Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1304.5198 · Journal of Experimental and Theoretical Physics Letters

Spectral analysis by the method of consistent constraints

Nikolay V. Prokof'ev, B. V. Svistunov

Published Aug 1, 2013• 26 citations
arXivHTMLPDF

Abstract

Two major challenges of numeric analytic continuation—restoring the spectral density, s (ω), from corresponding Matsubara correlator, g (τ)-are (i) producing the most smooth/featureless answer for s (ω), without compromising the error bars on g (τ), and (ii) quantifying possible deviations of the produced result from the actual answer. We introduce the method of consistent constraints that solves both problems.

DOI
10.1134/s002136401311009x

Related papers

Numerical analytic continuation: Answers to well-posed questions

Olga Goulko, Andrei S. Mishchenko, Lode Pollet +2 more

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.

arXiv:1609.01260•Physical review. B./Physical review. B•Jan 6, 2017•91 citations

Maximum entropy analytic continuation for frequency-dependent transport coefficients with nonpositive spectral weight

Alexis Reymbaut, Anne-Marie Gagnon, Dominic Bergeron +1 more

The computation of transport coefficients, even in linear response, is a major challenge for theoretical methods that rely on analytic continuation of correlation functions obtained numerically in Matsubara space. While maximum entropy methods can be used for certain correlation functions, this is not possible in general, important examples being the Seebeck, Hall, Nernst, and Reggi-Leduc coefficients. Indeed, positivity of the spectral weight on the positive real-frequency axis is not guaranteed in these cases. The spectral weight can even be complex in the presence of broken time-reversal symmetry. Various workarounds, such as the neglect of vertex corrections or the study of the infinite frequency or Kelvin limits, have been proposed. Here, we show that one can define auxiliary response functions that allow one to extract the desired real-frequency susceptibilities from maximum entropy methods in the most general multiorbital cases with no particular symmetry. As a benchmark case, we study the longitudinal thermoelectric response and corresponding Onsager coefficient in the single-band two-dimensional Hubbard model treated with dynamical mean-field theory and continuous-time quantum Monte Carlo. We thereby extend the maximum entropy analytic continuation with auxiliary functions (MaxEntAux method), developed for the study of the superconducting pairing dynamics of correlated materials, to transport coefficients.

arXiv:1612.02306•Physical review. B./Physical review. B•Mar 13, 2017•11 citations

Reconstruction of Spectral Densities from Incomplete Continued Fractions

Lecture notes in physics. M, Monographs•Jan 1, 1994•1 citation

Self-Consistent Treatment of Dynamical Correlation Functions Using a Spectral Representation Technique

Martin Letz, F. Marsiglio

arXiv:cond-mat/9905019•Journal of Low Temperature Physics•Oct 1, 1999•2 citations

Dynamical approach to analytic continuation of quantum Monte Carlo data

Mark Jarrell, Ofer Biham

A new method of analytic continuation from a Matsubara single-particle Green's function to a spectral function is presented. We recast this problem onto a new one where we dynamically minimize a suitably defined potential. Our method allows the imposition of physical constraints such as smoothness and adherence to the sum rule. The method is applied to the symmetric Anderson impurity model. We show how the spectral function changes with the Kondo temperature ${T}_{K}$, the hybridization witdth \ensuremath{\Delta}, and the Coulomb potential U.

Physical Review Letters•Nov 1, 1989•49 citations

Bogoliubov approximation as a second-order approximation in the spectral-density method.—II

Alvaro Caramico D’Auria, L. De Cesare, Ugo Esposito +1 more

Nuovo cimento della Società italiana di fisica. A, Nuclei, particles and fields•Oct 1, 1980•5 citations
Two major challenges of numeric analytic continuation—restoring the spectral density, s (ω), from corresponding Matsubara correlator, g (τ)-are (i) producing the most smooth/featureless answer for s (ω), without compromising the error bars on g (τ), and (ii) quantifying possible deviations of the produced result from the actual answer. We introduce the method of consistent constraints that solves both problems.
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.
The computation of transport coefficients, even in linear response, is a major challenge for theoretical methods that rely on analytic continuation of correlation functions obtained numerically in Matsubara space. While maximum entropy methods can be used for certain correlation functions, this is not possible in general, important examples being the Seebeck, Hall, Nernst, and Reggi-Leduc coefficients. Indeed, positivity of the spectral weight on the positive real-frequency axis is not guaranteed in these cases. The spectral weight can even be complex in the presence of broken time-reversal symmetry. Various workarounds, such as the neglect of vertex corrections or the study of the infinite frequency or Kelvin limits, have been proposed. Here, we show that one can define auxiliary response functions that allow one to extract the desired real-frequency susceptibilities from maximum entropy methods in the most general multiorbital cases with no particular symmetry. As a benchmark case, we study the longitudinal thermoelectric response and corresponding Onsager coefficient in the single-band two-dimensional Hubbard model treated with dynamical mean-field theory and continuous-time quantum Monte Carlo. We thereby extend the maximum entropy analytic continuation with auxiliary functions (MaxEntAux method), developed for the study of the superconducting pairing dynamics of correlated materials, to transport coefficients.
A new method of analytic continuation from a Matsubara single-particle Green's function to a spectral function is presented. We recast this problem onto a new one where we dynamically minimize a suitably defined potential. Our method allows the imposition of physical constraints such as smoothness and adherence to the sum rule. The method is applied to the symmetric Anderson impurity model. We show how the spectral function changes with the Kondo temperature TK{T}_{K}TK​, the hybridization witdth \ensuremath{\Delta}, and the Coulomb potential U.