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

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