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 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 where β = ℏ / k B T {displaystyle eta =hbar /k_{B}T} is the inverse temperature and the frequencies ω n {displaystyle omega _{n}} are usually taken from either of the following two sets (with n ∈ Z {displaystyle nin mathbb {Z} } ): The summation will converge if g ( z = i ω ) {displaystyle g(z=iomega )} tends to 0 in z → ∞ {displaystyle z o infty } limit in a manner faster than z − 1 {displaystyle z^{-1}} . The summation over bosonic frequencies is denoted as S B {displaystyle S_{B}} (with η = + 1 {displaystyle eta =+1} ), while that over fermionic frequencies is denoted as S F {displaystyle S_{F}} (with η = − 1 {displaystyle eta =-1} ). η {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 T = 0 K {displaystyle T=0,{ ext{K}}} , a certain Feynman diagram is represented by an integral ∫ T = 0 d ω   g ( ω ) {displaystyle int _{T=0}mathrm {d} omega g(omega )} , at finite temperature it is given by the sum S η {displaystyle S_{eta }} . The trick to evaluate Matsubara frequency summation is to use a Matsubara weighting function hη(z) that has simple poles located exactly at z = i ω {displaystyle z=iomega } . The weighting functions in the boson case η = +1 and fermion case η = −1 differ. The choice of weighting function will be discussed later. With the weighting function, the summation can be replaced by a contour integral in the complex plane. As in Fig. 1, the weighting function generates poles (red crosses) on the imaginary axis. The contour integral picks up the residue of these poles, which is equivalent to the summation. By deformation of the contour lines to enclose the poles of g(z) (the green cross in Fig. 2), the summation can be formally accomplished by summing the residue of g(z)hη(z) over all poles of g(z),