Kernels

Just as stated above, the spectral function and the imaginary time or Matsubara Green's function are related with each other by the Laplace transformation:

\[\begin{equation} G(x) = \int d\omega~K(x,\omega) A(\omega). \end{equation}\]

Here $K(x,\omega)$ is the so-called kernel function. It plays a key role in this equation. In this section, we would like to introduce the kernels that have been implemented in ACTest.

Fermionic kernels

For fermionic Green's function, we have

\[\begin{equation} G(\tau) = \int^{+\infty}_{-\infty} d\omega \frac{e^{-\tau\omega}}{1 + e^{-\beta\omega}} A(\omega), \end{equation}\]

and

\[\begin{equation} G(i\omega_n) = \int^{+\infty}_{-\infty} d\omega \frac{1}{i\omega_n - \omega} A(\omega). \end{equation}\]

The kernels are defined as

\[\begin{equation} K(\tau,\omega) = \frac{e^{-\tau\omega}}{1 + e^{-\beta\omega}}, \end{equation}\]

and

\[\begin{equation} K(\omega_n,\omega) = \frac{1}{i\omega_n - \omega}. \end{equation}\]

For fermionic systems, $A(\omega)$ is defined on $(-\infty,\infty)$. It is causal, i.e., $A(\omega) \ge 0$.

Bosonic kernels

For bosonic system, the spectral function obeys the following constraint:

\[\begin{equation} \text{sign}(\omega) A(\omega) \ge 0. \end{equation}\]

It is quite convenient to introduce a new variable $\tilde{A}(\omega)$:

\[\begin{equation} \tilde{A}(\omega) = \frac{A(\omega)}{\omega}. \end{equation}\]

Clearly, $\tilde{A}(\omega) \ge 0$. It means that $\tilde{A}(\omega)$ is positive definite. So, we have

\[\begin{equation} G(\tau) = \int^{+\infty}_{-\infty} d\omega \frac{e^{-\tau\omega}}{1 - e^{-\beta\omega}} A(\omega) =\int^{+\infty}_{-\infty} d\omega \frac{\omega e^{-\tau\omega}}{1 - e^{-\beta\omega}} \tilde{A}(\omega), \end{equation}\]

and

\[\begin{equation} G(i\omega_n) = \int^{+\infty}_{-\infty} d\omega \frac{1}{i\omega_n - \omega} A(\omega) = \int^{+\infty}_{-\infty} d\omega \frac{\omega}{i\omega_n - \omega} \tilde{A}(\omega). \end{equation}\]

The corresponding kernels read:

\[\begin{equation} K(\tau,\omega) = \frac{\omega e^{-\tau\omega}}{1 - e^{-\beta\omega}}, \end{equation}\]

and

\[\begin{equation} K(\omega_n,\omega) = \frac{\omega}{i\omega_n - \omega}. \end{equation}\]

Especially,

\[\begin{equation} K(\tau,\omega = 0) \equiv \frac{1}{\beta}, \end{equation}\]

and

\[\begin{equation} K(\omega_n = 0,\omega = 0) \equiv -1. \end{equation}\]

Symmetric bosonic kernels

This is a special case for the bosonic Green's function with Hermitian bosonic operators. Here, the spectral function $A(\omega)$ is an odd function. Let us introduce $\tilde{A}(\omega) = A(\omega)/\omega$ again. Since $\tilde{A}(\omega)$ is an even function, we can restrict it in $(0,\infty)$. Now we have

\[\begin{equation} G(\tau) = \int^{\infty}_{0} d\omega \frac{\omega [e^{-\tau\omega} + e^{-(\beta - \tau)\omega}]} {1 - e^{-\beta\omega}} \tilde{A}(\omega), \end{equation}\]

and

\[\begin{equation} G(i\omega_n) = \int^{\infty}_{0} d\omega \frac{-2\omega^2}{\omega_n^2 + \omega^2} \tilde{A}(\omega). \end{equation}\]

The corresponding kernel functions read:

\[\begin{equation} K(\tau,\omega) = \frac{\omega [e^{-\tau\omega} + e^{-(\beta - \tau)\omega}]} {1 - e^{-\beta\omega}}, \end{equation}\]

and

\[\begin{equation} K(\omega_n, \omega) = \frac{-2\omega^2}{\omega_n^2 + \omega^2}. \end{equation}\]

There are two special cases:

\[\begin{equation} K(\tau,\omega = 0) = \frac{2}{\beta}, \end{equation}\]

and

\[\begin{equation} K(\omega_n = 0,\omega = 0) = -2. \end{equation}\]