Parquet dual fermion approach for the Falicov-Kimball model

In the Falicov-Kimball model, a model for (annealed) disorder, we expect weak localization corrections to the optical conductivity. However, we get such weak localization effects only when employing a $pp$-ladder approximation in the dual fermion approach. In the full parquet approach, these $pp$ contributions are suppressed by $ph$-reducible diagrams. For the optical conductivity, we find that the $\overline{ph}$ channel yields the main contribution, even in the region where weak localization in the $pp$ ladder was indicated.

Figure 1

The two-particle Green's function $G^{\left(2\right)}$ consists of two pairs of disconnected Green's function lines and the connected vertex function $F^{\nu {\nu }^{\prime }\omega }$. In this figure, only the frequency arguments are shown.

Figure 2

A two-particle reducible diagram can be assigned to exactly one of three channels, according to which two of the four outer legs of the diagram can be separated from the other two. In the particle-hole channel, ${\mathrm{\Phi }}_{ph}$ legs (1,2) are separated from (3,4), in the transversal particle-hole channel ${\mathrm{\Phi }}_{\overline{ph}}$ (2,3) are separated from (1,4), and in the particle-particle channel ${\mathrm{\Phi }}_{pp}$ (2,4) are separated from (1,3).

Figure 3

DMFT spectral function $A\left(\omega \right)$ for the half-filled system, at $U=0.5$, $U=0.9$, and $U=1.5$. At $U=0.5$ and $U=0.9$, the system is still metallic. With increasing $U$, a gap forms and at $U=1$ the metal-to-insulator transition takes place; the spectrum is split into two subbands. This can be seen at $U=1.5$, where the system is already insulating.

Figure 4

Imaginary part (above) and real part (below) of the self-energy at $U=0.5$ (left) and $U=1.5$ (right) at half filling and $T=0.06$, as resulting from DMFT (${\mathrm{\Sigma }}_{\mathrm{DMFT}}$) and from the parquet DF approach (${\mathrm{\Sigma }}_{\mathrm{DF}}$). Right inset: Brillouin zone with the Fermi surface at half filling (black line). The red crosses on the $6\times 6$ grid of $\mathbf{k}$ points denote the $\mathbf{k}$ points for which self-energies are shown in the main panel.

Figure 5

Imaginary part of the dual self-energy at $U=0.5$ (left) and $U=1.5$ (right) at $\mathbf{k}=(\pi ,0)$ resulting from the full parquet DF approach (blue), a $ph+\overline{ph}$- (black dashed) and a $pp$-ladder (red) approximation for the half-filled system at $T=0.06$. The $ph$ ladder results also containing the $\overline{ph}$ contributions provide a good approximation to the dual self-energy as obtained from the full parquet calculation.

Figure 6

Bubble term ${\sigma }_0$ and total optical conductivity $\sigma$ at $U=0.5$ (above) and $U=0.9$ (below) as resulting from the $pp$-ladder approximation at $T=0.1$, $T=0.06$, and $T=0.02$ for a half-filled system. The effect of weak localization is clearly visible when employing only the $pp$ ladder.

Figure 7

Top: Optical conductivity for real frequencies (main panel) and the corresponding current-current correlation function in Matsubara frequencies (insets) for the half-filled FKM calculated now from parquet DF at $U=0.5$, $T=0.1$, and $T=0.06$. Shown are again the bare bubble (${\sigma }_0$) and the full conductivity ($\sigma$), including vertex corrections. Insets: The respective current-current correlation function ${\chi }_{jj}^0$ and ${\chi }_{jj}$. Bottom: Corresponding vertex correction to the current-current correlation function ${\chi }_{jj}$ separated into $ph$, $\overline{ph}$, and $pp$ contributions. Additionally, the contribution of a $\overline{ph}$ and a $pp$ ladder are shown. As can be seen, the full parquet calculation shows even bigger effects compared to Fig. 6. These do not originate from the $pp$ channel, but the $\overline{ph}$ channel.

Figure 8

Same as Fig. 7, but for the insulating system at $U=1.5$.

Figure 9

Vertex corrections to the current-current correlation function ${\chi }_{jj}$ resulting from the $\overline{ph}$ channel (black) for representative momentum differences ${\mathbf{k}}^{\prime }-\mathbf{k}$ for the half-filled FKM at $U=0.5$ and $T=0.06$.

Figure 10

Charge susceptibility ${\chi }_d$ at ${\omega }_n=0$ for a half-filled FKM at $U=0.5$ and $T=0.06$, both in momentum space (left) and real space (right). In momentum space, it can be seen that the dominating value lies at $\mathbf{q}=(\pi ,\pi )$, which corresponds to the formation of the checkerboard structure which is also visible in real space.

Figure 11

DMFT spectral function $A\left(\nu \right)$ of the FKM at filling $n_c=0.15$ and $n_f=0.5$. For all three $U$ shown, the system remains metallic, even when the spectrum is split into two subbands at $U\ge 1$.

Figure 12

Imaginary part (left) and real part (right) of the self-energy at $U=0.5$ and $T=0.06$ obtained in DMFT (${\mathrm{\Sigma }}_{\mathrm{DMFT}}$) and parquet DF (${\mathrm{\Sigma }}_{\mathrm{DF}}$) for the FKM at filling $n_c=0.15$, $n_f=0.5$. The nonlocal corrections resulting from the parquet DF approach are smaller at this filling compared to the half-filled case in Fig. 4. Right inset: Brillouin zone with the Fermi surface in DMFT for the $c$-doped system (black line).

Figure 13

Imaginary part of the dual self-energy at $U=0.5$, $T=0.06$ at $\mathbf{k}=(\frac{\pi }{3},\frac{\pi }{3})$ as resulting from the full parquet DF approach, a $ph+\overline{ph}$- and $pp$-ladder approximation for an occupation $n_c=0.15$. As in the half-filled case (Fig. 5), the $ph$ ladder approximates the full parquet results very well, indicating the dominance of the $ph$ channel also in the doped FKM.

Figure 14

Bubble term ${\sigma }_0$ and total optical conductivity $\sigma$ at $U=0.5$ (above) and $U=0.9$ (below) as resulting from the $pp$-ladder approximation at $T=0.1$, $T=0.06$, and $T=0.02$ for the $c$-doped FKM, $n_c=0.15$. The effect of weak localization is clearly visible when employing only the $pp$ ladder.

Figure 15

Top: Optical conductivity for real frequency (main panel) and the corresponding current-current correlation function in Matsubara frequencies (insets) for the filling $n_c=0.15$ at $U=0.5$ and at $T=0.1$ and $T=0.06$, showing the bare bubble (${\sigma }_0$) and the full conductivity ($\sigma$), including vertex corrections (in the insets ${\chi }_{jj}^0$ and ${\chi }_{jj}$, respectively). Bottom: Corresponding vertex correction to the current-current correlation function ${\chi }_{jj}$ separated into $ph$, $\overline{ph}$, and $pp$ contributions. Also the contributions of a $\overline{ph}$ and a $pp$ ladder are shown. As can be seen, the full parquet calculation shows very different effects compared to Fig. 14.

Figure 16

Vertex corrections to the current-current correlation function ${\chi }_{jj}$ resulting from the $\overline{ph}$ channel in the parquet DF calculation (black) for different representative ${\mathbf{k}}^{\prime }-\mathbf{k}$ for the doped FKM at doping $n_c=0.15$, $n_f=0.5$, interaction $U=0.5$, and $T=0.06$.

Figure 17

Charge susceptibility ${\chi }_d$ at ${\omega }_n=0$ for the $c$-doped FKM at $n_c=0.15$, $U=0.5$, $T=0.06$, both in momentum space (left) and real space (right). The visible features with $\mathbf{q}=(\frac{\pi }{3},0)$ as the strongest point are not as distinctive as in the half-filled system (compare Fig. 10). In real space, ${\chi }_d$ consists mostly of the on-site correlation of a $c$ electron, as the amplitude decreases rapidly around the origin.