# Supported integral kernels¶

Note

$$\beta$$ denotes the inverse temperature in all equations below.

## Fermionic thermal Green’s function¶

$$A(\epsilon) = -(1/\pi)\Im G^\mathrm{ret}(\epsilon)$$ is the spectral function to be found.

Enabled when Som() object is constucted with kind = “FermionGf”.

• In imaginary time, $$G(\tau)$$

$G(\tau) = -\int\limits_{-\infty}^\infty d\epsilon \frac{e^{-\tau\epsilon}}{1+e^{-\beta\epsilon}} A(\epsilon).$
• At Matsubara frequencies, $$G(i\omega_n)$$

$G(i\omega_n) = \int\limits_{-\infty}^\infty d\epsilon \frac{1}{i\omega_n-\epsilon} A(\epsilon).$
• In Legendre polynomial basis, $$G(\ell)$$

$G(\ell) = -\int\limits_{-\infty}^\infty d\epsilon \frac{\beta\sqrt{2\ell+1}(-\mathrm{sgn}(\epsilon))^\ell i_{\ell}(\beta|\epsilon|/2)} {2\cosh(\beta\epsilon/2)} A(\epsilon),$

where $$i_\ell(x)$$ is the modified spherical Bessel function of the first kind.

Norm of a solution $$A(\epsilon)$$ is defined as

$N = \int\limits_{-\infty}^\infty d\epsilon A(\epsilon).$

The default value $$N=1$$ can be overridden to perform analytical continuation of related fermionic quantities, such as self-energy.

The retared Green’s function of a real frequency is reconstructed according to

$G^\mathrm{ret}(\epsilon) = -\int\limits_{-\infty}^\infty d\epsilon' \frac{A(\epsilon')}{\epsilon' - \epsilon - i0}.$

## Correlator of boson-like operators¶

Enabled when Som() object is constucted with kind = “BosonCorr”.

$$A(\epsilon)$$ is the spectral function to be found. It is defined differently from the fermionic case, namely $$A(\epsilon) = \Im\chi(\epsilon)/\epsilon$$, where

$\chi(\epsilon) = \int\limits_{-\infty}^\infty dt\ e^{i\epsilon t}\chi_(t),\quad \chi(t) = i\theta(t)\langle[\hat O(t),\hat O^\dagger(0)]\rangle.$
• In imaginary time, $$\chi(\tau)$$

$\chi(\tau) = \int\limits_{-\infty}^\infty \frac{d\epsilon}{\pi} \frac{\epsilon e^{-\tau\epsilon}}{1-e^{-\beta\epsilon}} A(\epsilon).$
• At Matsubara frequencies, $$\chi(i\Omega_n)$$

$\chi(i\Omega_n) = \int\limits_{-\infty}^\infty d\epsilon \frac{1}{\pi}\frac{-\epsilon}{i\Omega_n - \epsilon} A(\epsilon).$
• In Legendre polynomial basis, $$\chi(\ell)$$

$\chi(\ell) = \int\limits_{-\infty}^\infty \frac{d\epsilon}{\pi} \frac{\beta\epsilon\sqrt{2\ell+1}(-\mathrm{sgn}(\epsilon))^\ell i_{\ell}(\beta|\epsilon|/2)} {2\sinh(\beta\epsilon/2)} A(\epsilon),$

where $$i_\ell(x)$$ is the modified spherical Bessel function of the first kind.

Norm of a solution $$A(\epsilon)$$ is defined as

$N = \int\limits_{-\infty}^\infty d\epsilon A(\epsilon) = \pi\chi(i\Omega_n=0).$

The correlator of a real frequency is reconstructed according to

$\chi(\epsilon) = \frac{1}{\pi}\int\limits_{-\infty}^\infty d\epsilon' \frac{\epsilon' A(\epsilon')}{\epsilon' - \epsilon - i0}.$

## Autocorrelator of a boson-like operator¶

Enabled when Som() object is constucted with kind = “BosonAutoCorr”.

$$A(\epsilon)$$ is the spectral function to be found. It is defined differently from the fermionic case, namely $$A(\epsilon) = \Im\chi(\epsilon)/\epsilon$$, where

$\chi(\epsilon) = \int\limits_{-\infty}^\infty dt\ e^{i\epsilon t}\chi(t),\quad \chi(t) = i\theta(t)\langle[\hat O(t),\hat O(0)]\rangle.$

Expressions in this section imply that $$A(-\epsilon) = A(\epsilon)$$, and, therefore, it is enough to find the spectral function on the positive energy half-axis only. This condition is fulfilled by the dynamical susceptibilities of a form $$\chi(\tau) = \langle\mathcal{T}\hat O(\tau)\hat O(0)\rangle$$, where $$\hat O$$ is a Hermitian operator.

• In imaginary time, $$\chi(\tau)$$

$\chi(\tau) = \int\limits_{-\infty}^\infty \frac{d\epsilon}{\pi} \frac{\epsilon (e^{-\tau\epsilon}+e^{-(\beta-\tau)\epsilon})} {1-e^{-\beta\epsilon}} A(\epsilon).$
• At Matsubara frequencies, $$\chi(i\Omega_n)$$

$\chi(i\Omega_n) = \int\limits_0^\infty d\epsilon \frac{1}{\pi}\frac{2\epsilon^2}{\Omega_n^2+\epsilon^2} A(\epsilon).$
• In Legendre polynomial basis, $$\chi(\ell)$$

$\begin{split}\chi(\ell) = \left\{ \begin{array}{ll} \int\limits_0^\infty \frac{d\epsilon}{\pi} \frac{\beta\epsilon\sqrt{2\ell+1} i_{\ell}(\beta\epsilon/2)} {\sinh(\beta\epsilon/2)} A(\epsilon), &\ell\ \mathrm{ even},\\ 0, &\ell\ \mathrm{odd}. \end{array}\right.,\end{split}$

where $$i_\ell(x)$$ is the modified spherical Bessel function of the first kind.

Norm of a solution $$A(\epsilon)$$ is defined as

$N = \int\limits_0^\infty d\epsilon A(\epsilon) = \frac{\pi}{2}\chi(i\Omega_n=0).$

The correlator of a real frequency is reconstructed according to

$\chi(\epsilon) = -\frac{1}{\pi}\int\limits_{-\infty}^0 d\epsilon' \frac{\epsilon' A(-\epsilon')}{\epsilon' - \epsilon - i0} + \frac{1}{\pi}\int\limits_0^\infty d\epsilon' \frac{\epsilon' A(\epsilon')}{\epsilon' - \epsilon - i0}.$

## Correlation function at zero temperature¶

$$A(\epsilon) = -(1/\pi)\Im G(\epsilon)$$ is the spectral function to be found.

Enabled when Som() object is constucted with kind = “ZeroTemp”.

• In imaginary time, $$G(\tau)$$, $$\tau\in[0;\tau_{max}]$$

$G(\tau) = -\int\limits_0^\infty d\epsilon\ e^{-\tau\epsilon} A(\epsilon).$
• At Matsubara frequencies, $$G(i\omega_n)$$, where $$\omega_n = (2n+1)\pi/\tau_{max}$$ for fermionic correlation functions and $$\omega_n = 2n\pi/\tau_{max}$$ for bosonic.

$G(i\omega_n) = \int\limits_0^\infty d\epsilon \frac{1}{i\omega_n-\epsilon} A(\epsilon).$
• In Legendre polynomial basis, $$G(\ell)$$

$G(\ell) = \int\limits_0^\infty d\epsilon \tau_{max}(-1)^{\ell+1}\sqrt{2\ell+1} i_{\ell}\left(\frac{\epsilon\tau_{max}}{2}\right) \exp\left(-\frac{\epsilon\tau_{max}}{2}\right) A(\epsilon),$

where $$i_\ell(x)$$ is the modified spherical Bessel function of the first kind.

Norm of a solution $$A(\epsilon)$$ is defined as

$N = \int\limits_0^\infty d\epsilon A(\epsilon).$

The correlation function of a real frequency is reconstructed according to

$G(\epsilon) = -\int\limits_0^\infty d\epsilon' \frac{A(\epsilon')}{\epsilon' - \epsilon - i0}.$