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]

Examples

A common analytic continuation problem is obtaining the spectral function ${\textstyle A(\omega )}$  at real frequencies ${\textstyle \omega }$  from the Green function values ${\textstyle {\mathcal {G}}(i\omega _{n})}$  at Matsubara frequencies ${\textstyle \omega _{n}}$  by numerically inverting the integral equation

${\displaystyle {\mathcal {G}}(i\omega _{n})=\int _{-\infty }^{\infty }{\frac {d\omega }{2\pi }}{\frac {1}{i\omega _{n}-\omega }}\;A(\omega )}$ 

where ${\textstyle \omega _{n}=(2n+1)\pi /\beta }$  for fermionic systems or ${\textstyle \omega _{n}=2n\pi /\beta }$  for bosonic ones and ${\textstyle \beta =1/T}$  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 ${\textstyle {\mathcal {G}}(\tau )}$  be applying the inverse Fourier transform to the above equation

${\displaystyle {\mathcal {G}}(\tau )\ \colon ={\frac {1}{\beta }}\sum _{\omega _{n}}e^{-i\omega _{n}\tau }{\mathcal {g}}(i\omega _{n})=\int _{-\infty }^{\infty }{\frac {d\omega }{2\pi }}A(\omega ){\frac {1}{\beta }}\sum _{\omega _{n}}{\frac {e^{-i\omega _{n}\tau }}{i\omega _{n}-\omega }}}$ 

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

${\displaystyle {\mathcal {G}}(\tau )=\int _{-\infty }^{\infty }{\frac {d\omega }{2\pi }}{\frac {-e^{-\tau \omega }}{1\pm e^{-\beta \omega }}}A(\omega )}$ 

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 ${\displaystyle \sigma (\omega )}$  from the current-current correlation function values ${\displaystyle \Pi (i\omega _{n})}$  at Matsubara frequencies. The two are related as following

${\displaystyle \Pi (i\omega _{n})=\int _{0}^{\infty }{\frac {d\omega }{\pi }}{\frac {2\omega ^{2}}{\omega _{n}^{2}+\omega ^{2}}}\;A(\omega )}$ 

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
• Fredholm integral equation
• Green's function
• Kramers–Kronig relations
• Quantum Monte Carlo

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. Bibcode:1990PhRvB..41.2380S. doi:10.1103/PhysRevB.41.2380. PMID 9993975.
2. Jarrell, Mark; Gubernatis, J. E. (1996-05-01). "Bayesian inference and the analytic continuation of imaginary-time quantum Monte Carlo data". Physics Reports. 269 (3): 133–195. Bibcode:1996PhR...269..133J. doi:10.1016/0370-1573(95)00074-7. ISSN 0370-1573.
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. arXiv:1507.01956. Bibcode:2015PhRvB..92f0509R. doi:10.1103/PhysRevB.92.060509. S2CID 56385057.
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. arXiv:1307.6106. Bibcode:2013PhRvL.111r2003B. doi:10.1103/PhysRevLett.111.182003. PMID 24237510.
5. White, S. R. (1991). "The Average Spectrum Method for the Analytic Continuation of Imaginary-Time Data". In Landau, David P.; Mon, K. K.; Schüttler, Heinz-Bernd (eds.). Computer Simulation Studies in Condensed Matter Physics III. Springer Proceedings in Physics. Vol. 53. Berlin, Heidelberg: Springer. pp. 145–153. doi:10.1007/978-3-642-76382-3_13. ISBN 978-3-642-76382-3.
6. Sandvik, Anders W. (1998-05-01). "Stochastic method for analytic continuation of quantum Monte Carlo data". Physical Review B. 57 (17): 10287–10290. Bibcode:1998PhRvB..5710287S. 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. arXiv:1912.01379. Bibcode:2020PhRvB.101h5111G. doi:10.1103/PhysRevB.101.085111. S2CID 208548627.
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. arXiv:2004.01155. Bibcode:2020PhRvB.102c5114G. doi:10.1103/PhysRevB.102.035114. S2CID 214775183.
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. arXiv:cond-mat/9908477. Bibcode:2000PhRvB..61.5147B. doi:10.1103/PhysRevB.61.5147. S2CID 17880539.
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. arXiv:1209.5283. Bibcode:2012PhRvB..86w5107O. doi:10.1103/PhysRevB.86.235107. S2CID 8434964.