Numerical analytic continuation: Answers to well-posed questions

We formulate the problem of numerical analytic continuation in a way that lets us draw meaningful conclusions about the properties of the spectral function based solely on the input data. Apart from ensuring consistency with the input data (within their error bars) and the a priori and a posteriori (conditional) constraints, it is crucial to reliably characterize the accuracy—or even ambiguity—of the output. We explain how these challenges can be met with two approaches: stochastic optimization with consistent constraints and the modified maximum entropy method. We perform illustrative tests for spectra with a double-peak structure, where we critically examine which spectral properties are accessible (second peak position and its spectral weight) and which ones are lost (second peak width/shape). For an important practical example, we apply our protocol to the Fermi polaron problem.

Figure 1

Challenging example of a spectrum $A\left(z\right)$ for the NAC problem (2). As shown in the main part of the text, the significant width of the first peak makes it essentially impossible to controllably restore the width of the second peak, even with small relative error bars ($\sim$${10}^{-5}$) on $g_n$. On the other hand, the first two moments of the second peak, characterizing its weight and position, can be extracted reliably.

Figure 2

(a) Spectral function parametrization by a set of rectangles. (b) Illustration of the treatment of intersections of rectangles. (c) Identical reparametrization of the spectrum.

Figure 3

Results for test 1, featuring two peaks of finite width with noise level ${10}^{-3}$. Shown is the comparison between the actual spectrum (red solid line), the smooth SOCC spectrum (blue short-dashed line), and the pulled-up high-energy peak SOCC solution (green dashed line). The error bars for the smooth SOCC spectrum $\left\{{\sigma }_m\right\}$ are determined from Eq. (7).

Figure 4

Results for test 1, featuring two peaks of finite width with noise level ${10}^{-3}$. Shown is the comparison between the actual spectrum (red solid line), the MEM spectrum with a flat default model (blue short dashed line), and the MEM spectrum with a pulled-up high-energy peak in the default model (green dashed line).

Figure 5

Distribution of the second moment for the high-frequency peak in test 1 (short-dashed black line), test 2 (dashed blue line), and test 3 (red solid line), among all basic solutions. The vertical line shows the second moment ${\sigma }_0=0.064$ of the original spectrum.

Figure 6

Results for test 2, featuring two peaks of finite width with noise level ${10}^{-5}$. Shown is the comparison between the actual spectrum (red solid line) and the smooth SOCC spectrum (blue short-dashed line), the pulled-up high-energy peak SOCC solution (green dashed line). The error bars for smooth SOCC spectrum $\left\{{\sigma }_m\right\}$ determined from Eq. (7).

Figure 7

Results for test 2, featuring two peaks of finite width with noise level ${10}^{-5}$. Shown is the comparison between the actual spectrum (red solid line), MEM with a flat default model (blue short-dashed line), and MEM with a pulled-up high-energy peak in the default model (green dashed line).

Figure 8

Results for test 3, characterized by a $\delta$-function at low frequency and a peak of finite width at high frequency with noise level ${10}^{-5}$. Shown is the comparison between the actual spectrum (red solid line), the smooth SOCC spectrum (blue short-dashed line), and the maximally pulled-up high-energy peak SOCC solution (green dashed line). The error bars for smooth SOCC spectrum $\left\{{\sigma }_m\right\}$ determined from Eq. (7).

Figure 9

Results for test 3, characterized by a $\delta$-function at low frequency and a peak of finite width at high frequency with noise level ${10}^{-5}$. Shown is the comparison between the actual spectrum (red solid line), the MEM spectrum with a flat default model (blue short-dashed line), and the MEM spectrum with a double Gaussian default model where the second peak is maximally pulled up (green dashed line).

Figure 10

Results for test 4 with a Fermi distribution kernel and noise level ${10}^{-5}$. Shown is the comparison between the actual spectrum (red solid line), the smooth SOCC spectrum (blue short-dashed line), and the maximally pulled up central peak SOCC solution (dashed green line). The logarithmic plot in the inset highlights the comparison of low-intensity features. The error bars for the smooth SOCC spectrum $\left\{{\sigma }_m\right\}$ are determined from Eq. (7).

Figure 11

Results for test 4 with a Fermi distribution kernel and noise level ${10}^{-5}$. Shown is the comparison between the actual spectrum (red solid line) and the MEM spectrum in the default setup (blue dashed line).

Figure 12

Spectral density of the resonant Fermi polaron for $k_{F}a=0.8$ at zero momentum and zero temperature. The smooth SOCC spectrum with the second peak dispersion ${\sigma }_2=0.48$ is shown by the blue short dashed line. However, a much narrower solution for the second peak with ${\sigma }_2=0.12$ (green dashed line) can also be obtained from the same set of basic solutions. The MEM spectra with broad and narrow second peaks are shown by solid lines.