Numerical analytic continuation: Answers to well-posed questions

Abstract

We formulate the problem of numerical analytic continuation in a way that lets us draw meaningful conclusions about 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.

Generalized method of Feynman-Pines diagram technique in the theory of energy spectrum of two-level quasiparticle renormalized due to multi-phonon processes at cryogenic temperature

“…The idea of a correct method of solution appeared at the base of the known new approaches, which were proposed and used in the theory of polarons and high-temperature superconductivity [17][18][19][20][21][22][23][24][25][26][27]. The diagrammatic Monte Carlo method and stochastic optimization (DMC) [17][18][19][20][21][22][23][24][25][26][27] and bold diagrammatic Monte Carlo (BDMC) [28,29] played an important role in investigation of high excited states and revealed the reasons of some questionable results obtained for the electron-phonon interaction (for example, in the concept of a relaxed excited state). These methods are essentially based on a computer algorithm of the calculation of high-order diagrams of mass operators of Matsubara Green's functions.…”

Polaron Mobility in the “Beyond Quasiparticles” Regime

“…The main difference is that we extract µ(ω) from the data parameterized by the Matsubara frequency rather than imaginary time. We also employ more conservative protocols for computing both the mobility and its errorbars from multiple solutions of the stochastic optimization with the consistent constraints method (see Supplemental material) [14,16].…”

Dielectric function and thermodynamic properties of jellium in the GW approximation

“…2 and 3 we show how our protocol works in practice by considering the case of r s = 1 at low temperature T / F = 0.02 and small momentum k/k F = 0.1. First, we performed analytic continuation of the imaginary frequency data for (k, ω n ) using a hybrid of stochastic optimization [17,18] and consistent constraints [19,20] methods to get the imaginary part (k, ω). Next, the real part (k, ω) is obtained from the Kramers-Kronig relation.…”