When we convert Matsubara's imaginary time Green's function to the retarded Green's function, we perform an analytical continuation by substituting $i\omega_n$ with $\omega + i\eta$, with $\eta\to0^+$. For instance, the non-interacting Green's function becomes:

$$ G^R(k,\omega) = \frac{1}{\omega - \epsilon(k) + i\eta} $$

Here, $\epsilon(k)$ denotes the dispersion relation.

I'm curious about the value of $\eta$, especially when computing the Green's function numerically. Is it on the order of $10^{-16}$ or closer to $10^{-1}$?