Nonlocal correlations and spectral properties of the Falicov-Kimball model

We derive an analytical expression for the local two-particle vertex of the Falicov-Kimball model, including its dependence on all three frequencies, the full vertex, and all reducible vertices. This allows us to calculate the self-energy in diagrammatic extensions of dynamical mean field theory, specifically in the dual fermion and the one-particle irreducible approach. Nonlocal correlations are thence included and originate here from charge-density wave fluctuations. At low temperatures and in two dimensions, they lead to a larger self-energy contribution at low frequencies and a more insulating spectrum.

Figure 1

Schematic representation of the structure of the full vertex $F$ in (Matsubara) frequency space [fermionic Matsubara frequencies, ${\nu }_{n_{\nu }}=(2n_{\nu }+1)\pi /\beta$; bosonic ones, ${\omega }_{n_{\omega }}=2n_{\omega }\pi /\beta$]. There are only $\omega =0$ and $\nu ={\nu }^{\prime }$ contributions.

Figure 2

Feynman-diagrammatic relation between the two-particle Green's function $G^{\left(2\right)}$ and the full vertex $F$ (top), $F$, and the irreducible vertex in the $ph$ channel ${\mathrm{\Gamma }}_{ph}$, given by the Bethe-Salpeter equation (middle), and the relation between $F, {\mathrm{\Gamma }}_c$, and the fully irreducible vertex $\mathrm{\Lambda }$ (bottom).

Figure 3

Divergencies of the $C$-factor, Eq. (15), and hence the local, irreducible vertices throughout the $U-T$ phase diagram of the $2D$ Falicov-Kimball model. Note that the divergencies are visible in the picture as jumps from negative (blue) to positive (red) values. The red area on the left side for $U<0.6$ is a consequence of the $1/T$ scaling and does not signify a divergency. In the lower right part, the lines are hardly visible due to emergent Moiré patterns.

Figure 4

From top to bottom: Full vertex, particle-particle irreducible, particle-hole irreducible, and fully irreducible vertex for $U=1$ (left) and $U=2$ (right) at $T=0.06$ for $\omega =0$ for the Matsubara frequency indices of the incoming electron ($n_{\nu }$) and hole ($n_{{\nu }^{\prime }}$).

Figure 5

DMFT phase diagram for the $2D$ Falicov-Kimball model. Shown is the CDW susceptibility as a function of $U$ and $T$. In the region where $\chi [\omega =0,\mathbf{q}=(\pi ,\pi \left)\right]$ is negative—indicated by the white region in the phase diagram—the system is already in the CDW ordered phase.

Figure 6

(a) Typical third-order diagram of DF. Note that apart from local vertex functions it contains just purely nonlocal propagators $\overline{G}=G_{\text{DMFT}}-G_{\text{loc}}$. (b) A diagram which can be constructed in DF only at the three-particle level (with the red part as a one-particle reducible three-particle vertex) [32], but is present in 1PI already at the two-particle level.

Figure 7

DF and 1PI self-energies for the $k$ points $(\pi ,0)$ (left) and $(\pi /2,\pi /2)$ (right) at $U=1$ for $T=0.055$ (bottom) and $T=0.08$ (top).

Figure 8

DF and 1PI self-energies for the $k$ points $(\pi ,0)$ (left) and $(\pi /2,\pi /2)$ (right) at $U=2$ for $T=0.05$ (bottom) and $T=0.07$ (top).

Figure 9

Green's functions for the $k$ points $(\pi ,0)$ (left) and $(\pi /2,\pi /2)$ (right) at $U=1$ for $T=0.07$. We attribute small nonanalyticities (positive imaginary parts) to the Padé fit.

Figure 10

Green's functions for the $k$ points $(\pi ,0)$ (left) and $(\pi /2,\pi /2)$ (right) at $U=1$ for $T=0.055$. We attribute small nonanalyticities (positive imaginary parts) to the Padé fit.

Figure 11

Spectral functions along a path through the Brillouin zone as a function of the real frequency $\omega$ for $U=1$ and $T=0.07$. For comparison, the eigenenergies of the square lattice (red, dark) and a simple noninteracting system with on-site energies $-U/2$ and $U/2$ on a checkerboard (green, light) are plotted.