Numerical analytic continuation

In many-body physics, the problem of analytic continuation is that of numerically extracting the spectral density of a Green function given its values on the imaginary axis. It is a necessary post-processing step for calculating dynamical properties of physical systems from quantum Monte Carlo simulations, which often compute Green function values only at imaginary-times or Matsubara frequencies. Mathematically, the problem reduces to solving a Fredholm integral equation of the first kind with an ill-conditioned kernel. As a result, it is an ill-posed inverse problem with no unique solution and where a small noise on the input leads to large errors in the unregularized solution. There are different methods for solving this problem including the maximum entropy method,[1][2][3][4] the average spectrum method[5][6][7][8] and Pade approximation methods.[9][10]

## Contents

* 1 Examples
* 2 Software
* 3 See also
* 4 References

## Examples

A common analytic continuation problem is obtaining the spectral function [math]\displaystyle{ A(\omega) }[/math] at real frequencies [math]\displaystyle{ \omega }[/math] from the Green function values [math]\displaystyle{ \mathcal{G}(i\omega_n) }[/math] at Matsubara frequencies [math]\displaystyle{ \omega_n }[/math] by numerically inverting the integral equation

[math]\displaystyle{ \mathcal{G}(i\omega_n) = \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} \frac{1}{i\omega_n - \omega}\; A(\omega) }[/math]

where [math]\displaystyle{ \omega_n = (2n+1) \pi/\beta }[/math] for fermionic systems or [math]\displaystyle{ \omega_n = 2n \pi/\beta }[/math] for bosonic ones and [math]\displaystyle{ \beta=1/ T }[/math] is the inverse temperature. This relation is an example of Kramers-Kronig relation.

The spectral function can also be related to the imaginary-time Green function [math]\displaystyle{ \mathcal{G}(\tau) }[/math] be applying the inverse Fourier transform to the above equation

[math]\displaystyle{ \mathcal{G}(\tau)\ \colon = \frac{1}{\beta}\sum_{\omega_n} e^{-i\omega_n \tau} \mathcal{g}(i\omega_n) = \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} A(\omega) \frac{1}{\beta}\sum_{\omega_n} \frac{e^{-i\omega_n \tau} }{i\omega_n - \omega} }[/math]

with [math]\displaystyle{ \tau \in [0,\beta] }[/math]. Evaluating the summation over Matsubara frequencies gives the desired relation

[math]\displaystyle{ \mathcal{G}(\tau) = \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} \frac{-e^{-\tau \omega}}{1\pm e^{-\beta\omega}} A(\omega) }[/math]

where the upper sign is for fermionic systems and the lower sign is for bosonic ones.

Another example of the analytic continuation is calculating the optical conductivity [math]\displaystyle{ \sigma(\omega) }[/math] from the current-current correlation function values [math]\displaystyle{ \Pi(i\omega_n) }[/math] at Matsubara frequencies. The two are related as following

[math]\displaystyle{ \Pi(i\omega_n) = \int_{0}^{\infty} \frac{d\omega}{\pi} \frac{2 \omega^2}{\omega_n^2 +\omega^2}\; A(\omega) }[/math]

## Software

* The Maxent Project: Open source utility for performing analytic continuation using the maximum entropy method.
* Spektra: Free online tool for performing analytic continuation using the average spectrum Method.
* SpM: Sparse modeling tool for analytic continuation of imaginary-time Green’s function.

## See also

* Analytic continuation
* Analytic continuation along a curve
* Green's function
* Kramers–Kronig relations
* Quantum Monte Carlo
* Fredholm integral equation

## References

1. ↑ Silver, R. N.; Sivia, D. S.; Gubernatis, J. E. (1990-02-01). "Maximum-entropy method for analytic continuation of quantum Monte Carlo data". Physical Review B 41 (4): 2380–2389. doi:10.1103/PhysRevB.41.2380. PMID 9993975. Bibcode: 1990PhRvB..41.2380S. https://link.aps.org/doi/10.1103/PhysRevB.41.2380.
2. ↑ Jarrell, Mark; Gubernatis, J. E. (1996-05-01). "Bayesian inference and the analytic continuation of imaginary-time quantum Monte Carlo data" (in en). Physics Reports 269 (3): 133–195. doi:10.1016/0370-1573(95)00074-7. ISSN 0370-1573. Bibcode: 1996PhR...269..133J. https://dx.doi.org/10.1016/0370-1573(95)00074-7.
3. ↑ Reymbaut, A.; Bergeron, D.; Tremblay, A.-M. S. (2015-08-27). "Maximum entropy analytic continuation for spectral functions with nonpositive spectral weight". Physical Review B 92 (6): 060509. doi:10.1103/PhysRevB.92.060509. Bibcode: 2015PhRvB..92f0509R. https://link.aps.org/doi/10.1103/PhysRevB.92.060509.
4. ↑ Burnier, Yannis; Rothkopf, Alexander (2013-10-31). "Bayesian Approach to Spectral Function Reconstruction for Euclidean Quantum Field Theories". Physical Review Letters 111 (18): 182003. doi:10.1103/PhysRevLett.111.182003. PMID 24237510. Bibcode: 2013PhRvL.111r2003B. https://link.aps.org/doi/10.1103/PhysRevLett.111.182003.
5. ↑ White, S. R. (1991). Landau, David P.; Mon, K. K.; Schüttler, Heinz-Bernd. eds. "The Average Spectrum Method for the Analytic Continuation of Imaginary-Time Data" (in en). Computer Simulation Studies in Condensed Matter Physics III. Springer Proceedings in Physics (Berlin, Heidelberg: Springer) 53: 145–153. doi:10.1007/978-3-642-76382-3_13. ISBN 978-3-642-76382-3. https://link.springer.com/chapter/10.1007/978-3-642-76382-3_13.
6. ↑ Sandvik, Anders W. (1998-05-01). "Stochastic method for analytic continuation of quantum Monte Carlo data". Physical Review B 57 (17): 10287–10290. doi:10.1103/PhysRevB.57.10287. Bibcode: 1998PhRvB..5710287S. https://link.aps.org/doi/10.1103/PhysRevB.57.10287.
7. ↑ Ghanem, Khaldoon; Koch, Erik (2020-02-10). "Average spectrum method for analytic continuation: Efficient blocked-mode sampling and dependence on the discretization grid". Physical Review B 101 (8): 085111. doi:10.1103/PhysRevB.101.085111. Bibcode: 2020PhRvB.101h5111G. https://link.aps.org/doi/10.1103/PhysRevB.101.085111.
8. ↑ Ghanem, Khaldoon; Koch, Erik (2020-07-06). "Extending the average spectrum method: Grid point sampling and density averaging". Physical Review B 102 (3): 035114. doi:10.1103/PhysRevB.102.035114. Bibcode: 2020PhRvB.102c5114G. https://link.aps.org/doi/10.1103/PhysRevB.102.035114.
9. ↑ Beach, K. S. D.; Gooding, R. J.; Marsiglio, F. (2000-02-15). "Reliable Pad\'e analytical continuation method based on a high-accuracy symbolic computation algorithm". Physical Review B 61 (8): 5147–5157. doi:10.1103/PhysRevB.61.5147. Bibcode: 2000PhRvB..61.5147B. https://link.aps.org/doi/10.1103/PhysRevB.61.5147.
10. ↑ Östlin, A.; Chioncel, L.; Vitos, L. (2012-12-06). "One-particle spectral function and analytic continuation for many-body implementation in the exact muffin-tin orbitals method". Physical Review B 86 (23): 235107. doi:10.1103/PhysRevB.86.235107. Bibcode: 2012PhRvB..86w5107O. https://link.aps.org/doi/10.1103/PhysRevB.86.235107.

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