Comparison between methods of analytical continuation for bosonic functions

In this paper we perform a critical assessment of different known methods for the analytical continuation of bosonic functions, namely, the maximum entropy method, the non-negative least-squares method, the non-negative Tikhonov method, the Padé approximant method, and a stochastic sampling method. Four functions of different shape are investigated, corresponding to four physically relevant scenarios. They include a simple two-pole model function; two flavors of the tight-binding model on a square lattice, i.e., a single-orbital metallic system and a two-orbital insulating system; and the Hubbard dimer. The effect of numerical noise in the input data on the analytical continuation is discussed in detail. Overall, the stochastic method by A. S. Mishchenko et al. [Phys. Rev. B 62, 6317 (2000)] is shown to be the most reliable tool for input data whose numerical precision is not known. For high-precision input data, this approach is slightly outperformed by the Padé approximant method, which combines a good-resolution power with a good numerical stability. Although none of the methods retrieves all features in the spectra in the presence of noise, our analysis provides a useful guideline for obtaining reliable information of the spectral function in cases of practical interest.

Figure 1

Schematic illustration of the analytic continuation of a bosonic response function $\chi \left(z\right)$. The values $\chi \left(i{\omega }_n\right)$ at the Matsubara frequencies $i{\omega }_n$ are used to reconstruct the function $\chi (E+i{\delta }^{+})$ just above the axis of real energies.

Figure 2

Spectra for the two-pole test function. Left: $a_1=0.1,a_2=0.335663,E_1=0.7$, and $E_2=1.2$; right: $a_1=0.1,a_2=0.7,E_1=0.7$, and $E_2=2.5$, as defined in Eq. (9).

Figure 3

Integrated real-axis error for the two-pole model as a function of the highest Matsubara point index $n_{\text{max}}$ used. Top: $a_1=0.1,a_2=0.335663,E_1=0.7$ and $E_2=1.2$; bottom: $a_1=0.1,a_2=0.7,E_1=0.7$, and $E_2=2.5$, as defined in Eq. (9). Matsubara noise level $\sigma ={10}^{-4}$ is used.

Figure 4

Spectrum of the Lindhard function $\left(-\frac{1}{\pi }\text{Im}\left[\chi (\omega +i\delta )\right]\right)$ of the two-dimensional doped tight-binding model along the Brillouin zone path $\mathrm{\Gamma }\to X\to M\to \mathrm{\Gamma }$, obtained using Padé, NNLS, NNT, MEM, and Mishchenko's methods. The noise level varies among the values $\sigma ={10}^{-3},\sigma ={10}^{-4}$, and $\sigma ={10}^{-10}$. All panels on the right-hand side show the exact spectrum for an easier comparison. The color map denotes the intensity of the spectra, and contour lines are added at $-1/\pi \text{Im}\left(\chi \right)=0,0.15,0.30,0.45,0.60,0.75$. For the NNLS plots not all contour lines are displayed (see main text). The spectra for the $M$ point separately are also reported in Fig. 5.

Figure 5

Spectrum at the $M$ point for the Lindhard function of the doped tight-binding model reported in Fig. 4. Data for two noise levels, $\sigma ={10}^{-4}$ and $\sigma ={10}^{-10}$, are shown.

Figure 6

Spectrum of the Lindhard function for the band gap model $\left(-\frac{1}{\pi }\text{Im}\left[\chi (\omega +i\delta )\right]\right)$ along the Brillouin zone path $\mathrm{\Gamma }\to X\to M\to \mathrm{\Gamma }$, obtained using Padé, NNLS, NNT, MEM, and Mishchenko's method. The noise level varies among the values $\sigma ={10}^{-3},\sigma ={10}^{-4}$, and $\sigma ={10}^{-10}$. All right panels show the exact spectrum. The color map denotes the intensity of the spectra, and contour lines are added at $-1/\pi \text{Im}\left(\chi \right)=0,0.1,0.2,0.3,0.4,0.5,0.6$. The spectra at the $M$ point and at the $\frac{1}{2}(\mathrm{\Gamma }\to X)$ point are reported separately in Fig. 7.

Figure 7

Spectrum at (a) the $\frac{1}{2}(\mathrm{\Gamma }\to X)$ and (b) $M$ points of the Lindhard function for the band gap model in Fig. 6 for two noise levels. $\sigma ={10}^{-4}$ and $\sigma ={10}^{-10}$.

Figure 8

The noise characteristics of the input data for the analytical continuations for the Hubbard dimer. (a) The standard deviation of the average QMC data between different simulations normalized by the average values (blue) compared to the relative Gaussian noise that is chosen to have $\sigma =8\times {10}^{-3}$ (red). (b) Correlation matrix (normalized covariance matrix) of the QMC data. The color coding denotes the amount of correlation between two Matsubara frequencies $i,j$.

Figure 9

Spectra for the Hubbard dimer obtained with different methods of analytic continuation. The blue spectra are obtained from the exact data on the Matsubara axis with added relative Gaussian noise $\sigma =8\times {10}^{-3}$. The red and green spectra are obtained from QMC data, where for the red spectra we used the Cholesky decomposition of the inverse covariance matrix to consider the frequency-dependent standard deviation and the correlation between different input data points [39, 44]. The exact solution is given in black.

Figure 10

The dependence on the number of iterations (see legend) performed with the Mishchenko method. (a) The two-pole function in the presence of input noise with $\sigma ={10}^{-4}$. (b) The band gap model at the $k$ point $\frac{1}{2}(\mathrm{\Gamma }\to X)$ for two different noise levels of $\sigma ={10}^{-3}$ (left) and $\sigma ={10}^{-4}$ (right). (c) The doped tight-binding model at the $M$ point and Matsubara noise level $\sigma ={10}^{-4}$.

Figure 11

Integrated real-axis error for the two-pole model as a function of the highest Matsubara point index $n_{\text{max}}$ used in Mishchenko's method. Different numbers of local iterations are investigated and are indicated by the legend. Top: $a_1=0.1,a_2=0.335663,E_1=0.7$, and $E_2=1.2$; bottom: $a_1=0.1,a_2=0.7,E_1=0.7$, and $E_2=2.5$, as defined in Eq. (9). Matsubara noise level $\sigma ={10}^{-4}$ is used.