Analytic continuation by averaging Padé approximants

The ill-posed analytic continuation problem for Green's functions and self-energies is investigated by revisiting the Padé approximants technique. We propose to remedy the well-known problems of the Padé approximants by performing an average of several continuations, obtained by varying the number of fitted input points and Padé coefficients independently. The suggested approach is then applied to several test cases, including Sm and Pr atomic self-energies, the Green's functions of the Hubbard model for a Bethe lattice and of the Haldane model for a nanoribbon, as well as two special test functions. The sensitivity to numerical noise and the dependence on the precision of the numerical libraries are analyzed in detail. The present approach is compared to a number of other techniques, i.e., the nonnegative least-squares method, the nonnegative Tikhonov method, and the maximum entropy method, and is shown to perform well for the chosen test cases. This conclusion holds even when the noise on the input data is increased to reach values typical for quantum Monte Carlo simulations. The ability of the algorithm to resolve fine structures is finally illustrated for two relevant test functions.

Figure 1

Typical example of analytic continuation of the Green's function in the complex plane. The Matsubara frequencies on the imaginary axis are labeled following the convention explained in Sec. 2.

Figure 2

Illustration of the different improvements done on plain Padé for a Sm atom, with relative noise of magnitude $\sigma ={10}^{-6}$ on the Matsubara points. Description of the five obtained spectra (from left to right panel): Plain Padé, using $N=M=70,n_0=1$, and characterized by unphysical features; Padé average with $w_c^D$ and $n_0=1$; Padé average with $w_c^{LS}$ and $n_0=1$; Padé average with $w_c^S$ and $n_0=1$; Padé average with $w_c^S$ and with mirror symmetry imposed by using $n_0\in \left\{-5:1:0\right\}$.

Figure 3

Illustration of configurations in $(N,M)$ space with $n_0=1$ for the self-energy of a Sm atom, with relative noise of magnitude $\sigma ={10}^{-6}$ on Matsubara points. A black cross denotes an unphysical, ${\rho }_c<0$, continuation. A circular dot denotes a physical continuation, ${\rho }_c\ge 0$, and the color denotes the deviation ${\mathrm{\Delta }}_c$.

Figure 4

Average error $\langle E\rangle$ versus standard deviation $\sigma$ of the numerical noise. The error deviation ${\sigma }_E$ (see main text) is represented as error bars. The five circles in the figure indicate those continuations corresponding to the spectra reported in Fig. 2. Top panel: Analytic continuations performed with $n_0=1$. Bottom panel: Analytic continuations performed with $n_0\in \left\{-5:1:0\right\}$, except for plain Padé in red where $n_0=-2$ is used.

Figure 5

In the bottom panel, the real-axis error $\langle E\rangle$ as a function of input noise $\sigma$ is reported for various precisions of the numerical routines (MPACK library). ${\sigma }_E$ is represented by the error bars. The circles indicate continuations whose spectra are plotted in the top panel, using the same lines as in the bottom panel, and shifted with an offset for a better visualization. Exact results are also reported (lowest curve, in gray).

Figure 6

Comparison of the Padé scheme with three other well-known methods for analytic continuation. In the bottom panels, the real-axis error $\langle E\rangle$ as a function of the Matsubara noise $\sigma$ is shown. The errors bars represent ${\sigma }_E$. The circles indicate continuations whose spectra are plotted in the top panels. Spectra shifted with an offset for a better visualization. Exact results are also reported (lowest curves, in gray). In the middle panels the energy-resolved errors of the spectra in the top panels with respect to the exact result are reported.

Figure 7

Comparing the Padé scheme with three other well-known methods for the analytic continuation of the noninteracting Green's function of the Hubbard model on a Bethe lattice with infinite nearest neighbors [3]. In the bottom panel, the real-axis error $\langle E\rangle$ as a function of the Matsubara noise $\sigma$ is shown. The error bars represent ${\sigma }_E$. The circles indicate continuations whose spectra are plotted in the top panel, using the same lines as in the bottom panel, and shifted with an offset for a better visualization. Exact spectrum is also reported (lowest curve, in gray).

Figure 8

Comparing the Padé scheme with three other well-known methods for the analytic continuation of the noninteracting Green's function of a nanoribbon. In the bottom panel, the real-axis error $\langle E\rangle$ as a function of the Matsubara noise $\sigma$ is shown. The error bars represent ${\sigma }_E$. The circles indicate continuations whose spectra are plotted in the top panel, using the same lines as in the bottom panel. Exact spectrum is also reported (in gray).

Figure 9

Left panel: Highest possible noise magnitude to resolve the two-peak structure in Eq. (19) as a function of the center position ${\omega }_0$. Results for the Padé scheme as well as NNLS, NNT, and MEM are reported. Right panel: Same as left panel but for a function with an additional peak at zero energy.

Figure 10

Real-axis error as a function of the residual static self-energy $\mathrm{\Delta }{\mathrm{\Sigma }}_0$. Both positive and negative $\mathrm{\Delta }{\mathrm{\Sigma }}_0$ values are considered, hence the two lines per method. The test is for the self-energy of a Sm atom with double-precision Matsubara data.