Matsubara frequency

In thermal quantum field theory, the Matsubara frequency summation (named after Takeo Matsubara) is the summation over discrete imaginary frequencies. It takes the following form

${\displaystyle S_{\eta }={\frac {1}{\beta }}\sum _{i\omega _{n}}g(i\omega _{n}),}$

where ${\displaystyle \beta =\hbar /k_{\rm {B}}T}$ is the inverse temperature and the frequencies ${\displaystyle \omega _{n}}$ are usually taken from either of the following two sets (with ${\displaystyle n\in \mathbb {Z} }$):

bosonic frequencies: ${\displaystyle \omega _{n}={\frac {2n\pi }{\beta }},}$

fermionic frequencies: ${\displaystyle \omega _{n}={\frac {(2n+1)\pi }{\beta }},}$

The summation will converge if ${\displaystyle g(z=i\omega )}$ tends to 0 in ${\displaystyle z\to \infty }$ limit in a manner faster than ${\displaystyle z^{-1}}$. The summation over bosonic frequencies is denoted as ${\displaystyle S_{\rm {B}}}$ (with ${\displaystyle \eta =+1}$), while that over fermionic frequencies is denoted as ${\displaystyle S_{\rm {F}}}$ (with ${\displaystyle \eta =-1}$). ${\displaystyle \eta }$ is the statistical sign.

In addition to thermal quantum field theory, the Matsubara frequency summation method also plays an essential role in the diagrammatic approach to solid-state physics, namely, if one considers the diagrams at finite temperature.

Generally speaking, if at ${\displaystyle T=0\,{\text{K}}}$, a certain Feynman diagram is represented by an integral ${\displaystyle \int _{T=0}\mathrm {d} \omega \ g(\omega )}$, at finite temperature it is given by the sum ${\displaystyle S_{\eta }}$.