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
\phi(\tau)
\tau
\beta=\hbar/k\rmT
\phi(\tau)=
1 | |
\sqrt{\beta |
The frequencies
\omegan
n\inZ
bosonic frequencies:
\omega | ||||
|
,
fermionic frequencies:
\omega | ||||
|
,
\phi(\tau)
Once such substitutions have been made, certain diagrams contributing to the action take the form of a so-called Matsubara summation
Sη=
1 | |
\beta |
\sum | |
i\omegan |
g(i\omegan).
The summation will converge if
g(z=i\omega)
z\toinfty
z-1
S\rm
η=+1
S\rm
η=-1
η
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
T=0K
\intT=0d\omega g(\omega)
Sη
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\omegan
S | ||||
|
\sumi\omegag(i\omega)=
1 | |
2\pii\beta |
\ointg(z)hη(z)dz,
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),
S | ||||
|
\sum | |
z0\ing(z)poles |
\operatorname{Res}g(z0)hη(z0).
Note that a minus sign is produced, because the contour is deformed to enclose the poles in the clockwise direction, resulting in the negative residue.
To produce simple poles on boson frequencies
z=i\omegan
(1) | ||
h | (z)= | |
\rmB |
\beta | |
1-e-\beta |
=-\betan\rm(-z)=\beta(1+n\rm(z)),
(2) | ||
h | (z)= | |
\rmB |
-\beta | |
1-e\beta |
=\betan\rm(z),
(1) | |
h | |
\rmB |
(z)
(2) | |
h | |
\rmB |
(z)
n\rm(z)=(e\beta-1)-1
The case is similar for fermion frequencies. There are also two types of Matsubara weighting functions that produce simple poles at
z=i\omegam
(1) | ||
h | (z)= | |
\rmF |
\beta | |
1+e-\beta |
=\betan\rm(-z)=\beta(1-n\rm(z)),
(2) | ||
h | (z)= | |
\rmF |
-\beta | |
1+e\beta |
=-\betan\rm(z).
(1) | |
h | |
\rmF |
(z)
(2) | |
h | |
\rmF |
(z)
n\rm(z)=(e\beta+1)-1
In the application to Green's function calculation, g(z) always have the structure
g(z)=G(z)e-z\tau,
hη(z)=h
(1) | |
η |
(z)
The following table contains
S | ||||
|
\sumi\omegag(i\omega)
g(i\omega) | Sη | |||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
(i\omega-\xi)-1 | -ηnη(\xi) | |||||||||||||||||||||||||||||||||||||||||||||||||
(i\omega-\xi)-2 | -η
nη(\xi)(η+nη(\xi)) | |||||||||||||||||||||||||||||||||||||||||||||||||
(i\omega-\xi)-n |
nη(\xi) | |||||||||||||||||||||||||||||||||||||||||||||||||
|
| |||||||||||||||||||||||||||||||||||||||||||||||||
|
| |||||||||||||||||||||||||||||||||||||||||||||||||
| ηcη(\xi1,\xi2)\equiv-η
| |||||||||||||||||||||||||||||||||||||||||||||||||
| η
(1+2ηnη(\xi)) | |||||||||||||||||||||||||||||||||||||||||||||||||
|
(1+2ηnη(\xi)) | |||||||||||||||||||||||||||||||||||||||||||||||||
|
(cη(0,\xi)+
| |||||||||||||||||||||||||||||||||||||||||||||||||
|
(cη(0,\xi)-
| |||||||||||||||||||||||||||||||||||||||||||||||||
| -η
nη(\xi)(η+nη(\xi)) | |||||||||||||||||||||||||||||||||||||||||||||||||
|
| |||||||||||||||||||||||||||||||||||||||||||||||||
\right)2 |
cη(0,\xi1)-
\right)+(1\leftrightarrow2) | |||||||||||||||||||||||||||||||||||||||||||||||||
\right)2 |
cη(0,\xi1)-
\right)+(1\leftrightarrow2) | |||||||||||||||||||||||||||||||||||||||||||||||||
[1] Since the summation does not converge, the result may differ upon different choice of the Matsubara weighting function.
[2] (1 ↔ 2) denotes the same expression as the before but with index 1 and 2 interchanged.
In this limit
\beta → infty
1 | |
\beta |
\sumi\omega
iinfty | |
=\int | |
-iinfty |
d(i\omega) | |
2\pii |
.
\Omega
\Omega
\Omega → infty
η\lim\Omega → infty
i\Omega | |
\left[ \int | |
-i\Omega |
d(i\omega) | \left(ln(-i\omega+\xi)- | |
2\pii |
\pi\xi | \right)- | |
2\Omega |
\Omega | |
\pi |
(ln\Omega-1)\right] =\left\{\begin{array}{cc} 0&\xi\geq0,\\ -η\xi&\xi<0, \end{array} \right.
η\lim\Omega → infty
i\Omega | |
\int | |
-i\Omega |
d(i\omega) | \left( | |
2\pii |
1 | - | |
-i\omega+\xi |
\pi | |
2\Omega |
\right) =\left\{\begin{array}{cc} 0&\xi\geq0,\\ -η&\xi<0, \end{array} \right.
Consider a function G(τ) defined on the imaginary time interval (0,β). It can be given in terms of Fourier series,
G(\tau)= | 1 |
\beta |
\sumi\omegaG(i\omega)e-i\omega\tau,
where the frequency only takes discrete values spaced by 2/β.
The particular choice of frequency depends on the boundary condition of the function G(τ). In physics, G(τ) stands for the imaginary time representation of Green's function
G(\tau)=-\langlel{T}\tau\psi(\tau)\psi*(0)\rangle.
It satisfies the periodic boundary condition G(τ+β)=G(τ) for a boson field. While for a fermion field the boundary condition is anti-periodic G(τ + β) = −G(τ).
Given the Green's function G(iω) in the frequency domain, its imaginary time representation G(τ) can be evaluated by Matsubara frequency summation. Depending on the boson or fermion frequencies that is to be summed over, the resulting G(τ) can be different. To distinguish, define
Gη(\tau)=\begin{cases} G\rm(\tau),&ifη=+1,\\ G\rm(\tau),&ifη=-1, \end{cases}
G\rm(\tau)=
1 | |
\beta |
\sum | |
i\omegan |
-i\omegan\tau | |
G(i\omega | |
n)e |
,
G\rm(\tau)=
1 | |
\beta |
\sum | |
i\omegam |
-i\omegam\tau | |
G(i\omega | |
m)e |
.
Note that τ is restricted in the principal interval (0,β). The boundary condition can be used to extend G(τ) out of the principal interval. Some frequently used results are concluded in the following table.
G(i\omega) | Gη(\tau) | |||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
(i\omega-\xi)-1 | -e\xi(\beta-\tau)nη(\xi) | |||||||||||||||||||||||||||
(i\omega-\xi)-2 | e\xi(\beta-\tau)nη(\xi)\left(\tau+η\betanη(\xi)\right) | |||||||||||||||||||||||||||
(i\omega-\xi)-3 |
e\xi(\beta-\tau)
| |||||||||||||||||||||||||||
|
| |||||||||||||||||||||||||||
(\omega2+m2)-1 |
\cosh{m\tau} nη(m) | |||||||||||||||||||||||||||
i\omega(\omega2+m2)-1 |
-η\sinh{m\tau} nη(m) | |||||||||||||||||||||||||||
The small imaginary time plays a critical role here. The order of the operators will change if the small imaginary time changes sign.
\langle\psi\psi*\rangle=\langlel{T}\tau\psi(\tau=0+)
| ||||
\psi | ||||
η(\tau=0 |
\sumi\omega
-i\omega0+ | |
G(i\omega)e |
\langle\psi*\psi\rangle=η\langlel{T}\tau\psi(\tau=0-)\psi*(0)\rangle =-η
| ||||
G | ||||
η(\tau=0 |
\sumi\omega
i\omega0+ | |
G(i\omega)e |
The evaluation of distribution function becomes tricky because of the discontinuity of Green's function G(τ) at τ = 0. To evaluate the summation
G(0)=\sumi\omega(i\omega-\xi)-1,
(1) | |
h | |
η |
(z)
G(\tau=0+)
(2) | |
h | |
η |
(z)
G(\tau=0-)
Bosons
G\rm
| ||||
(\tau=0 |
\sum | |
i\omegan |
| |||||
i\omegan-\xi |
=-n\rm(\xi),
G\rm
| ||||
(\tau=0 |
\sum | |
i\omegan |
| |||||
i\omegan-\xi |
=-(n\rm(\xi)+1).
G\rm
| ||||
(\tau=0 |
\sum | |
i\omegam |
| |||||
i\omegam-\xi |
=n\rm(\xi),
G\rm
| ||||
(\tau=0 |
\sum | |
i\omegam |
| |||||
i\omegam-\xi |
=n\rm(\xi)-1.
Bosons
1 | |
\beta |
\sum | |
i\omegan |
ln(\beta(-i\omega | ||||
|
ln(1-e-\beta\xi),
- | 1 |
\beta |
\sum | |
i\omegam |
ln(\beta(-i\omega | ||||
|
ln(1+e-\beta\xi).
Frequently encountered diagrams are evaluated here with the single mode setting. Multiple mode problems can be approached by a spectral function integral.Here
\omegam
\omegan
\Sigma(i\omega | ||||
|
\sum
i\omegan |
1 | |
i\omegam+i\omegan-\varepsilon |
1 | = | |
i\omegan-\Omega |
n\rm(\varepsilon)+n\rm(\Omega) | |
i\omegam-\varepsilon+\Omega |
.
\Pi(i\omegan)=
1 | |
\beta |
\sum
i\omegam |
1 | |
i\omegam+i\omegan-\varepsilon |
1 | =- | |
i\omegam-\varepsilon' |
n\rm(\varepsilon)-n\rm\left(\varepsilon'\right) | |
i\omegan-\varepsilon+\varepsilon' |
.
\Pi(i\omegan)=-
1 | |
\beta |
\sum
i\omegam |
1 | |
i\omegam+i\omegan-\varepsilon |
1 | = | |
-i\omegam-\varepsilon' |
1-n\rm\left(\varepsilon'\right)-n\rm(\varepsilon) | |
i\omegan-\varepsilon-\varepsilon' |
.
The general notation
nη
n | ||||
|
.
nη(\xi)=\begin{cases} n\rm(\xi),&ifη=+1,\\ n\rm(\xi),&ifη=-1. \end{cases}
The Bose distribution function is related to hyperbolic cotangent function by
n\rm(\xi)=
1 | \left(\operatorname{coth} | |
2 |
\beta\xi | |
2 |
-1\right).
n\rm(\xi)=
1 | \left(1-\operatorname{tanh} | |
2 |
\beta\xi | |
2 |
\right).
Both distribution functions do not have definite parity,
nη(-\xi)=-η-nη(\xi).
cη
nη(-\xi)=nη(\xi)+2\xicη(0,\xi).
Bose and Fermi distribution functions transmute under a shift of the variable by the fermionic frequency,
nη(i\omegam+\xi)=-n-η(\xi).
| ||||
n | ||||
\rmB |
| ||||
csch |
,
| ||||
n | ||||
\rmF |
| ||||
sech |
.
\prime(\xi)= | |
n | |
η |
-\betanη(\xi)(1+ηnη(\xi)).
\prime(\xi)=η\delta(\xi) | |
n | |
η |
as\beta → infty.
\prime\prime | ||
n | (\xi)= | |
\rmB |
\beta2 | |
4 |
| |||||
\operatorname{csch} | \operatorname{coth} |
\beta\xi | |
2 |
,
\prime\prime | ||
n | (\xi)= | |
\rmF |
\beta2 | |
4 |
| |||||
\operatorname{sech} | \operatorname{tanh} |
\beta\xi | |
2 |
.
nη(a+b)-n
|
.
n\rm(b)-n\rm(-b)=coth
\betab | |
2 |
,
n\rm(b)-n\rm(-b)=-tanh
\betab | |
2 |
.
n\rm(a+b)-n\rm(a-b)=\operatorname{coth}
\betab | |
2 |
\prime\prime | |
+n | |
\rmB |
(b)a2+ … ,
n\rm(a+b)-n\rm(a-b)=-\operatorname{tanh}
\betab | |
2 |
\prime\prime | |
+n | |
\rmF |
(b)a2+ … .
n\rm(a+b)-n\rm
\prime(a)b+ … , | |
(a-b)=2n | |
\rmB |
n\rm(a+b)-n\rm
\prime(a)b+ … . | |
(a-b)=2n | |
\rmF |
Definition:
c | ||||
|
.
c\rm(a,b)\equivc+(a,b),
c\rm(a,b)\equivc-(a,b).
c | ||||
|
.
c\rm(a,b)
To avoid overflow in the numerical calculation, the tanh and coth functions are used
c\rm(a,b)=
1 | \left(\operatorname{coth} | |
4b |
\beta(a-b) | |
2 |
-\operatorname{coth}
\beta(a+b) | |
2 |
\right),
c\rm(a,b)=
1 | \left(\operatorname{tanh} | |
4b |
\beta(a+b) | |
2 |
-\operatorname{tanh}
\beta(a-b) | |
2 |
\right).
c\rm(0,b)=-
1 | \operatorname{coth} | |
2b |
\betab | |
2 |
,
c\rm(0,b)=
1 | \operatorname{tanh} | |
2b |
\betab | |
2 |
.
c\rm(a,0)=
\beta | |
4 |
| ||||
\operatorname{csch} |
,
c\rm(a,0)=
\beta | |
4 |
| ||||
\operatorname{sech} |
.
For a = 0:
c\rm(0,b)=
1 | |
2|b| |
.
For b = 0:
c\rm(a,0)=\delta(a).
In general,
c\rm(a,b)=\begin{cases}
1 | |
2|b| |
,&if|a|<|b|\\ 0,&if|a|>|b| \end{cases}
Agustin Nieto: Evaluating Sums over the Matsubara Frequencies. arXiv:hep-ph/9311210
Github repository: MatsubaraSum A Mathematica package for Matsubara frequency summation.