Max-Planck-Institut für Festkörperforschung
Andersen Group El.-Phon. QMC C60 Resistivity saturation

Analytical continuation of spectral functions from imaginary axis data

For strongly correlated systems analytical methods usually involve uncontrolled approximations. Therefore stochastical methods such as quantum Monte-Carlo (QMC) methods are often used. Apart from statistical errors, these methods can produce essentially exact results, but the results are obtained on the imaginary axis. This leaves the problem of analytically continuing the results to the real axis, which is an ill-posed problem. Small changes in the data on the imaginary axis can lead to large changes on the real axis. Since the imaginary axis data contain statistical noise, the analytical continuation is very difficult.
One method for treating this problem is to combine the Bayesian theory with the maximum entropy approach (MaxEnt). Although this method often works quite well, it sometimes puts too much emphasize on the noise, leading to unphysical structures. We have shown how this problem can be removed by a "batching" method. The samples of the imaginary axis data obtained from a QMC calculation are split up into batches. Instead of doing a MaxEnt calculation using all the samples, we perform a MaxEnt calculation for each batch and then average the resulting spectra. This strongly reduces the influence of the statistical noise at the cost of increasing a systematic error. We discuss how to optimize the number of batches ( Phys. Rev. B 81, 155107 (2010) ).
We have developed an alternative method based on a statistical sampling over the space of real frequency spectral functions weighted by a likelyhood function (Phys. Rev. B 76, 035115 (2007)). This method gives spectra of a comparable quality as the MaxEnt approach.
Other alternative methods for analytically continuing data have been proposed. These include the Pade approximation and the singular value decomposition (SVD). We have compared these methods with with the modified MaxEnt and sampling methods described above for analytically continuing optical conductivity data. We find that the SVD method gives comparable accuracy as the modified MaxEnt and sampling methods while the Pade approximation is usually less reliable ( Phys. Rev. B 82 , 165125 (2010)).


K. Vafayi and O. Gunnarsson:
Analytical continuation of spectral data from imaginary time axis to real frequency axis using statistical sampling,
Phys. Rev. B i76, 035115 (2007).

O. Gunnarsson, M.W. Haverkort, G. Sangiovanni:
Analytical continuation of imaginary axis data using maximum entropy,
Phys. Rev. B 81, 155107 (2010) .

O. Gunnarsson, M.W. Haverkort, G. Sangiovanni:
Analytical continuation of imaginary axis data for optical conductivity,
Phys. Rev. B 82 , 165125 (2010).

For further information contact Olle Gunnarsson (

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Last Update: September 2012
Andersen Group