Analytic continuation

The goal for the analytic continuation¹⁹ is to find a suitable trial solution for the dynamic structure factor Strial(q,ω) that, when it is being inserted into Eq. (³) of the main text, reproduces the PIMC data for the imaginary-time correlation function F(q,τ) for all τ∈[0,β]. Yet, these constraints are typically not sufficient to fully constrain the space of possible solutions of S(q,ω), and additional information are needed^39,72. For this purpose, we consider the frequency moments of the dynamic structure factor, which are defined as

Two moments are known from the corresponding sum rules: (1) the first moment is determined by the well-known f-sum rule⁵⁴

and the inverse moment is given by⁷³

In addition, the zero-moment is automatically satisfied, since it holds

The final exact property of S(q,ω) that we consider in this work is the detailed balance relation between positive and negative frequencies⁷⁴,

which is automatically fulfilled by our trial solutions Strial(q,ω).

Many practical approaches have been suggested to accomplish goals of the analytic continuation. One family of methods is based on Bayes’ theorem, and is capable of producing smooth solutions without any unphysical sawtooth instabilities¹⁹. Yet, such maximum entropy methods²¹ might introduce an artificial a-priori bias into the solutions, although notable progress is continually being made in this area^20,75. A second paradigm for the analytic continuation is based on the averaging over NS∼103–104 noisy, independent trial solutions to compose a smooth result^22,58,65. While computationally more demanding, this method has the advantage that unexpected physical features of S(q,ω) might still be recovered because no explicit bias is introduced into the solution.

In the present work, we pursue the latter strategy and employ a genetic algorithm that maximizes the fitness function,

We note that F(q,τ) is symmetric around β/2 so that we only have to consider the interval τ∈[0,β/2]. Furthermore, the weights aχ and a1 control the respective influence of the individual constraints and are chosen empirically. We find that a reasonable choice is given by aχ=a1=1/2, which means that the frequency moments are of the same importance as F(q,τ). The f-sum rule, Eq. (⁸), is actually known exactly, and we empirically set Δω1=ω1×10-3.

A practical example for the analytical continuation is shown in the left panel of Fig. 8 for the case considered in Fig. 4b of the main text. The red (blue) curve has been composed by averaging over Ns∼102 (Ns∼104) individual noisy trial solutions Strial,i(q,ω). A frequency grid with δω≈0.6K and Nω=400 points was employed.

Left: Averaging of the dynamic structure factor S(q,ω) for T=1.2K and q=1.62 Å-1. Red (blue) curve: average of Ns∼102 (Ns∼104) noisy individual solutions. Right: Comparing PIMC results for F(q,τ) (black) and the reconstructed solution (red) for the same case as in Fig. 8. Note that F(q,τ) is symmetric with respect to β/2, i.e., F(q,τ)=F(q,β-τ).

To conclude this discussion, we consider the main input of the analytic continuation method: the τ-dependence of the imaginary-time correlation function F(q,τ) for a specific wave number. This is shown for the present example in the right panel of Fig. 8, with the black stars and red line corresponding to the original PIMC results and the reconstructed solution, respectively. Evidently, the latter curve perfectly agrees with the PIMC data within the given Monte Carlo error bars that are on the order of ΔF∼10-3.

All S(q,ω) results in this work have been obtained with the procedure that we just described but a note of caution is warranted. Despite being defined as an average value, it is not straightforward to estimate the true uncertainty in the final result for the reconstructed solution for S(q,ω). Rather, the space of possible trial solutions remains a property of a specific optimization method (genetic algorithm, simulated annealing, etc.) and might, or might not, include the true, physical curve with finite probability. Therefore, we abstain from using the variance of the trial solutions as an error bar, as they might underestimate the true uncertainties.