Interplay between local response and vertex divergences in many-fermion systems with on-site attraction

We investigate the divergences appearing in the two-particle-irreducible vertex functions of many-fermion systems with attractive on-site interactions. By means of dynamical mean-field theory calculations, we determine the location of singularity lines in the phase diagram of the attractive Hubbard model at half-filling, where the local Bethe-Salpeter equations are noninvertible. We find that divergences appear both in the magnetic and in the density scattering channels. The former affect a sector of suppressed fluctuations and comply with the mapping of the physical susceptibilities of the repulsive case. At the same time, the appearance of singularities in the density channel of the attractive model demonstrates that vertex divergences can also plague the dominant scattering sectors associated with enhanced local susceptibilities. This constitutes a counterexample to previously proposed interpretations. Eventually, by exploiting the underlying physical symmetries and a spectral representation of the susceptibilities, we clarify the relation between vertex divergences and the local response of the system in different channels.

Figure 1

Left: location of the divergences of the irreducible vertex in the different channels along the whole phase diagram of the attractive and the repulsive half-filled Hubbard model, computed in DMFT. The comparison of the negative and positive $U$ sectors yields perfectly mirrored divergences in the density channel (red). The simultaneous divergences in the density and particle-particle channel (orange) in the repulsive model are mapped into divergences of the magnetic channel (green) on the attractive side. Stars refers to the position in the phase diagram of the data shown in Fig. 4. Right: schematic sketch comparing the mapping of the singular eigenvalues ${\lambda }^S\left({\lambda }^A\right)$ associated to symmetric (antisymmetric) eigenvectors to the mapping of different physical degrees of freedoms (DOF).

Figure 2

Comparison of the singular eigenvectors $V_{\alpha }^r$ in the repulsive and the attractive case, plotted as a function of the Matsubara index $N=\nu \frac{\beta }{\pi }$. The upper (lower) panel shows perfectly identical singular antisymmetric (symmetric) eigenvectors located at different temperatures along the first (second) attractive (left) and first (second) repulsive (right) divergence line.

Figure 3

Evolution of the diagonal $\left(D\right)$ and the off-diagonal $\left(O\right)$ innermost matrix elements of the generalized susceptibilities, computed in DMFT, as a function of the local (attractive and repulsive) Hubbard interaction (left panel: density channel; right panel: magnetic and $pp$ channels) in the high-$T$ regime $\left(\beta =5\right)$. The vertical bars in light gray mark the $U$ values of the vertex divergences in the corresponding sectors, while the black arrows highlight the fulfillment of the singularity conditions for the $2\times 2$ innermost frequency matrix of ${\chi }_r^{\nu ,{\nu }^{\prime }}\left(\mathrm{\Omega }\right)$, namely, ${\chi }_r^D={\chi }_r^O$ (associated to antisymmetric singular eigenvectors) and ${\chi }_r^D=-{\chi }_r^O$ (symmetric eigenvectors).

Figure 4

Comparison of susceptibility densities [Eq. (22)] for the magnetic (green), density (red), and $pp$ (orange) sectors of the attractive $U<0$ and repulsive $U>0$ case, respectively. Each $\delta$ function in Eq. (22) is represented by horizontal bars [gray (colored) if corresponding to an antisymmetric (symmetric) EV]. The corresponding densities ${\rho }_r\left(\chi \right)$ are plotted as colored circles. Data shown here were obtained for $T=0.2$ and $U=2.16$. The position in the phase diagram (Fig. 1) is indicated by light blue stars.

Figure 5

Comparison of the static density ${\chi }_d(\mathrm{\Omega }=0)$ (red) and magnetic ${\chi }_m(\mathrm{\Omega }=0)$ (green) susceptibility with the contribution of the largest eigenvalue only, as a function of the attractive/repulsive Hubbard interaction $U$ at $T=0.2$. In the bottom of the plot, the lowest eigenvalue of ${\chi }_d^{\nu {\nu }^{\prime }}$ is shown. The evolution of the lowest eigenvalues $({\lambda }_d^{min}$, in dark gray) is completely decoupled from the behavior of the static susceptibility.

Figure 6

Eigenvalues of the generalized susceptibility of the DMFT solution of the BM model, from Eq. (C3), with $D=1$.

Figure 7

Intensity map (in logarithmic scale) of the susceptibility density ${\rho }_d\left(\chi \right)$ for the charge/density sector of the DMFT solution of the BM model at $T=0$ as function of $W$. The green curve displays the corresponding evolution of physical local susceptibility.