Supported observables

This page lists all kinds of dynamical observables currently supported by SOM and explicitly states the integral equation being solved for each of them.

Thermal Green’s function of fermions

A Green’s function of fermions at temperature \(T = 1/\beta\) is defined as

\[G(\tau) = -\langle \mathbb{T}_\tau c(\tau) c^\dagger(0)\rangle.\]

Its real-frequency counterpart is the retarded Green’s function \(G^\mathrm{ret}(\epsilon)\), which is directly connected to the spectral function \(A(\epsilon)\),

\[A(\epsilon) = -\frac{1}{\pi}\Im G^\mathrm{ret}(\epsilon)\]

and can be recovered from it using the Hilbert transform,

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

The spectral function is normalized according to

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

Note

\(\mathcal{N} = 1\) for a fermionic Green’s function.

Using the same integral equations, one can also continue fermionic self-energies as long as they do not contain a static Hartree-Fock contribution (i.e. they decay to 0 as \(\omega\to\infty\)). In this case norms must be computed separately by the user as first spectral moments of the self-energy. For derivation of the spectral moments see, for instance,

M. Potthoff, T. Wegner, and W. Nolting, Phys. Rev. B 55, 16132 (1997).

This observable kind is selected and one of the following integral equations is solved when the Som object is constructed with kind="FermionGf".

Imaginary time Imaginary frequencies Legendre orthogonal polynomials
\[G(\tau) = -\int\limits_{-\infty}^\infty d\epsilon \frac{e^{-\tau\epsilon}}{1+e^{-\beta\epsilon}} A(\epsilon).\]
\[G(i\omega_n) = \int\limits_{-\infty}^\infty d\epsilon \frac{1}{i\omega_n-\epsilon} A(\epsilon).\]
\[G(\ell) = -\int\limits_{-\infty}^\infty d\epsilon \frac{\beta\sqrt{2\ell+1} (-\mathrm{sgn}(\epsilon))^\ell i_{\ell}(\beta|\epsilon|/2)} {2\cosh(\beta\epsilon/2)} A(\epsilon),\]

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

In many model calculations fermionic Green’s functions obey the particle-hole symmetry, which manifests itself in the symmetry of the spectral function, \(A(-\epsilon) = A(\epsilon)\). One can enforce this symmetry by constructing Som with kind="FermionGfSymm". More precisely, SOM will

  • Use symmetrized integral kernels (see below) sensitive only to the symmetric part of \(A(\epsilon)\);
  • Symmetrize calculated final solution \(A(\epsilon)\) before recovering \(G^\mathrm{ret}(\epsilon)\).
Imaginary time Imaginary frequencies Legendre orthogonal polynomials
\[G(\tau) = -\int\limits_{-\infty}^\infty \frac{d\epsilon}{2} \frac{e^{-\tau\epsilon} + e^{-(\beta-\tau)\epsilon}} {1+e^{-\beta\epsilon}} A(\epsilon).\]
\[G(i\omega_n) = -\int\limits_{-\infty}^\infty d\epsilon \frac{i\omega_n}{\omega_n^2+\epsilon^2} A(\epsilon).\]
\[\begin{split}G(\ell) = \left\{ \begin{array}{ll} -\int\limits_{-\infty}^\infty d\epsilon \frac{\beta\sqrt{2\ell+1} i_{\ell}(\beta|\epsilon|/2)} {2\cosh(\beta\epsilon/2)} A(\epsilon), &\ell\ \mathrm{ even},\\ 0, &\ell\ \mathrm{odd}. \end{array}\right.,\end{split}\]

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

Thermal Green’s function of bosons, dynamical susceptibilities and conductivity

The following correlators share the same general form \(\chi_{OO^\dagger}(\tau) = \langle \mathbb{T}_\tau \hat O(\tau) \hat O^\dagger(0)\rangle\), where \(\hat O\) is a bosonic or boson-like (fermion-number-conserving) operator.

  • Matsubara Green’s function of bosons \(G(\tau) = \langle \mathbb{T}_\tau a(\tau) a^\dagger(0)\rangle\);
  • Charge susceptibility \(\chi_{NN}(\tau) = \langle \mathbb{T}_\tau \hat N(\tau) \hat N(0)\rangle\);
  • Longitudinal magnetic susceptibility \(\chi_{zz}(\tau) = \langle \mathbb{T}_\tau \hat S_z(\tau) \hat S_z(0)\rangle\);
  • Transverse magnetic susceptibility \(\chi_{-+}(\tau) = \langle \mathbb{T}_\tau \hat S_-(\tau) \hat S_+(0)\rangle\);
  • Optical conductivity \(\sigma(\tau) = \langle \mathbb{T}_\tau \hat j(\tau) \hat j(0)\rangle\).

The real-time and real-frequency counterparts of \(\chi_{OO^\dagger}(\tau)\) are

\[ \begin{align}\begin{aligned}\chi_{OO^\dagger}(t) = i\theta(t)\langle[\hat O(t),\hat O^\dagger(0)]\rangle,\\\chi_{OO^\dagger}(\epsilon) = \int\limits_{-\infty}^\infty dt\ e^{i\epsilon t}\chi_{OO^\dagger}(t).\end{aligned}\end{align} \]

The imaginary part of \(\chi_{OO^\dagger}(\epsilon)\) obeys \(\mathrm{sgn}(\Im\chi_{OO^\dagger}(\epsilon)) = \mathrm{sgn}(\epsilon)\), which allows to introduce a non-negative auxiliary function \(A(\epsilon) = \Im\chi_{OO^\dagger}(\epsilon) / \epsilon\). It plays the role of the spectral function for this class of continuation problem.

Norm of the spectral function is defined as

\[\mathcal{N} = \int\limits_{-\infty}^\infty d\epsilon A(\epsilon) = \pi\chi_{OO^\dagger}(i\Omega_n=0).\]

The correlator of a real frequency is recovered according to

\[\chi_{OO^\dagger}(\epsilon) = \frac{1}{\pi}\int\limits_{-\infty}^\infty d\epsilon' \frac{\epsilon' A(\epsilon')}{\epsilon' - \epsilon - i0}.\]

This observable kind is selected and one of the following integral equations is solved when the Som object is constructed with kind="BosonCorr".

Imaginary time Imaginary frequencies Legendre orthogonal polynomials
\[\chi_{OO^\dagger}(\tau) = \int\limits_{-\infty}^\infty \frac{d\epsilon}{\pi} \frac{\epsilon e^{-\tau\epsilon}}{1-e^{-\beta\epsilon}} A(\epsilon).\]
\[\chi_{OO^\dagger}(i\Omega_n) = \int\limits_{-\infty}^\infty \frac{d\epsilon}{\pi}\frac{-\epsilon}{i\Omega_n - \epsilon} A(\epsilon).\]
\[\chi_{OO^\dagger}(\ell) = \int\limits_{-\infty}^\infty \frac{d\epsilon}{\pi} \frac{\beta\epsilon\sqrt{2\ell+1}(-\mathrm{sgn}(\epsilon))^\ell i_{\ell}(\beta|\epsilon|/2)} {2\sinh(\beta\epsilon/2)} A(\epsilon),\]

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

When \(\hat O^\dagger = \hat O\), such as in the case of the charge susceptibility, the longitudinal magnetic susceptibility and the optical conductivity, it is recommended to exploit the additional spectral symmetry \(A(-\epsilon) = A(\epsilon)\). If Som is constructed with kind="BosonAutoCorr", it

  • Uses symmetrized integral kernels (see below) sensitive only to the symmetric part of \(A(\epsilon)\);
  • Symmetrizes calculated final solution \(A(\epsilon)\) before recovering \(\chi_{OO^\dagger}(\epsilon)\).
Imaginary time Imaginary frequencies Legendre orthogonal polynomials
\[\chi_{OO}(\tau) = \int\limits_{-\infty}^\infty \frac{d\epsilon}{2\pi} \frac{\epsilon (e^{-\tau\epsilon}+e^{-(\beta-\tau)\epsilon})} {1-e^{-\beta\epsilon}} A(\epsilon).\]
\[\chi_{OO}(i\Omega_n) = \int\limits_{-\infty}^\infty \frac{d\epsilon}{\pi} \frac{\epsilon^2}{\Omega_n^2+\epsilon^2} A(\epsilon).\]
\[\begin{split}\chi_{OO}(\ell) = \left\{ \begin{array}{ll} \int\limits_{-\infty}^\infty \frac{d\epsilon}{\pi} \frac{\beta\epsilon\sqrt{2\ell+1} i_{\ell}(\beta|\epsilon|/2)} {2\sinh(\beta\epsilon/2)} A(\epsilon),&\ell\ \mathrm{ even},\\ 0, &\ell\ \mathrm{odd}. \end{array}\right.,\end{split}\]

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

Dynamical response functions at zero temperature

In the limit of zero temperature (\(\beta=1/T\to\infty\)), both Fermi-Dirac and Bose-Einstein distributions approach the shape of a step function \(\theta(\epsilon)\). It is, therefore, sufficient to consider a spectral function \(A(\epsilon)\) defined for \(\epsilon\geq 0\) regardless of operator statistics. Formally, the Matsubara time segment \(\tau\in[0; \beta]\) becomes infinitely long in this limit. Practically, however, it is still possible to consider \(\tau\) varying on a reduced segment \([0; \tau_\mathrm{max}]\). It is also possible to introduce a countable sequence of fictitious Matsubara frequencies \(\omega_n = (2n+1)\pi/\tau_\mathrm{max}\) or \(\omega_n = 2n\pi/\tau_\mathrm{max}\).

In the \(T=0\) case, SOM defines the non-negative spectral function as \(A(\epsilon) = -(1/\pi)\Im G(\epsilon)\) and its norm as

\[\mathcal{N} = \int\limits_0^\infty d\epsilon A(\epsilon).\]

The original response function of a real frequency is recovered according to

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

This observable kind is selected and one of the following integral equations is solved when the Som object is constructed with kind="ZeroTemp".

Imaginary time Imaginary frequencies Legendre orthogonal polynomials
\[G(\tau) = -\int\limits_0^\infty d\epsilon e^{-\tau\epsilon} A(\epsilon).\]
\[G(i\omega_n) = \int\limits_0^\infty d\epsilon \frac{1}{i\omega_n-\epsilon} A(\epsilon).\]
\[G(\ell) = \int\limits_0^\infty d\epsilon \tau_\mathrm{max}(-1)^{\ell+1}\sqrt{2\ell+1} i_{\ell}\left(\frac{\epsilon\tau_\mathrm{max}}{2}\right) \exp\left(-\frac{\epsilon\tau_\mathrm{max}}{2}\right) A(\epsilon),\]

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