Numerical analytic continuation explained

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 $A(\backslash omega)$ at real frequencies $\backslash omega$ from the Green function values $\backslash mathcal(i\backslash omega\_n)$ at Matsubara frequencies $\backslash omega\_n$ by numerically inverting the integral equation

l{G}(i\omega _{n)} = \int ^{inf ty} _{-inf ty} (d\omega)/(2 \pi) (1)/(i\omega _{n} - \omega) A(\omega)

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

l{G}(\tau) \colon = (1)/(\beta) \sum _{\omega _{n}} e ^{- i\omega _{n} \tau} l{g}(i\omega _{n)} = \int ^{inf ty} _{-inf ty} (d\omega)/(2 \pi) A(\omega) (1)/(\beta) \sum _{\omega _{n}} (e ^{- i\omega _{n} \tau})/(i\omega _{n} - \omega)

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

l{G}(\tau) = \int ^{inf ty} _{-inf ty} (d\omega)/(2 \pi) (- e ^{-\tau})/(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

\sigma(\omega) from the current-current correlation function values \Pi(i\omega _{n)} at Matsubara frequencies. The two are related as following\Pi(i\omega _{n)} = \int ^{inf ty} _{0} (d\omega)/(\pi) (2 \omega ^{2})/(\omega ^{2} _{n} + \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

Notes and 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. 10.1103/PhysRevB.41.2380. 9993975. 1990PhRvB..41.2380S.
2. Jarrell. Mark. Gubernatis. J. E.. 1996-05-01. Bayesian inference and the analytic continuation of imaginary-time quantum Monte Carlo data. Physics Reports. en. 269. 3. 133–195. 10.1016/0370-1573(95)00074-7. 1996PhR...269..133J. 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. 10.1103/PhysRevB.92.060509. 1507.01956. 2015PhRvB..92f0509R. 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. 10.1103/PhysRevLett.111.182003. 24237510. 1307.6106. 2013PhRvL.111r2003B.
5. Book: White, S. R.. Computer Simulation Studies in Condensed Matter Physics III . The Average Spectrum Method for the Analytic Continuation of Imaginary-Time Data . 1991. Landau. David P.. Mon. K. K.. Schüttler. Heinz-Bernd . https://link.springer.com/chapter/10.1007/978-3-642-76382-3_13 . Springer Proceedings in Physics. 53. en. Berlin, Heidelberg. Springer. 145–153. 10.1007/978-3-642-76382-3_13. 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. 10.1103/PhysRevB.57.10287. 1998PhRvB..5710287S.
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. 10.1103/PhysRevB.101.085111. 1912.01379. 2020PhRvB.101h5111G. 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. 10.1103/PhysRevB.102.035114. 2004.01155. 2020PhRvB.102c5114G. 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. 10.1103/PhysRevB.61.5147. cond-mat/9908477. 2000PhRvB..61.5147B . 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. 10.1103/PhysRevB.86.235107. 1209.5283. 2012PhRvB..86w5107O. 8434964.