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]

ExamplesEdit

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

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

This article uses material from the Wikipedia article  Metasyntactic variable, which is released under the  Creative Commons Attribution-ShareAlike 3.0 Unported License.