Summation over Matsubara axis

In many cases, we want to perform the summation of a Greens-function-like object \(f(\mathrm{i}\nu)\) over the Matsubara axis. Here, \(\nu = n\pi/\beta\) runs over the fermionic (odd \(n\)) or bosonic (even \(n\)) Matsubara frequencies; the letter \(\omega\) is reserved for real frequencies and \(A\) for the spectral function (see Notation and conventions).

The Fourier transform of \(f\) reads

\[ f(\tau) = \frac{1}{\beta} \sum_{\nu} f(\mathrm{i}\nu) e^{-\mathrm{i}\nu \tau}. \]

This leads to the following the two formula:

\[\begin{split} \begin{align} \sum_{\nu} f(\mathrm{i}\nu) e^{\mathrm{i}\nu 0^+} &= \beta f(\tau=0^-), \\ \sum_{\nu} f(\mathrm{i}\nu) e^{\mathrm{i}\nu 0^-} &= \beta f(\tau=0^+). \\ \end{align} \end{split}\]

We now expand \(f(\mathrm{i}\nu)\) at high frequencies as

\[ f(\mathrm{i}\nu) = \frac{c_1}{\mathrm{i}\nu} + \frac{c_2}{(\mathrm{i}\nu)^2} + \cdots. \]

As discussed in Sec. B3 of E. Gull’s Ph. D thesis, \(f(\tau=0^+) = f(\tau=0^-)\) if and only if \(c_1 = 0\). This condition is equivalent that \(f(\mathrm{i}\nu)\) vanishes at high frequencies faster than \(O(1/{\mathrm{i}\nu})\). More precisely, \(f(\tau=0^+) - f(\tau=0^-) = -c_1\): the Green’s function of an elementary operator has \(c_1 = 1\) and jumps by \(-1\) at \(\tau = 0\) for both statistics. If \(c_1 \neq 0\), the summation does NOT converge without a convergence factor and thus \( \sum_{\nu} f(\mathrm{i}\nu) e^{\mathrm{i}\nu 0^+} \neq \sum_{\nu} f(\mathrm{i}\nu) e^{\mathrm{i}\nu 0^-}\).