Counting Feynman diagrams via many-body relations

We present an iterative algorithm to count Feynman diagrams via many-body relations. The algorithm allows us to count the number of diagrams of the exact solution for the general fermionic many-body problem at each order in the interaction. Further, we apply it to different parquet-type approximations and consider spin-resolved diagrams in the Hubbard model. Low-order results and asymptotics are explicitly discussed for various vertex functions and different two-particle channels. The algorithm can easily be implemented and generalized to many-body relations of different forms and levels of approximation.

Figure 1

Graphical representation of many-body relations, where solid lines represent dressed propagators $G$ and dots represent bare four-point vertices ${\mathrm{\Gamma }}_0^{\left(4\right)}$. (a) Schwinger-Dyson equation (3b) for the self-energy. (b) To perform the functional derivative $\delta \mathrm{\Sigma }/\delta G$ in Eq. (3c), one sums all copies of diagrams where one $G$ line is removed. Conversely, the self-energy differentiated w.r.t. a scalar parameter (see main text), $\dot{\mathrm{\Sigma }}$, is obtained by contracting [cf. Eq. (5a)] the vertex $I_t$ with $\dot{G}$ (line with double dash) or [cf. Eq. (5b)] the full vertex ${\mathrm{\Gamma }}^{\left(4\right)}$ with the singled-scale propagator $S$ [cf. Eq. (4), line with one dash]. (c) ${\mathrm{\Gamma }}^{\left(4\right)}$ deduced from the Bethe-Salpeter equation (BSE) in the transverse channel (3c). (d)–(e) BSEs (7) for the reducible vertices in (d) the antiparallel channel and (e) the parallel channel. (f) Dyson equation (3a) involving the bare propagator $G_0$ (gray line). Note that the relations (a)–(c) suffice to generate all skeleton diagrams for the self-energy and the vertex (with all signs and prefactors written explicitly). Relations (c)–(e) together with Eq. (6) enable the parquet decomposition of the four-point vertex. Finally, the Dyson equation (f) makes the connection between bare and skeleton diagrams.

Figure 2

Examples and translation from Hugenholtz to Feynman diagrams. (a) Bare (antisymmetric) four-point vertex (dot) as used for Hugenholtz diagrams expressed by direct and exchange interactions [cf. Eq. (2b), wavy lines] as used for Feynman diagrams. (b)–(d) Diagrams for the reducible vertices ${\gamma }_r$ in the two-particle channels $a, p, t$, respectively. Whereas ${\gamma }_a$ and ${\gamma }_t$ have four Feynman diagrams, ${\gamma }_p$ has only two. In fact, inserting the direct and exchange interactions from (a) into the Hugenholtz diagram containing two equivalent propagators (parallel lines connected to antisymmetric vertices) yields only two topologically distinct diagrams, properly canceling the factor of $1/2$. (e) First- and (f) second-order diagrams for the self-energy. The prefactor of $1/2$ is again canceled upon decomposing ${\mathrm{\Gamma }}_0$. Note that, if the electron propagators (lines) are considered as dressed ones, the above diagrams comprise all skeleton diagrams of the four-point vertex and the self-energy up to second order.

Figure 3

Plots for the rescaled number of (a) bare and (b) skeleton diagrams with $n$ ranging up to 1500. Numbers are rescaled as ${\tilde{\mathcal{N}}}_{{\mathrm{\Gamma }}^{\left(m\right)}}\left(n\right)={\mathcal{N}}_{{\mathrm{\Gamma }}^{\left(m\right)}}\left(n\right)/(n!n^{(m-1)/2}{2}^{(m-2)/2})$ [Eq. (22)]; $G$ is rescaled in the same way as $\mathrm{\Sigma }={\mathrm{\Gamma }}^{\left(2\right)}$ [Eq. (24)]; $R$ and ${\gamma }_r$ ($r=a,p$, dotted) in the same way as ${\mathrm{\Gamma }}^{\left(4\right)}$. Dashed lines for ${\gamma }_r$ account for the correct asymptote, showing ${\mathcal{N}}_{{\gamma }_r}/\left(4\right|{\sigma }_r|n!n^{1/2})$ [Eq. (25)].

Figure 4

Ratio of subsequent elements of (a) ${\mathcal{N}}_X$ and (b) ${\mathcal{N}}_X^{\mathrm{sk}}$ in the parquet-type approximations with $n_p=30$ and $n_p=12$ (see main text). We use the same color coding as in Fig. 3; dashed lines represent ${\gamma }_r$. The inset shows an analogous plot for ${\mathcal{N}}_G$, obtained from a finite-order self-energy ($n_s=20$) [cf. Eq. (29)]. The cusp for ${\mathrm{\Gamma }}^{\left(4\right)}, \mathrm{\Sigma }, G$ occurs at $1/n_p$ (inset: $1/n_s$), and for ${\gamma }_r$ at $1/n_p+1$, due to the structure of the BSEs [cf. (19b)].

Figure 5

Spin-resolved diagrams of the Hubbard model in the Hugenholtz and Feynman representation up to second order. Blue (dark) lines denote spin-up and red (light) lines spin-down propagators; dashed lines symbolize a sum over spin. Panels (a)–(c) give diagrams for ${\mathrm{\Gamma }}_0^{\uparrow \downarrow }, {\gamma }_a^{\uparrow \downarrow }$, and ${\gamma }_p^{\uparrow \downarrow }$; (d)–(e) for ${\gamma }_a^{\uparrow \uparrow }$ and ${\gamma }_t^{\uparrow \uparrow }$; and (f)–(g) for $\mathrm{\Sigma }$. Viewed with full propagators, these are all skeleton diagrams entering ${\mathrm{\Gamma }}^{\left(4\right)}$ and $\mathrm{\Sigma }$ up to second order. We explicitly see that the numbers of Hugenholtz and Feynman diagrams are equal.