Divergences of the irreducible vertex functions in correlated metallic systems: Insights from the Anderson impurity model

In this work, we analyze in detail the occurrence of divergences in the irreducible vertex functions for one of the fundamental models of many-body physics: the Anderson impurity model (AIM). These divergences, a surprising hallmark of the breakdown of many-electron perturbation theory, have been recently observed in several contexts, including the dynamical mean-field solution of the Hubbard model. The numerical calculations for the AIM presented in this work, as well as their comparison with the corresponding results for the Hubbard model, allow us to clarify several open questions about the properties of vertex divergences in a particularly interesting context, the correlated metallic regime at low temperatures. Specifically, our analysis (i) rules out explicitly the transition to a Mott-insulating phase, but not the more general suppression of charge fluctuations (proposed in [O. Gunnarsson et al., Phys. Rev. B 93, 245102 (2016)]), as a necessary condition for the occurrence of vertex divergences, (ii) clarifies their relation with the underlying Kondo physics, and, eventually, (iii) individuates which divergences might also appear on the real-frequency axis in the limit of zero temperature, through the discovered scaling properties of the singular eigenvectors.

Figure 1

Schematic representation of the Bethe-Salpeter equation in the charge channel (see text).

Figure 2

Visualization of the relation between the singular eigenvalues ${\lambda }_{\alpha }$ and eigenvectors $V_{\alpha }^c\left(i{\nu }_n\right)$ of ${\mathbf{\chi }}_c$ and the full frequency structure of ${\mathbf{\Gamma }}_c$ in proximity of a divergence (the specific calculation, shown in the main panel as an example, has been performed for an AIM with $U=3.321444$ and $\beta =40$, yielding ${\lambda }_{\alpha }=0.00025$). The color of the singular eigenvectors (not related to the color scale of the main plot) highlights the connections to the sign/intensity structure of ${\mathbf{\Gamma }}_c$ as a function of ${\nu }_n, {\nu }_{n^{\prime }}$, defined by Eq. (6). The values of ${\mathbf{\Gamma }}_c$ are rescaled by $1/{\beta }^2$ in the main panel for a better readability.

Figure 3

Left panel: $T-U$ diagram of the AIM at half-filling, showing the first divergence lines along which the irreducible vertex functions diverge. For red lines, this divergence takes place in the charge channel ${\mathrm{\Gamma }}_c^{{\nu }_n{\nu }_{n^{\prime }}({\mathrm{\Omega }}_n=0)}$, along the orange lines simultaneous divergences in the charge and the particle-particle up-down channel, ${\mathrm{\Gamma }}_c^{{\nu }_n{\nu }_{n^{\prime }}({\mathrm{\Omega }}_n=0)}$ and ${\mathrm{\Gamma }}_{pp,\uparrow \downarrow }^{{\nu }_n{\nu }_{n^{\prime }}({\mathrm{\Omega }}_n=0)}$, are observed. The dashed blue box marks the parameter region where the “atomic” ordering of divergence lines is violated (see text). Right panel: divergence lines of the half-filled unfrustrated Hubbard model (square lattice dispersion with $4t=1$), solved with DMFT. The lines are plotted with the same color code; the blue solid line represents the Mott-Hubbard MIT [64]. Readapted from Ref. [14].

Figure 4

A zoom of the $T-U$ diagram of the AIM (left panel Fig. 3) at half-filling is shown. The blue solid line marks the Kondo temperature ($T_K$), estimated from the rescaling of our numerical data for the magnetic susceptibility to the universal function given in Ref. [68]. The black dotted line represents an estimate for $T_K$ obtained from an analytic expression [67] valid in the limit $D\gg U,T$. An additional scale related to the Kondo screening, the half-bandwidth of the $T\to 0$ Kondo peak ($\frac{\pi }{2}Z\mathrm{\Delta }$) [69], is marked with a gray dotted line, and is roughly five times larger than $T_K$. The light gray shaded area can be regarded, thus, as the parameter region where the effects of the Kondo screening become visible.

Figure 5

Cuts of the irreducible vertex function in the charge channel ${\mathrm{\Gamma }}_c^{{\nu }_n{\nu }_{n^{\prime }}({\mathrm{\Omega }}_n=0)}$ for the first Matsubara frequency ${\nu }_1=\pi /\beta$ at $T=0.05$ and two $U$ values are shown. In the left insets, the lowest two eigenvalues of ${({\chi }_c/{\chi }_{ph,0})}^{{\nu }_n{\nu }_{n^{\prime }}({\mathrm{\Omega }}_n=0)}$ are reported vs $U$, in the right insets the eigenvector $V_{\alpha }^c\left(i{\nu }_n\right)$ corresponding to the lowest eigenvalue ${\lambda }_{\alpha }$ is shown as a function of $i{\nu }_n$. Left panel: at $U=4.5$ the lowest eigenvalue is corresponding to the second red divergence line (red dot), hence, $V_{\alpha }^c\left(i{\nu }_n\right)$ is antisymmetric. Right panel: for $U=4.59$ the eigenvalue of the first orange line is the smallest (orange dot), $V_{\alpha }^c\left(i{\nu }_n\right)$ is symmetric.

Figure 6

Singular eigenvectors of ${({\chi }_c/{\chi }_{ph,0})}^{{\nu }_n{\nu }_{n^{\prime }}({\mathrm{\Omega }}_n=0)}$ (numerically equivalent to the ones of ${\chi }_c^{{\nu }_n{\nu }_{n^{\prime }}({\mathrm{\Omega }}_n=0)}$), corresponding to the five divergence lines (left panel Fig. 3), shown as a function of Matsubara index $n$ for the temperature $T=0.025$. In gray, the eigenvectors of the red divergence lines for a higher temperature $T=0.5$, properly rescaled, are plotted, indicating the broadening of $V_{\alpha }^c\left(i{\nu }_n\right)$ for lower temperatures. Top panel: eigenvectors of the first red divergence line (red, antisymmetric) and the first orange divergence line (orange, symmetric). Middle (bottom) panel: same as top panel, but for the second (third) red and orange (red) divergence lines.

Figure 7

Left top panel: singular eigenvectors $V_{\alpha }^c\left(i{\nu }_n\right)$ for several temperatures along the first red divergence line, plotted as a function of the Matsubara index $n$. Right top panel: singular eigenvectors of the left top panel plotted as a function of Matsubara frequency $i{\nu }_n$ instead. Bottom panels: as top panels, but showing the data corresponding to the first orange divergence line.

Figure 8

Zoom on the first red divergence line showing the low-$T$ behavior (for $T\ll T_K$) which shows that within the error bars obtained by a jackknife analysis (see Appendix pp3), the line bends towards the $U$ axis for $T\to 0$. Inset: further zoom on the lowest temperatures, emphasizing the growth of the error bar with decreasing temperature.

Figure 9

Left panel: eigenvectors of the first red divergence line for a given temperature multiplied with the inverse of the scaling factor $f\left(T\right)$ yielding the eigenvector at $T=0$ governing the frequency structure of the eigenvectors at all temperatures in the $T\ll T_K$ regime. Right panel: as left panel, but for the first orange divergence line. Here, the minor discrepancies can be ascribed to a stronger $U$ dependence of the singular eigenvectors in comparison to those of the red lines.

Figure 10

Schematic representation of the energy scale(s) controlling the border(s) of the perturbative/nonperturbative regimes (a) in the AL/FK case, which is relevant for the large $T$ and $U$ limits of the AIM and the DMFT solution of the Hubbard model, (b) for the ring scenario which was proposed in Ref. [14] for the Hubbard model in the correlated metallic regime (see text). Note that for the AIM with a large bandwidth, as studied here, only one energy scale exists.

Figure 11

Comparison of $\mathrm{Im}G\left(i{\nu }_n\right)$ (rescaled by a proper factor) and ${\partial }^2\mathrm{Im}G\left(i{\nu }_n\right)=\frac{{\partial }^2\mathrm{Im}G\left(i{\nu }_n\right)}{\partial {\nu }_n^2}$ for three different temperatures along the third red divergence line of the AIM.

Figure 12

The shifted data for $T{\chi }_s(\omega =0)$ for several temperatures at $U=4.2$, compared to the universal result for $T{\chi }_s\left(T\right)$ (see Ref. [68]).

Figure 13

The singular eigenvalue ${\lambda }_{\alpha }$ of ${({\chi }_c/{\chi }_{ph,0})}^{{\nu }_n{\nu }_{n^{\prime }}\mathrm{\Omega }=0}$ for $T=0.003\overline{3}$ and two different $U$ values (3.383, 3.42) is shown. A linear interpolation is used for the determination of $\tilde{U}$ (red solid line). The jackknife error analysis provided error bars for the two results for ${\lambda }_{\alpha }$ (red dashed), which are, then, used to estimate the error bar of $\tilde{U}$ (gray dashed lines).