Local correlation functions of arbitrary order for the Falicov-Kimball model

Local $n$-particle vertex functions represent the crucial ingredient for diagrammatic extensions of dynamical mean field theory (DMFT). Hitherto their application has been restricted—with a few exceptions—to the $n=2$-particle vertex while higher-order vertices have been neglected. In this paper we derive a general analytical expression for the local $n$-particle (one-particle-reducible) vertex of the Falicov-Kimball model (FKM). We observe that the magnitude of such vertex functions itself strongly increases with the number of particles $n$. On the other hand, their effect on generic Feynman diagrams remains rather moderate due to the damping effect of the Green's functions present in such diagrams. Nevertheless, they yield important contributions to the self-energy corrections calculated in diagrammatic extensions of DMFT as we explicitly demonstrate using the example of dual-fermion calculations for the two-dimensional FKM at quarter filling of the stationary $f$ electrons. Here corrections to the self-energy obtained from the three-particle vertex are indeed comparable in magnitude to corresponding corrections stemming from the two-particle vertex.

Figure 1

Left: All diagrams where only the frequency ${\nu }_1$ is 1PD. Right: All the diagrams contributing to the $n$-particle one-particle-connected propagator $T^{\left(n\right)}$ where frequency ${\nu }_1$ is connected only to the frequency ${\nu }_2$.

Figure 2

Coefficients ${\mathcal{C}}_n$ (${\mathcal{F}}_n$) for the connected (one-particle-connected) $n$-particle Green's function at half filling $p_1=p_2=0.5$ for $n$ up to 10. The inset shows the behavior up to $n=20$.

Figure 3

Same as Fig. 2 but for quarter filling $p_1=0.25$.

Figure 4

Coefficients $abs\left({\mathcal{C}}_n\right)\left[abs\left({\mathcal{F}}_n\right)\right]$ for the (one-particle) connected $n$-particle vertex at half filling $p_1=p_2=0.5$ for even $n$ up to 200. Note the logarithmic scale.

Figure 5

Same as Fig. 4, but for quarter filling $p_1=0.25$ and also including odd numbers for $n$.

Figure 6

Ratio $r_n={\mathcal{C}}_n/\left[(n-1)(n-2){\mathcal{C}}_{n-2}\right]$ at half filling $p_1=p_2=0.5$. Note that $r_n$ converges towards a negative constant, implying an alternating sign of ${\mathcal{C}}_n$ (for even $n$).

Figure 7

Same as Fig. 6, but for $p_1=0.25$. Away from half filling, quasiperiodic structures arise in the ratio.

Figure 8

An exemplary diagram (“necklace” diagram) contributing to the partition function, which can be generated within a diagrammatic extension of DMFT such as DF or 1PI including the local 4-particle vertex function. Generalizations to corresponding diagrams with higher-order local vertices are straightforward.

Figure 9

Diagrammatic representation of correction to the dual $\mathrm{\Sigma }$ due to a 3-particle vertex.

Figure 10

Diagrammatic representation of the (local) dual $\mathrm{\Sigma }$ on the two-particle level.

Figure 11

Upper left: Self-energy $\mathrm{\Sigma }$ in DMFT for the square lattice FKM at quarter filling $p_1=0.25,U=1$, and $T=0.05$. Upper right: Contribution of the local diagrams based on the 3-particle vertex in Fig. 9 (denoted as ${\mathrm{\Sigma }}^3$) and the comparison to the local 1PI correction (${\mathrm{\Sigma }}_{1\mathrm{PI}}^{\mathrm{loc}}$) in a standard calculation based on the 2-particle vertex. Lower panels: Standard DF correction to the self-energy (${\mathrm{\Sigma }}_{DF}$) for four different $\mathbf{k}$ points as calculated from the two-particle vertex. Note that the correction term from the 3-particle vertex is of similar magnitude as the contributions from the conventional 2-particle vertex in DF; the DF self-energies are those for the dual, not the real fermions.