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 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) ).

Many alternative methods for analytically continuing data have been proposed. These include the Pade approximation, singular value decomposition (SVD) and sampling methods. We have compared these methods with with the modified MaxEnt method described above for analytically continuing optical conductivity data. We find that the SVD and sampling methods give comparable accuracy as the modified MaxEnt method described above ( Phys. Rev. B , 165125 (2010)).

Publications:

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

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

For further information contact Olle Gunnarsson (gunnar@fkf.mpg.de).

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