Analytical continuation of imaginary axis data for optical conductivity

We compare different methods for performing analytical continuation of spectral data from the imaginary time or frequency axis to the real frequency axis for the optical conductivity $\sigma \left(\omega \right)$. We compare the maximum entropy (MaxEnt), singular value decomposition (SVD), sampling, and Padé methods for analytical continuation. We also study two direct methods for obtaining $\sigma \left(0\right)$. For the MaxEnt approach we focus on a recent modification. The data are split up in batches, a separate MaxEnt calculation is done for each batch and the results are averaged. For the problems studied here, we find that typically the SVD, sampling, and modified MaxEnt methods give comparable accuracy while the Padé approximation is usually less reliable.

Figure 1

(Color online) The current-current correlation function $\pi \left(\nu \right)$ multiplied by ${\nu }^2$ as a function of $\nu$ as well as the function ${\pi }_0\left(\nu \right)$ calculated without vertex corrections. Results are shown for a large number $n\beta =160$ time slices as well as for a small $n\beta =60$, illustrating the resulting poor accuracy for large $\nu$ in the latter case. The figure also shows $\gamma \left(\nu \right)=\left[\pi \left(0\right)-\pi \left(\nu \right)\right]/\nu$ [Eq. (5)] and its extrapolation to $\nu =0$ (thin dotted line), which provides an estimate of $\sigma \left(0\right)$.

Figure 2

(Color online) The optical conductivity for model [Eq. (21)] using ${\Gamma }_1=0.60$ and ${\sigma }_0=0.01$ according to (a) the Padé, (b) the SVD method, (c) the sampling, and (d) the MaxEnt methods compared with the exact results. (a) and (c) show results for different values of the maximum frequency ${\nu }_{\text{max}}$ considered and (b) for different values of $n_{\nu }$. (d) shows results both for each individual sample (thin lines) and the average over all ten samples as well as the model used. The thick line in (b) indicates the optimum value of $n_{\nu }=5$ and the thick line in (c) the largest value of ${\nu }_{\text{max}}=30$ considered here. The $x$ in the main part of (c) shows the estimate of $\sigma \left(0\right)$ by extrapolating $\left[\Pi \left(0\right)-\Pi \left(\nu \right)\right]/\nu$ to zero. This is illustrated by the inset in (c), where the symbol $\times$ gives the exact value of $\sigma \left(0\right)$. The symbol $o$ in (d) is the estimate of $\sigma \left(0\right)$ based on $\Pi \left(\beta /2\right)$ [Eq. (6)].

Figure 3

(Color online) The same as Fig. 2 but for ${\Gamma }_1=0.3$ $\left({\sigma }_0=0.01\right)$.

Figure 4

(Color online) The same as Fig. 2 $\left({\Gamma }_1=0.6\right)$ but for ${\sigma }_0=0.001$.

Figure 5

(Color online) The same as Fig. 2 but for ${\Gamma }_1=0.3$ and ${\sigma }_0=0.001$.

Figure 6

(Color online) Comparison of the different methods for ${\Gamma }_1=0.6$ and 0.3 and for $\sigma =0.01$ and 0.001. The insets show a magnified view in the range $0\le \omega \le 0.25$.

Figure 7

(Color online) Difference between the spectrum calculated using one of the methods and the exact spectrum for the parameters ${\Gamma }_1=0.6$ and 0.3 and for $\sigma =0.01$ and 0.001.