Single-boson exchange formulation of the Schwinger-Dyson equation and its application to the functional renormalization group

Abstract

We extend the recently introduced single-boson exchange formulation to the computation of the self-energy from the Schwinger--Dyson equation (SDE). In particular, we derive its expression both in diagrammatic and in physical channels. The simple form of the single-boson exchange SDE, involving only the bosonic propagator and the fermion-boson vertex, but not the rest function, allows for an efficient numerical implementation. We furthermore discuss its implications in a truncated unity solver, where a restricted number of form factors introduces an information loss in the projection of the momentum dependence that in general affects the equivalence between the different channel representations. In the application to the functional renormalization group, we find that the convergence in the number of form factors depends on the channel representation of the SDE. For the two-dimensional Hubbard model at weak coupling, the pseudogap opening driven by antiferromagnetic fluctuations is captured already by a single ($s$-wave) form factor in the magnetic channel representation, differently to the density and superconducting channels.

Figures (7)

• Figure 1: Diagrammatic representation of the SDE for the self-energy: We show the diagram in the conventional form and the corresponding respresentation in single-boson exchange formalism in the ${\overline{ph}}$ and $pp$ channel (without the Hartree term).
• Figure 2: The same self-energy diagram drawn as $\lambda_{\overline{ph}}w_{\overline{ph}}G$ (left), as $\lambda_{pp}w_{pp}G$ (center), and as $\lambda_{ph}w_{ph}G$ (on the right). The dashed line indicates the closing Green's function. Using only an $s$-wave form factor is exact for the computation via $\lambda_{\overline{ph}}w_{\overline{ph}}G$, but not for $\lambda_{pp/ph}w_{pp/ph}G$, due to the information loss induced by the form-factor projections.
• Figure 3: Imaginary part of the self-energy as a function of Matsubara frequencies at half filling ($\mu=0$), $U=2$, and various temperatures, as determined by its expression in the magnetic channel \ref{['eq:sigmaM']}. At the antinodal point $\textbf{k}=(\pi,0)$ displayed in the main panel, the pseudogap opens at higher temperatures as compared to the nodal point $\textbf{k}=(\pi/2,\pi/2)$, see inset.
• Figure 4: Same as Fig. \ref{['fig:omf_rep_m_2']}, but determined by the density and superconducting channel using Eqs. \ref{['eq:set_d']} and \ref{['eq:set_sc']}. It can be clearly seen that these representations fail to capture the pseudogap opening both at the antinodal and the nodal point. Note also the different scales with respect to Fig. \ref{['fig:omf_rep_m_2']}.
• Figure 5: Fluctuation diagnostics of the imaginary part of the self-energy at the antinodal point, for the repulsive Hubbard model at half filling, $U=2$ and $T=0.13$. The histogram bars display the contributions of the different bosonic momenta $Q=(\textbf{Q},\Omega=0)$ in the magnetic (red), density (blue) and superconducting (green) representations. The pronounced red bar at $\bf{Q}=(\pi,\pi)$ clearly shows the dominant contributions of the antiferromagnetic spin fluctuations.