Multiloop Functional Renormalization Group That Sums Up All Parquet Diagrams

We present a multiloop flow equation for the four-point vertex in the functional renormalization group (FRG) framework. The multiloop flow consists of successive one-loop calculations and sums up all parquet diagrams to arbitrary order. This provides substantial improvement of FRG computations for the four-point vertex and, consequently, the self-energy. Using the x-ray–edge singularity as an example, we show that solving the multiloop FRG flow is equivalent to solving the (first-order) parquet equations and illustrate this with numerical results.

Figure 1

Low-order diagrams for (a) the vertex reducible in antiparallel lines, ${\gamma }_a$, and (b) the totally irreducible vertex $R$. Solid (dashed) lines denote $G^c$ ($G^d$), and a dot the bare vertex $-U$. The first-order or so-called parquet approximation only retains the bare vertex for $R$.

Figure 2

(a) Bethe-Salpeter equations in the antiparallel ($a$) and parallel ($p$) channels. A full square denotes the full vertex ${\mathrm{\Gamma }}^{(4)}$. (b) FRG flow equation for both channels relating ${\partial }_{\mathrm{\Lambda }}{\mathrm{\Gamma }}^{(4)}$ to ${\mathrm{\Gamma }}^{(4)}$ and ${\mathrm{\Gamma }}^{(6)}$. The conventional approximation is to set ${\mathrm{\Gamma }}^{(6)}=0$.

Figure 3

Multiloop FRG flow equations, ${\partial }_{\mathrm{\Lambda }}{\gamma }_r=\sum _{\ell \ge 1}{\dot{\gamma }}_r^{(\ell )}$, for the four-point vertex reducible in channel $r$, with $r=a$ or $p$, and $\overline{r}=p$ or $a$. The subscript $r$ in the diagrams further symbolizes antiparallel or parallel $c\text{−}d$ lines, respectively. (a) One-loop, (b) two-loop, (c) three- and higher-loop flows. (d) One-loop calculation of ${\dot{\gamma }}_{r,\mathrm{C}}^{(\ell +2)}$, using the previously computed ${\dot{\gamma }}_{r,\mathrm{R}}^{(\ell +1)}$ or ${\dot{\gamma }}_{r,\mathrm{L}}^{(\ell +1)}$.

Figure 4

(a) Noninteracting “standard” particle-hole bubble ${\mathrm{\Pi }}_0^G$ and propagator $G^d$ (inset) for different regulators (cf. Eq. (S8) of Ref. [26]) and $\mathrm{\Lambda }/{\xi }_0=0.2$. (b) Same as (a) for the “single-scale” bubble ${\mathrm{\Pi }}_0^S$ and propagator $S^d$. (c) Double-logarithmic plot for the particle-hole susceptibility $\mathrm{\Pi }$, obtained from solving the parquet equations. (d) ${\mathrm{\Pi }}_{\overline{\omega }=0}(u)$ computed via the parquet equations [${\epsilon }_d$, $\beta$ as in (c)] and according to Eq. (2) with different choices for $\alpha (u)$. The comparison between these guide-to-the-eye lines and the numerical solution confirms that $\alpha (u)\approx 2u$, but also shows that subleading contributions become sizable for larger $u$. These are present since internal numerical calculations go beyond logarithmic accuracy.

Figure 5

(a)–(d) Numerical solutions for the particle-hole susceptibility $\mathrm{\Pi }$, obtained from the parquet equations and from MFRG with different regulators [cf. Figs. 4 and 4], using the parameters of Fig. 4. Insets: Relative deviation between parquet and MFRG results for $\mathrm{\Pi }$ (solid line) and between $\mathrm{\Pi }$ and $\underset{|\omega |,|\nu |\to \infty }{\lim }{\gamma }_a/U^2$ (dashed line), all evaluated at $\overline{\omega }=0$.