Summation over Matsubara axis

In many cases, we want to perform the summation of a Greens-function-like object \(A(\mathrm{i}\omega)\) over the Matsubara axis.

The Fourier transform of \(A\) reads

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

This leads to the following the two formula:

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

We now expand \(A(\mathrm{i}\omega)\) at high frequencies as

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

As discussed in Sec. B3 of E. Gull’s Ph. D thesis, \(A(\tau=0^+) = A(\tau=0^-)\) if and only if \(c_1 = 0\). This condition is equivalent that \(A(\mathrm{i}\omega)\) vanishes at high frequencies faster than \(O(1/{\mathrm{i}\omega})\). If \(c_1 \neq 0\), the summation does NOT converge without a convergence factor and thus \( \sum_{\omega} A(\mathrm{i}\omega) e^{\mathrm{i}\omega 0^+} \neq \sum_{\omega} A(\mathrm{i}\omega) e^{\mathrm{i}\omega 0^-}\).