$F$-electron spectral function of the Falicov-Kimball model in infinite dimensions: The half-filled case

The $f$-electron spectral function of the Falicov-Kimball model is calculated via a Keldysh-based many-body formalism originally developed by Brandt and Urbanek. We provide results for both the Bethe lattice and the hypercubic lattice at half filling. Since the numerical computations are quite sensitive to the discretization along the Kadanoff-Baym contour and to the maximum cutoff in time that is employed, we analyze the accuracy of the results using a variety of different moment sum rules and spectral formulas. We find that the $f$-electron spectral function has interesting temperature dependence, becoming a narrow single-peaked function for small $U$ and developing a gap, with two broader peaks for large $U$.

Figure 1

Kadanoff-Baym contour for evaluating the equilibrium Green’s function for $t⩽t^{\prime }$. For $t⩾t^{\prime }$ the contour starts at $t^{\prime }$ and runs to $t$, then goes back to $t^{\prime }$ and ends at $t^{\prime }-i\beta$.

Figure 2

Real part of the (greater) $f$-electron Green’s function as a function of time for $T=1$ and five different values of $U$ on the Bethe lattice.

Figure 3

Logarithm of the absolute value of the real part of the (greater) $f$-electron Green’s function as a function of time for $T=0.2$ and $U=1.5$ on the Bethe lattice. Three different values of $\Delta t_{\mathrm{real}}$ are shown.

Figure 4

$f$-electron DOS for different discretization sizes. Also plotted is the $\delta$-extrapolated result using a three-point (quadratic) Lagrange interpolation formula. The parameters are $U=1.5$ on the Bethe lattice, with $T=5.0$.

Figure 5

$f$-electron DOS for different discretization sizes. Also plotted is the $\delta$-extrapolated result using a linear interpolation formula for the smallest two $\Delta t_{\text{real}}$ values and the Matsubara extrapolated result. The parameters are $U=1.5$ on the Bethe lattice, with $T=0.1$.

Figure 6

$f$-electron DOS for different temperatures. Also plotted is the conduction-electron DOS (which is temperature independent). The parameters are $U=1.5$ on the Bethe lattice. Inset is a plot of $1∕A_f(\omega =0)$ versus $T$. Note how it appears to behave linearly at small $T$, allowing us to extrapolate to the $T=0$ result, so we can predict the maximal height of the $f$-electron DOS at $T=0$.

Figure 7

$f$-electron DOS for different temperatures with $U=2.5$ on the Bethe lattice. Also plotted is the conduction-electron DOS (which is temperature independent). The $T=5$ data uses $\delta$ extrapolation, the $T=1$ and 0.6 data use the Matsubara-extrapolation procedure, and the lower temperatures are not extrapolated (but have $\Delta t_{\text{real}}=0.05$). Note how the $f$-electron DOS develops a gap as $T$ is lowered. Note further that a kink starts to develop near the upper and lower band edges of the conduction DOS as expected too. Our computational accuracy is worst for the subgap DOS at low temperature.

Figure 8

$f$-electron DOS for different temperatures with $U=1$ on the hypercubic lattice. Also plotted is the conduction-electron DOS. All of the data were extrapolated with one of the two extrapolation techniques discussed in the text. Note the similarity with Fig. 6 for the Bethe lattice. In the inset we plot the inverse of the DOS at the chemical potential. Here, the low-temperature results don’t appear to behave in quite the linear fashion we saw on the Bethe lattice, but we can still attempt to extrapolate to $T=0$ with the prediction that the peak in the DOS will also be around 4.5 at $T=0$.

Figure 9

$f$-electron DOS for different temperatures with $U=2$ on the hypercubic lattice. Also plotted is the conduction-electron DOS. The data are either extrapolated with one of the two extrapolation techniques discussed in the text, or we work with a fixed value of the discretization on the real-time axis. Note the similarity with Fig. 7 for the Bethe lattice.

Figure 10

$f$-electron DOS for different temperatures with $U=4$ on the hypercubic lattice (the $T=5$ and $T=2$ data overlap). Also plotted is the conduction-electron DOS. The $T=5$ data are calculated with the $\delta$-extrapolation technique; all other temperatures work with $\Delta t_{\text{real}}=0.05$. These results are similar to those found for the Bethe lattice (Ref. 10), but for the Bethe lattice the difference between the $f$-electron DOS and the conduction electron DOS is more dramatic.