Numerical analytic continuation

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

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

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