Numerical analytic continuation

<h2 id="examples">Examples</h2>
A common analytic continuation problem is obtaining the spectral function 
 
 
 
 A
 (
 ω
 )
 
 
 {\textstyle A(\omega )}
 
 at real frequencies 
 
 
 
 ω
 
 
 {\textstyle \omega }
 
 from the Green function values 
 
 
 
 
 
 G
 
 
 (
 i
 
 ω
 
 n
 
 
 )
 
 
 {\textstyle {\mathcal {G}}(i\omega _{n})}
 
 at <a href="/facts/Matsubara_frequency/DJpTWrlE">Matsubara frequencies</a> 
 
 
 
 
 ω
 
 n
 
 
 
 
 {\textstyle \omega _{n}}
 
 by numerically inverting the integral equation

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

where 
 
 
 
 
 ω
 
 n
 
 
 =
 (
 2
 n
 +
 1
 )
 π
 
 /
 
 β
 
 
 {\textstyle \omega _{n}=(2n+1)\pi /\beta }
 
 for <a href="/facts/Fermion/LEckdDsR">fermionic</a> systems or 
 
 
 
 
 ω
 
 n
 
 
 =
 2
 n
 π
 
 /
 
 β
 
 
 {\textstyle \omega _{n}=2n\pi /\beta }
 
 for <a href="/facts/Boson/XtPFcjb5">bosonic</a> ones and 
 
 
 
 β
 =
 1
 
 /
 
 T
 
 
 {\textstyle \beta =1/T}
 
 is the inverse temperature. This relation is an example of <a href="/facts/Kramers%E2%80%93Kronig_relations/S2UemwVo">Kramers-Kronig relation</a>.

The spectral function can also be related to the <a href="/facts/Imaginary_time/UTTYSUw7">imaginary-time</a> Green function 
 
 
 
 
 
 G
 
 
 (
 τ
 )
 
 
 {\textstyle {\mathcal {G}}(\tau )}
 
 be applying the <a href="/facts/Inverse_Fourier_transform/aM1aMkf0">inverse Fourier transform</a> to the above equation

 
 
 
 
 
 G
 
 
 (
 τ
 )
  
 :
 =
 
 
 1
 β
 
 
 
 ∑
 
 
 ω
 
 n
 
 
 
 
 
 e
 
 −
 i
 
 ω
 
 n
 
 
 τ
 
 
 
 
 g
 
 
 (
 i
 
 ω
 
 n
 
 
 )
 =
 
 ∫
 
 −
 ∞
 
 
 ∞
 
 
 
 
 
 d
 ω
 
 
 2
 π
 
 
 
 A
 (
 ω
 )
 
 
 1
 β
 
 
 
 ∑
 
 
 ω
 
 n
 
 
 
 
 
 
 
 e
 
 −
 i
 
 ω
 
 n
 
 
 τ
 
 
 
 i
 
 ω
 
 n
 
 
 −
 ω
 
 
 
 
 
 {\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 
 
 
 
 τ
 ∈
 [
 0
 ,
 β
 ]
 
 
 {\textstyle \tau \in [0,\beta ]}
 
. Evaluating the <a href="/facts/Matsubara_frequency/DJpTWrlE">summation over Matsubara frequencies</a> gives the desired relation

 
 
 
 
 
 G
 
 
 (
 τ
 )
 =
 
 ∫
 
 −
 ∞
 
 
 ∞
 
 
 
 
 
 d
 ω
 
 
 2
 π
 
 
 
 
 
 
 −
 
 e
 
 −
 τ
 ω
 
 
 
 
 1
 ±
 
 e
 
 −
 β
 ω
 
 
 
 
 
 A
 (
 ω
 )
 
 
 {\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 
 
 
 
 σ
 (
 ω
 )
 
 
 {\displaystyle \sigma (\omega )}
 
 from the current-current correlation function values 
 
 
 
 Π
 (
 i
 
 ω
 
 n
 
 
 )
 
 
 {\displaystyle \Pi (i\omega _{n})}
 
 at Matsubara frequencies. The two are related as following

 
 
 
 Π
 (
 i
 
 ω
 
 n
 
 
 )
 =
 
 ∫
 
 0
 
 
 ∞
 
 
 
 
 
 d
 ω
 
 π
 
 
 
 
 
 2
 
 ω
 
 2
 
 
 
 
 
 ω
 
 n
 
 
 2
 
 
 +
 
 ω
 
 2
 
 
 
 
 
 
 A
 (
 ω
 )
 
 
 {\displaystyle \Pi (i\omega _{n})=\int _{0}^{\infty }{\frac {d\omega }{\pi }}{\frac {2\omega ^{2}}{\omega _{n}^{2}+\omega ^{2}}}\;A(\omega )}

<h2 id="software">Software</h2>
<ul><li><a href="https://github.com/CQMP/Maxent">The Maxent Project</a>: Open source utility for performing analytic continuation using the maximum entropy method.</li>
<li><a href="https://www.spektra.app">Spektra</a>: Free online tool for performing analytic continuation using the average spectrum Method.</li>
<li><a href="https://spm-lab.github.io/SpM/manual/build/html/index.html">SpM</a>: Sparse modeling tool for analytic continuation of imaginary-time Green’s function.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Analytic_continuation/aBCuEgvt">Analytic continuation</a></li>
<li><a href="/facts/Monodromy_theorem/nUmMg3Fj">Monodromy theorem</a></li>
<li><a href="/facts/Fredholm_integral_equation/euHGJbQo">Fredholm integral equation</a></li>
<li><a href="/facts/Green%27s_function_(many-body_theory)/1v3ethbc">Green's function</a></li>
<li><a href="/facts/Kramers%E2%80%93Kronig_relations/S2UemwVo">Kramers–Kronig relations</a></li>
<li><a href="/facts/Quantum_Monte_Carlo/qenhfw9u">Quantum Monte Carlo</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn: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. <a href="https://link.aps.org/doi/10.1103/PhysRevB.41.2380" target="_blank">https://link.aps.org/doi/10.1103/PhysRevB.41.2380</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://dx.doi.org/10.1016%2F0370-1573%2895%2900074-7" target="_blank">https://dx.doi.org/10.1016%2F0370-1573%2895%2900074-7</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://link.aps.org/doi/10.1103/PhysRevB.92.060509" target="_blank">https://link.aps.org/doi/10.1103/PhysRevB.92.060509</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://link.aps.org/doi/10.1103/PhysRevLett.111.182003" target="_blank">https://link.aps.org/doi/10.1103/PhysRevLett.111.182003</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="978-3-642-76382-3" target="_blank">978-3-642-76382-3</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://link.aps.org/doi/10.1103/PhysRevB.57.10287" target="_blank">https://link.aps.org/doi/10.1103/PhysRevB.57.10287</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://link.aps.org/doi/10.1103/PhysRevB.101.085111" target="_blank">https://link.aps.org/doi/10.1103/PhysRevB.101.085111</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://link.aps.org/doi/10.1103/PhysRevB.102.035114" target="_blank">https://link.aps.org/doi/10.1103/PhysRevB.102.035114</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://link.aps.org/doi/10.1103/PhysRevB.61.5147" target="_blank">https://link.aps.org/doi/10.1103/PhysRevB.61.5147</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://link.aps.org/doi/10.1103/PhysRevB.86.235107" target="_blank">https://link.aps.org/doi/10.1103/PhysRevB.86.235107</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
</ol>

Numerical analytic continuation open-in-new

Numerical analytic continuation