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:1012.5934 · Physical Review B

Analytical continuation of imaginary axis data for optical conductivity

O. Gunnarsson, M. W. Haverkort, Giorgio Sangiovanni

Published Oct 28, 2010• 66 citations
arXivHTMLPDF

Abstract

We compare different methods for performing analytical continuation of spectral data from the imaginary time or frequency axis to the real frequency axis for the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$. We compare the maximum entropy (MaxEnt), singular value decomposition (SVD), sampling, and Pad\'e methods for analytical continuation. We also study two direct methods for obtaining $\ensuremath{\sigma}(0)$. For the MaxEnt approach we focus on a recent modification. The data are split up in batches, a separate MaxEnt calculation is done for each batch and the results are averaged. For the problems studied here, we find that typically the SVD, sampling, and modified MaxEnt methods give comparable accuracy while the Pad\'e approximation is usually less reliable.

DOI
10.1103/physrevb.82.165125

Related papers

Analytical continuation of imaginary axis data using maximum entropy

O. Gunnarsson, M. W. Haverkort, Giorgio Sangiovanni

We study the maximum entropy (MaxEnt) approach for analytical continuation of spectral data from imaginary times to real frequencies. The total error is divided in a statistical error, due to the noise in the input data, and a systematic error, due to deviations of the default function, used in the MaxEnt approach, from the exact spectrum. We find that the MaxEnt approach in its classical formulation can lead to a nonoptimal balance between the two types of errors, leading to an unnecessary large statistical error. The statistical error can be reduced by splitting up the data in several batches, performing a MaxEnt calculation for each batch and averaging. This can outweigh an increase in the systematic error resulting from this approach. The output from the MaxEnt calculation can be used as a default function for a new MaxEnt calculation. Such iterations often lead to worse results due to an increase in the statistical error. By splitting up the data in batches, the statistical error is reduced and the increase resulting from iterations can be outweighed by a decrease in the systematic error. Finally we consider a linearized version to obtain a better understanding of the method.

arXiv:1001.4351•Physical Review B•Apr 9, 2010•47 citations

Analytical continuation of spectral data from imaginary time axis to real frequency axis using statistical sampling

Kiamars Vafayi, O. Gunnarsson

We present a method for performing analytical continuation of spectral data from imaginary time to real frequencies based on a statistical sampling method. Compared with the maximum entropy method (MEM), an advantage is that no default model needs to be introduced. For the problems studied here, the statistical sampling method gives comparable or slightly better results than MEM using quite accurate default models.

Physical Review B•Jul 19, 2007•46 citations

Analytical continuation approaches to electronic transport: The resonant level model

Eli Y. Wilner, Tal J. Levy, Eran Rabani

The analytical continuation average spectrum method (ASM) and maximum entropy (MaxEnt) method are applied to the dynamic response of a noninteracting resonant level model within the framework of the Kubo formula for electric conductivity. The frequency dependent conductivity is inferred from the imaginary time current-current correlation function for a wide range of temperatures, gate voltages, and spectral densities representing the leads, and compared with exact results. We find that the MaxEnt provides more accurate results compared to the ASM over the full spectral range.

arXiv:1205.5576•The Journal of Chemical Physics•Dec 5, 2012•4 citations

The simulation of electronic absorption spectrum of a chromophore coupled to a condensed phase environment: Maximum entropy versus singular value decomposition approaches

S. A. Egorov, Emilio Gallicchio, B. J. Berne

We consider the problem of calculating the electronic absorption spectrum of a chromophore with intramolecular degrees of freedom coupled to a condensed phase environment. We approach this calculation in the framework of the imaginary-time path integral Monte Carlo techniques, and focus on the problem of the analytic continuation of the imaginary-time data to the real-time axis. Two alternative analytic continuation methods are considered: the maximum entropy method and the singular value decomposition method. An exactly solvable model is introduced to test the accuracy of these methods. Exact numerical results for the absorption spectra are compared to the spectra reconstructed by the analytic continuation methods; it is found that the singular value decomposition method gives systematically higher resolution than the maximum entropy method and is capable of reproducing the fine vibronic structure of the absorption spectrum.

The Journal of Chemical Physics•Dec 8, 1997•39 citations

On the application of numerical analytic continuation methods to the study of quantum mechanical vibrational relaxation processes

Emilio Gallicchio, S. A. Egorov, B. J. Berne

A major problem still confronting molecular simulations is how to determine time-correlation functions of many-body quantum systems. In this paper the results of the maximum entropy (ME) and singular value decomposition (SVD) analytic continuation methods for calculating real time quantum dynamics from path integral Monte Carlo calculations of imaginary time time-correlation functions are compared with analytical results for quantum mechanical vibrational relaxation processes. This system studied is an exactly solvable system: a harmonic oscillator bilinearly coupled to a harmonic bath. The ME and SVD methods are applied to exact imaginary-time correlation functions with various level of added random noise, and also to imaginary-time data from path integral Monte Carlo (PIMC) simulations. The information gathered in the present benchmark study is valuable for the application of the analytic continuation of PIMC data to complex systems.

The Journal of Chemical Physics•Nov 8, 1998•48 citations

Comparison between methods of analytical continuation for bosonic functions

Johan Schött, Erik G. C. P. van Loon, Inka L. M. Locht +2 more

In this paper we perform a critical assessment of different known methods for the analytical continuation of bosonic functions, namely, the maximum entropy method, the non-negative least-squares method, the non-negative Tikhonov method, the Pad\'e approximant method, and a stochastic sampling method. Four functions of different shape are investigated, corresponding to four physically relevant scenarios. They include a simple two-pole model function; two flavors of the tight-binding model on a square lattice, i.e., a single-orbital metallic system and a two-orbital insulating system; and the Hubbard dimer. The effect of numerical noise in the input data on the analytical continuation is discussed in detail. Overall, the stochastic method by A. S. Mishchenko et al. [Phys. Rev. B 62, 6317 (2000)] is shown to be the most reliable tool for input data whose numerical precision is not known. For high-precision input data, this approach is slightly outperformed by the Pad\'e approximant method, which combines a good-resolution power with a good numerical stability. Although none of the methods retrieves all features in the spectra in the presence of noise, our analysis provides a useful guideline for obtaining reliable information of the spectral function in cases of practical interest.

arXiv:1607.04212•Physical review. B./Physical review. B•Dec 27, 2016•18 citations
We compare different methods for performing analytical continuation of spectral data from the imaginary time or frequency axis to the real frequency axis for the optical conductivity \ensuremathσ(\ensuremathω)\ensuremath{\sigma}(\ensuremath{\omega})\ensuremathσ(\ensuremathω). We compare the maximum entropy (MaxEnt), singular value decomposition (SVD), sampling, and Pad'e methods for analytical continuation. We also study two direct methods for obtaining \ensuremathσ(0)\ensuremath{\sigma}(0)\ensuremathσ(0). For the MaxEnt approach we focus on a recent modification. The data are split up in batches, a separate MaxEnt calculation is done for each batch and the results are averaged. For the problems studied here, we find that typically the SVD, sampling, and modified MaxEnt methods give comparable accuracy while the Pad'e approximation is usually less reliable.
We study the maximum entropy (MaxEnt) approach for analytical continuation of spectral data from imaginary times to real frequencies. The total error is divided in a statistical error, due to the noise in the input data, and a systematic error, due to deviations of the default function, used in the MaxEnt approach, from the exact spectrum. We find that the MaxEnt approach in its classical formulation can lead to a nonoptimal balance between the two types of errors, leading to an unnecessary large statistical error. The statistical error can be reduced by splitting up the data in several batches, performing a MaxEnt calculation for each batch and averaging. This can outweigh an increase in the systematic error resulting from this approach. The output from the MaxEnt calculation can be used as a default function for a new MaxEnt calculation. Such iterations often lead to worse results due to an increase in the statistical error. By splitting up the data in batches, the statistical error is reduced and the increase resulting from iterations can be outweighed by a decrease in the systematic error. Finally we consider a linearized version to obtain a better understanding of the method.
The analytical continuation average spectrum method (ASM) and maximum entropy (MaxEnt) method are applied to the dynamic response of a noninteracting resonant level model within the framework of the Kubo formula for electric conductivity. The frequency dependent conductivity is inferred from the imaginary time current-current correlation function for a wide range of temperatures, gate voltages, and spectral densities representing the leads, and compared with exact results. We find that the MaxEnt provides more accurate results compared to the ASM over the full spectral range.
We consider the problem of calculating the electronic absorption spectrum of a chromophore with intramolecular degrees of freedom coupled to a condensed phase environment. We approach this calculation in the framework of the imaginary-time path integral Monte Carlo techniques, and focus on the problem of the analytic continuation of the imaginary-time data to the real-time axis. Two alternative analytic continuation methods are considered: the maximum entropy method and the singular value decomposition method. An exactly solvable model is introduced to test the accuracy of these methods. Exact numerical results for the absorption spectra are compared to the spectra reconstructed by the analytic continuation methods; it is found that the singular value decomposition method gives systematically higher resolution than the maximum entropy method and is capable of reproducing the fine vibronic structure of the absorption spectrum.
A major problem still confronting molecular simulations is how to determine time-correlation functions of many-body quantum systems. In this paper the results of the maximum entropy (ME) and singular value decomposition (SVD) analytic continuation methods for calculating real time quantum dynamics from path integral Monte Carlo calculations of imaginary time time-correlation functions are compared with analytical results for quantum mechanical vibrational relaxation processes. This system studied is an exactly solvable system: a harmonic oscillator bilinearly coupled to a harmonic bath. The ME and SVD methods are applied to exact imaginary-time correlation functions with various level of added random noise, and also to imaginary-time data from path integral Monte Carlo (PIMC) simulations. The information gathered in the present benchmark study is valuable for the application of the analytic continuation of PIMC data to complex systems.
In this paper we perform a critical assessment of different known methods for the analytical continuation of bosonic functions, namely, the maximum entropy method, the non-negative least-squares method, the non-negative Tikhonov method, the Pad\'e approximant method, and a stochastic sampling method. Four functions of different shape are investigated, corresponding to four physically relevant scenarios. They include a simple two-pole model function; two flavors of the tight-binding model on a square lattice, i.e., a single-orbital metallic system and a two-orbital insulating system; and the Hubbard dimer. The effect of numerical noise in the input data on the analytical continuation is discussed in detail. Overall, the stochastic method by A. S. Mishchenko et al. [Phys. Rev. B 62, 6317 (2000)] is shown to be the most reliable tool for input data whose numerical precision is not known. For high-precision input data, this approach is slightly outperformed by the Pad\'e approximant method, which combines a good-resolution power with a good numerical stability. Although none of the methods retrieves all features in the spectra in the presence of noise, our analysis provides a useful guideline for obtaining reliable information of the spectral function in cases of practical interest.