Fast and efficient stochastic optimization for analytic continuation

The analytic continuation of imaginary-time quantum Monte Carlo data to extract real-frequency spectra remains a key problem in connecting theory with experiment. Here we present a fast and efficient stochastic optimization method (FESOM) as a more accessible variant of the stochastic optimization method introduced by Mishchenko et al. [Phys. Rev. B 62, 6317 (2000)], and we benchmark the resulting spectra with those obtained by the standard maximum entropy method for three representative test cases, including data taken from studies of the two-dimensional Hubbard model. We generally find that our FESOM approach yields spectra similar to the maximum entropy results. In particular, while the maximum entropy method yields superior results when the quality of the data is strong, we find that FESOM is able to resolve fine structure with more detail when the quality of the data is poor. In addition, because of its stochastic nature, the method provides detailed information on the frequency-dependent uncertainty of the resulting spectra, while the maximum entropy method does so only for the spectral weight integrated over a finite frequency region. We therefore believe that this variant of the stochastic optimization approach provides a viable alternative to the routinely used maximum entropy method, especially for data of poor quality.

Figure 1

Example 1. The synthetic spectral function $A\left(\omega \right)$ (black line) is used to generate different samples for the input data $G\left(i{\omega }_n\right)$ using Eq. (1) and compared to the results of 5 (a) and 100 (b) independent realizations of stochastic optimal spectral function samples (blue dashes curves). Here we have used a ${\chi }^2$ threshold $\epsilon =0.05$.

Figure 2

Example 1. Averaged spectrum $\overline{A}\left(\omega \right)$ obtained from the stochastic optimization with 500 realizations and noncorrelated noise proposals ($\alpha =0$, magenta dash-dotted line) compared with the result from 100 realizations and noncorrelated noise proposals ($\alpha =0$, green dash-dotted line) and 100 realizations and correlated noise proposals ($\alpha =0.5$, blue dash-dotted line). Here we have used a ${\chi }^2$ threshold $\epsilon =0.05$.

Figure 3

Example 1, case 1. Comparison between the true spectrum $\overline{A}\left(\omega \right)$ (black solid curve) with the results of FESOM (blue dash-dotted line), using a ${\chi }^2$ threshold $\epsilon =0.05$, and MaxEnt (red dash-dotted line) for low-quality data. Here we have used a noise amplitude of 0.1 to generate the 1000 samples for the input data.

Figure 4

Example 1, case 2. Comparison between the true spectrum $\overline{A}\left(\omega \right)$ (black solid curve) with the results of FESOM (blue dash-dotted line), using a ${\chi }^2$ threshold $\epsilon =0.05$, and MaxEnt (red dash-dotted line) for high-quality data. Here we have used a noise amplitude of 0.001 to generate the 1000 samples for the input data.

Figure 5

Example 1. True spectrum $\overline{A}\left(\omega \right)$ (black solid line), FESOM simulated spectrum $\overline{A}\left(\omega \right)$ (blue dash-dotted line), and 95% confidence band in FESOM (green dash-dotted line). Here we have used a ${\chi }^2$ threshold $\epsilon =0.05$.

Figure 6

Example 2. Comparisons between the true spectrum (black solid line) and the results of MaxEnt (a) and FESOM (b), for which we have used a ${\chi }^2$ threshold $\epsilon =0.001$. Panel (c) shows the 95% confidence band obtained in the FESOM. The real spectral function is obtained from a DMFT/NCA calculation of a 2D Hubbard model in the antiferromagnetic state with $U=16t, \langle n\rangle =0.95$, and $T=0.29$.

Figure 7

Example 3. Spectral function $A\left(\omega \right)$ obtained from MaxEnt (a) and FESOM (b) (using a ${\chi }^2$ threshold of $\epsilon =0.001$) analytic continuation of DCA QMC data for a 2D Hubbard model with $U=8t, \langle n\rangle =0.95$, and $T=0.08t$.