Numerical analytic continuation
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In many-body physics, the problem of analytic continuation refers to 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][11]
See also
References
- ^ 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.
- ^ 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. doi:10.1016/0370-1573(95)00074-7. ISSN 0370-1573.
- ^ 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.
- ^ 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.
- ^ 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". Computer Simulation Studies in Condensed Matter Physics III. Springer Proceedings in Physics. Berlin, Heidelberg: Springer: 145–153. doi:10.1007/978-3-642-76382-3_13. ISBN 978-3-642-76382-3.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.