Grids for Green's functions

As mentioned before, Green's functions in quantum many-body physics are usually defined on imaginary time or Matsubara frequency axes. Imaginary time Green's functions $G(\tau)$ and Matsubara Green's functions $G(i\omega_n)$ are related via Fourier transformation:

\[\begin{equation} G(\tau) = \frac{1}{\beta} \sum_n e^{i\omega_n \tau} G(i\omega_n), \end{equation}\]

and

\[\begin{equation} G(i\omega_n) = \int^{\beta}_0 d\tau\ e^{-i\omega_n \tau} G(\tau), \end{equation}\]

where $\beta$ denotes inverse temperature ($\beta \equiv 1/T$). The grids for Green's functions are linear. Specifically, $\tau_i = i \beta/N_{\tau}$, where $N_{\tau}$ denotes number of time slices and $i \in [0,N_{\tau}]$. $\omega_n = (2n+1)\pi/\beta$ for fermions and $2n\pi/\beta$ for bosons, where $n \in [0, N]$ and $N$ means number of Matsubara frequency points.