diagrammatic representation of the BSE in all scattering channels. Each $k^{\prime \prime }$ is integrated over.

Figure 4

The so-called “eye” diagrams in the $pp$ channel.

Figure 5

Diagrammatic representation of the asymptotic functions for the particle-particle channel. For a more rigorous definition see Appendix pp2.

Figure 6

Sketch of a reducible vertex function in frequency space as a function of $k$ and $k^{\prime }$ for fixed $q$. It consists mainly of two extensive stripes and a more dynamical localized structure centered at a position determined by the transfer frequency. The two stripes are described by ${\mathcal{K}}_{2,r}^{kq}$ and ${\mathcal{K}}_{2,r}^{k^{\prime }q}$, the local structure is contained in the rest function $\mathcal{R}$. The nearly constant background is described by ${\mathcal{K}}_{1,r}^q$.

Figure 7

Schematic diagrammatic representation of the localized structure presented in Eq. (29).

Figure 8

Diagrammatic representation of the particle-particle contribution ${\mathcal{T}}_{pp}$ (34a) in the vertex flow equation. The dashed line denotes the single-scale propagator $S^{\mathrm{\Lambda }}$.

Figure 9

Derivative of the lowest-order contribution to ${\mathcal{K}}_{2,ph}^{\mathrm{\Lambda }}$.

Figure 10

The Feynman diagrams for the reducible vertex function ${\mathrm{\Phi }}_d^{k{k}^{\prime }q}$ generated in the first two iterations of a parquet approximation calculation. They can be attributed to (I) ${\mathcal{K}}_{1,d}^q$, (II) ${\mathcal{K}}_{2,d}^{kq}$, and (III) ${\mathcal{R}}_d^{k{k}^{\prime }q}$, respectively.

Figure 11

${\mathcal{K}}_{1,\uparrow \downarrow }^{\mathrm{\Omega }}$ for all three scattering channels. We present results obtained by fRG (left, solid), PA (right, solid), and ED (right, dashed) for the SIAM with $U=1, \beta =20$, and $D=1$.

Figure 12

${\mathcal{K}}_{2,\uparrow \downarrow }^{\nu \mathrm{\Omega }}$ for all three scattering channels as a function of $\nu$ and for different values of $\mathrm{\Omega }$. We present results obtained by fRG (left, solid), PA (right, solid), and ED (right, dashed) for the SIAM with $U=1, \beta =20$, and $D=1$.

Figure 13

Rest function ${\mathcal{R}}_{\uparrow \downarrow }^{\nu {\nu }^{\prime }\mathrm{\Omega }}$ for all three scattering channels as a function of $\nu$ and ${\nu }^{\prime }$ plotted for $\mathrm{\Omega }=0$. We present results obtained by fRG (first row, left) and PA (second row, left) for the SIAM with $U=1, \beta =20$, and $D=1$. The right side always shows the corresponding ED result.

Figure 14

Same as Fig. 13, but for a finite transfer frequency $\mathrm{\Omega }=8\frac{2\pi }{\beta }$.

Figure 15

$Im\mathrm{\Sigma }\left(i\nu \right)$ as obtained by fRG (red, solid), PA (green, solid), and ED (blue, dotted) for the SIAM with $U=1, \beta =20$, and $D=1$.

Figure 16

Same as Fig. 11, but for $U=4$.

Figure 17

Same as Fig. 12, but for $U=4$.

Figure 18

Comparison of ${\mathcal{R}}_{ph,\uparrow \downarrow }$ obtained by means of PA with the exact result. Here, $U=4$.

Figure 19

Same as Fig. 15, but for $U=4$.

Figure 20

Comparison of $Im\mathrm{\Sigma }\left(i{\nu }_0\right)/U^2$ for the SIAM with $\beta =20$ and $D=1$ as obtained by fRG and PA with and without (${\mathcal{K}}_1={\mathcal{K}}_2=0$) high-frequency asymptotics, compared with ED for ${\nu }_0=\frac{\pi }{\beta }$ and different values of $U$.

Figure 21

(Top) Comparison of $Im\mathrm{\Sigma }\left(i{\nu }_0\right)$ for the SIAM with $\beta =20, U=1$, and $D=1$ as obtained by fRG and PA with and without (${\mathcal{K}}_1={\mathcal{K}}_2=0$) high-frequency asymptotics for ${\nu }_0=\frac{\pi }{\beta }$ and different values of the grid size $N_{\mathrm{Grid}}$. (Bottom) Deviation of the data presented in the top panel from the converged result ($N_{\mathrm{Grid}}=\infty$) using a log-log plot. Convergence w.r.t. the grid size follows a power law $Im\mathrm{\Sigma }(i{\nu }_0,N_{\mathrm{Grid}})=Im\mathrm{\Sigma }(i{\nu }_0,\infty )+\mathcal{O}\left(N_{\mathrm{Grid}}^{-\alpha }\right)$. Estimates for the scaling exponents are shown.

Figure 22

Comparison of ${\mathcal{K}}_{1,ph,\uparrow \downarrow }^{\mathrm{\Omega }=0}/U^3, {\mathcal{K}}_{2,ph,\uparrow \downarrow }^{{\nu }_0(\mathrm{\Omega }=0)}/U^3$, and ${max}_{\nu {\nu }^{\prime }}\left|{\mathcal{R}}_{ph,\uparrow \downarrow }^{\nu {\nu }^{\prime }(\mathrm{\Omega }=0)}\right|/U^4$ (${\nu }_0=\frac{\pi }{\beta }$) for fRG in the one-loop ($1\ell$) and two-loop ($2\ell$) implementation, for both the $\mathrm{\Omega }$ and $U$ flow, with ED and PA. We note that the one-loop $U$ flow diverges for $U=3$ or larger.

Figure 23

Left two panels: Comparison of $Im\mathrm{\Sigma }\left(i{\nu }_0\right)/U^2$ and ${lim}_{\nu \to \infty }Im\nu \mathrm{\Sigma }\left(i\nu \right)/U^2$ for fRG one-loop and two-loop with the corresponding simplified schemes introduced previously [${\mathcal{K}}_{\mathrm{eff}}$, see Eq. (38)], PA and ED. Right panel: Comparison of the susceptibility $\chi$ in the $ph$ channel for $U=4$ as obtained by means of Eq. (B1b) after the two-loop fRG flow (${\chi }_{\mathrm{VC}}$) in the conventional and the simplified scheme (eff). This is compared to the susceptibility $\chi$ obtained directly from Eq. (10) as well as the PA and ED result.

Figure 24

Diagrammatic representation of the ${\mathcal{K}}_1$ functions in the three different channels. As denoted in the first diagram, the external lines are to be excluded, making the $k$ and $k^{\prime }$ arguments redundant. Here, $\overline{\sigma }$ denotes the opposite spin of $\sigma$, and $SU\left(2\right)$ symmetry is explicitly assumed.

Figure 25

Diagrammatic representation of the ${\mathcal{K}}_2$ functions in the three different channels. Here, $\overline{\sigma }$ denotes the opposite spin of $\sigma$, while $SU\left(2\right)$ symmetry is explicitly assumed.