Matsubara frequency

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In thermal quantum field theory, the Matsubara frequency summation (named after Takeo Matsubara) is a technique used to simplify calculations involving Euclidean (imaginary time) path integrals.^[1]

In thermal quantum field theory, bosonic and fermionic quantum fields ${\displaystyle \phi (\tau )}$ are respectively periodic or antiperiodic in imaginary time ${\displaystyle \tau }$, with periodicity ${\displaystyle \beta =\hbar /k_{\rm {B}}T}$. Matsubara summation refers to the technique of expanding these fields in Fourier series

${\displaystyle \phi (\tau )={\frac {1}{\sqrt {\beta }}}\sum _{n}e^{-i\omega _{n}\tau }\phi (i\omega _{n})\Leftrightarrow \phi (i\omega _{n})={\frac {1}{\sqrt {\beta }}}\int _{0}^{\beta }d\tau \ e^{i\omega _{n}\tau }\phi (\tau ).}$

The frequencies ${\displaystyle \omega _{n}}$ are called the Matsubara frequencies, taking values from either of the following sets (with ${\displaystyle n\in \mathbb {Z} }$):

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

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

which respectively enforce periodic and antiperiodic boundary conditions on the field ${\displaystyle \phi (\tau )}$.

Once such substitutions have been made, certain diagrams contributing to the action take the form of a so-called Mastubara summation

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

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.^{[2] [3] [4]}

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 ${\displaystyle S_{\eta }}$.