Matsubara frequency

Figure 1.

Figure 2.

The trick to evaluate Matsubara frequency summation is to use a Matsubara weighting function h_η(z) that has simple poles located exactly at ${\displaystyle z=i\omega _{n}}$.[3] 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 surrounding the imaginary axis.

${\displaystyle S_{\eta }={\frac {1}{\beta }}\sum _{i\omega }g(i\omega )={\frac {1}{2\pi i\beta }}\oint g(z)h_{\eta }(z)\,dz,}$

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. This procedure is sometimes called Sommerfeld-Watson transformation.[4]

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)h_η(z) over all poles of g(z),

${\displaystyle S_{\eta }=-{\frac {1}{\beta }}\sum _{z_{0}\in g(z){\text{ poles}}}\operatorname {Res} g(z_{0})h_{\eta }(z_{0}).}$

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 ${\displaystyle z=i\omega _{n}}$, either of the following two types of Matsubara weighting functions can be chosen

${\displaystyle h_{\rm {B}}^{(1)}(z)={\frac {\beta }{1-e^{-\beta z}}}=-\beta n_{\rm {B}}(-z)=\beta (1+n_{\rm {B}}(z)),}$

${\displaystyle h_{\rm {B}}^{(2)}(z)={\frac {-\beta }{1-e^{\beta z}}}=\beta n_{\rm {B}}(z),}$

depending on which half plane the convergence is to be controlled in. ${\displaystyle h_{\rm {B}}^{(1)}(z)}$ controls the convergence in the left half plane (Re z < 0), while ${\displaystyle h_{\rm {B}}^{(2)}(z)}$ controls the convergence in the right half plane (Re z > 0). Here ${\displaystyle n_{\rm {B}}(z)=(e^{\beta z}-1)^{-1}}$ is the Bose–Einstein distribution function.

The case is similar for fermion frequencies. There are also two types of Matsubara weighting functions that produce simple poles at ${\displaystyle z=i\omega _{m}}$

${\displaystyle h_{\rm {F}}^{(1)}(z)={\frac {\beta }{1+e^{-\beta z}}}=\beta n_{\rm {F}}(-z)=\beta (1-n_{\rm {F}}(z)),}$

${\displaystyle h_{\rm {F}}^{(2)}(z)={\frac {-\beta }{1+e^{\beta z}}}=-\beta n_{\rm {F}}(z).}$

${\displaystyle h_{\rm {F}}^{(1)}(z)}$ controls the convergence in the left half plane (Re z < 0), while ${\displaystyle h_{\rm {F}}^{(2)}(z)}$ controls the convergence in the right half plane (Re z > 0). Here ${\displaystyle n_{\rm {F}}(z)=(e^{\beta z}+1)^{-1}}$ is the Fermi–Dirac distribution function.

In the application to Green's function calculation, g(z) always have the structure

${\displaystyle g(z)=G(z)e^{-z\tau },}$

which diverges in the left half plane given 0 < τ < β. So as to control the convergence, the weighting function of the first type is always chosen ${\displaystyle h_{\eta }(z)=h_{\eta }^{(1)}(z)}$. However, there is no need to control the convergence if the Matsubara summation does not diverge, in that case, any choice of the Matsubara weighting function will lead to identical results.

The following table contains ${\displaystyle S_{\eta }={\frac {1}{\beta }}\sum _{i\omega }g(i\omega )}$ for some simple rational functions g(z). The symbol η = ±1 is the statistical sign, +1 for bosons and -1 for fermions.

${\displaystyle g(i\omega )}$	${\displaystyle S_{\eta }}$
${\displaystyle (i\omega -\xi )^{-1}}$	${\displaystyle -\eta n_{\eta }(\xi )}$[1]
${\displaystyle (i\omega -\xi )^{-2}}$	${\displaystyle -\eta n_{\eta }^{\prime }(\xi )=\beta n_{\eta }(\xi )(\eta +n_{\eta }(\xi ))}$
${\displaystyle (i\omega -\xi )^{-n}}$	${\displaystyle -{\frac {\eta }{(n-1)!}}\partial _{\xi }^{n-1}n_{\eta }(\xi )}$
${\displaystyle {\frac {1}{(i\omega -\xi _{1})(i\omega -\xi _{2})}}}$	${\displaystyle -{\frac {\eta (n_{\eta }(\xi _{1})-n_{\eta }(\xi _{2}))}{\xi _{1}-\xi _{2}}}}$
${\displaystyle {\frac {1}{(i\omega -\xi _{1})^{2}(i\omega -\xi _{2})^{2}}}}$	${\displaystyle {\frac {\eta }{(\xi _{1}-\xi _{2})^{2}}}\left({\frac {2(n_{\eta }(\xi _{1})-n_{\eta }(\xi _{2}))}{\xi _{1}-\xi _{2}}}-(n_{\eta }^{\prime }(\xi _{1})+n_{\eta }^{\prime }(\xi _{2}))\right)}$
${\displaystyle {\frac {1}{(i\omega -\xi _{1})^{2}-\xi _{2}^{2}}}}$	${\displaystyle \eta c_{\eta }(\xi _{1},\xi _{2})}$
${\displaystyle {\frac {1}{(i\omega )^{2}-\xi ^{2}}}}$	${\displaystyle \eta c_{\eta }(0,\xi )=-{\frac {1}{2\xi }}(1+2\eta n_{\eta }(\xi ))}$
${\displaystyle {\frac {(i\omega )^{2}}{(i\omega )^{2}-\xi ^{2}}}}$	${\displaystyle -{\frac {\xi }{2}}(1+2\eta n_{\eta }(\xi ))}$[1]
${\displaystyle {\frac {1}{((i\omega )^{2}-\xi ^{2})^{2}}}}$	${\displaystyle -{\frac {\eta }{2\xi ^{2}}}(c_{\eta }(0,\xi )+n_{\eta }^{\prime }(\xi ))}$
${\displaystyle {\frac {(i\omega )^{2}}{((i\omega )^{2}-\xi ^{2})^{2}}}}$	${\displaystyle {\frac {\eta }{2}}(c_{\eta }(0,\xi )-n_{\eta }^{\prime }(\xi ))}$
${\displaystyle {\frac {(i\omega )^{2}+\xi ^{2}}{((i\omega )^{2}-\xi ^{2})^{2}}}}$	${\displaystyle -\eta n_{\eta }^{\prime }(\xi )=\beta n_{\eta }(\xi )(\eta +n_{\eta }(\xi ))}$
${\displaystyle {\frac {1}{((i\omega )^{2}-\xi _{1}^{2})((i\omega )^{2}-\xi _{2}^{2})}}}$	${\displaystyle {\frac {\eta (c_{\eta }(0,\xi _{1})-c_{\eta }(0,\xi _{2}))}{\xi _{1}^{2}-\xi _{2}^{2}}}}$
${\displaystyle \left({\frac {1}{(i\omega )^{2}-\xi _{1}^{2}}}+{\frac {1}{(i\omega )^{2}-\xi _{2}^{2}}}\right)^{2}}$	${\displaystyle \eta \left({\frac {3\xi _{1}^{2}+\xi _{2}^{2}}{2\xi _{1}^{2}(\xi _{1}^{2}-\xi _{2}^{2})}}c_{\eta }(0,\xi _{1})-{\frac {n_{\eta }^{\prime }(\xi _{1})}{2\xi _{1}^{2}}}\right)+(1\leftrightarrow 2)}$[2]
${\displaystyle \left({\frac {1}{(i\omega )^{2}-\xi _{1}^{2}}}-{\frac {1}{(i\omega )^{2}-\xi _{2}^{2}}}\right)^{2}}$	${\displaystyle \eta \left(-{\frac {5\xi _{1}^{2}-\xi _{2}^{2}}{2\xi _{1}^{2}(\xi _{1}^{2}-\xi _{2}^{2})}}c_{\eta }(0,\xi _{1})-{\frac {n_{\eta }^{\prime }(\xi _{1})}{2\xi _{1}^{2}}}\right)+(1\leftrightarrow 2)}$[2]

[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 ${\displaystyle \beta \rightarrow \infty }$, the Matsubara frequency summation is equivalent to the integration of imaginary frequency over imaginary axis.

${\displaystyle {\frac {1}{\beta }}\sum _{i\omega }=\int _{-i\infty }^{i\infty }{\frac {\mathrm {d} (i\omega )}{2\pi i}}.}$

Some of the integrals do not converge. They should be regularized by introducing the frequency cutoff ${\displaystyle \Omega }$, and then subtracting the divergent part (${\displaystyle \Omega }$-dependent) from the integral before taking the limit of ${\displaystyle \Omega \rightarrow \infty }$. For example, the free energy is obtained by the integral of logarithm,

${\displaystyle \eta \lim _{\Omega \rightarrow \infty }\left[\int _{-i\Omega }^{i\Omega }{\frac {\mathrm {d} (i\omega )}{2\pi i}}\left(\ln(-i\omega +\xi )-{\frac {\pi \xi }{2\Omega }}\right)-{\frac {\Omega }{\pi }}(\ln \Omega -1)\right]=\left\{{\begin{array}{cc}0&\xi \geq 0,\\-\eta \xi &\xi <0,\end{array}}\right.}$

meaning that at zero temperature, the free energy simply relates to the internal energy below the chemical potential. Also the distribution function is obtained by the following integral

${\displaystyle \eta \lim _{\Omega \rightarrow \infty }\int _{-i\Omega }^{i\Omega }{\frac {\mathrm {d} (i\omega )}{2\pi i}}\left({\frac {1}{-i\omega +\xi }}-{\frac {\pi }{2\Omega }}\right)=\left\{{\begin{array}{cc}0&\xi \geq 0,\\-\eta &\xi <0,\end{array}}\right.}$

which shows step function behavior at zero temperature.

Consider a function G(τ) defined on the imaginary time interval (0,β). It can be given in terms of Fourier series,

${\displaystyle G(\tau )={\frac {1}{\beta }}\sum _{i\omega }G(i\omega )e^{-i\omega \tau },}$

where the frequency only takes discrete values spaced by 2π/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

${\displaystyle G(\tau )=-\langle {\mathcal {T}}_{\tau }\psi (\tau )\psi ^{*}(0)\rangle .}$

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

${\displaystyle G_{\eta }(\tau )={\begin{cases}G_{\rm {B}}(\tau ),&{\text{if }}\eta =+1,\\G_{\rm {F}}(\tau ),&{\text{if }}\eta =-1,\end{cases}}}$

with

${\displaystyle G_{\rm {B}}(\tau )={\frac {1}{\beta }}\sum _{i\omega _{n}}G(i\omega _{n})e^{-i\omega _{n}\tau },}$

${\displaystyle G_{\rm {F}}(\tau )={\frac {1}{\beta }}\sum _{i\omega _{m}}G(i\omega _{m})e^{-i\omega _{m}\tau }.}$

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.

${\displaystyle G(i\omega )}$	${\displaystyle G_{\eta }(\tau )}$
${\displaystyle (i\omega -\xi )^{-1}}$	${\displaystyle -e^{\xi (\beta -\tau )}n_{\eta }(\xi )}$
${\displaystyle (i\omega -\xi )^{-2}}$	${\displaystyle e^{\xi (\beta -\tau )}n_{\eta }(\xi )\left(\tau +\eta \beta n_{\eta }(\xi )\right)}$
${\displaystyle (i\omega -\xi )^{-3}}$	${\displaystyle -{\frac {1}{2}}e^{\xi (\beta -\tau )}n_{\eta }(\xi )\left(\tau ^{2}+\eta \beta (\beta +2\tau )n_{\eta }(\xi )+2\beta ^{2}n_{\eta }^{2}(\xi )\right)}$
${\displaystyle (i\omega -\xi _{1})^{-1}(i\omega -\xi _{2})^{-1}}$	${\displaystyle -{\frac {e^{\xi _{1}(\beta -\tau )}n_{\eta }(\xi _{1})-e^{\xi _{2}(\beta -\tau )}n_{\eta }(\xi _{2})}{\xi _{1}-\xi _{2}}}}$
${\displaystyle (\omega ^{2}+m^{2})^{-1}}$	${\displaystyle {\frac {e^{-m\tau }}{2m}}+{\frac {\eta }{m}}\cosh {m\tau }\;n_{\eta }(m)}$
${\displaystyle i\omega (\omega ^{2}+m^{2})^{-1}}$	${\displaystyle {\frac {e^{-m\tau }}{2}}-\eta \,\sinh {m\tau }\;n_{\eta }(m)}$

The small imaginary time plays a critical role here. The order of the operators will change if the small imaginary time changes sign.

${\displaystyle \langle \psi \psi ^{*}\rangle =\langle {\mathcal {T}}_{\tau }\psi (\tau =0^{+})\psi ^{*}(0)\rangle =-G_{\eta }(\tau =0^{+})=-{\frac {1}{\beta }}\sum _{i\omega }G(i\omega )e^{-i\omega 0^{+}}}$

${\displaystyle \langle \psi ^{*}\psi \rangle =\eta \langle {\mathcal {T}}_{\tau }\psi (\tau =0^{-})\psi ^{*}(0)\rangle =-\eta G_{\eta }(\tau =0^{-})=-{\frac {\eta }{\beta }}\sum _{i\omega }G(i\omega )e^{i\omega 0^{+}}}$

The evaluation of distribution function becomes tricky because of the discontinuity of Green's function G(τ) at τ = 0. To evaluate the summation

${\displaystyle G(0)=\sum _{i\omega }(i\omega -\xi )^{-1},}$

both choices of the weighting function are acceptable, but the results are different. This can be understood if we push G(τ) away from τ = 0 a little bit, then to control the convergence, we must take ${\displaystyle h_{\eta }^{(1)}(z)}$ as the weighting function for ${\displaystyle G(\tau =0^{+})}$, and ${\displaystyle h_{\eta }^{(2)}(z)}$ for ${\displaystyle G(\tau =0^{-})}$.

Bosons

${\displaystyle G_{\rm {B}}(\tau =0^{-})={\frac {1}{\beta }}\sum _{i\omega _{n}}{\frac {e^{i\omega _{n}0^{+}}}{i\omega _{n}-\xi }}=-n_{\rm {B}}(\xi ),}$

${\displaystyle G_{\rm {B}}(\tau =0^{+})={\frac {1}{\beta }}\sum _{i\omega _{n}}{\frac {e^{-i\omega _{n}0^{+}}}{i\omega _{n}-\xi }}=-(n_{\rm {B}}(\xi )+1).}$

Fermions

${\displaystyle G_{\rm {F}}(\tau =0^{-})={\frac {1}{\beta }}\sum _{i\omega _{m}}{\frac {e^{i\omega _{m}0^{+}}}{i\omega _{m}-\xi }}=n_{\rm {F}}(\xi ),}$

${\displaystyle G_{\rm {F}}(\tau =0^{+})={\frac {1}{\beta }}\sum _{i\omega _{m}}{\frac {e^{-i\omega _{m}0^{+}}}{i\omega _{m}-\xi }}=-(1-n_{\rm {F}}(\xi )).}$

Bosons

${\displaystyle {\frac {1}{\beta }}\sum _{i\omega _{n}}\ln(\beta (-i\omega _{n}+\xi ))={\frac {1}{\beta }}\ln(1-e^{-\beta \xi }),}$

Fermions

${\displaystyle -{\frac {1}{\beta }}\sum _{i\omega _{m}}\ln(\beta (-i\omega _{m}+\xi ))=-{\frac {1}{\beta }}\ln(1+e^{-\beta \xi }).}$

Frequently encountered diagrams are evaluated here with the single mode setting. Multiple mode problems can be approached by a spectral function integral.

${\displaystyle \Sigma (i\omega _{m})=-{\frac {1}{\beta }}\sum _{i\omega _{n}}{\frac {1}{i\omega _{m}+i\omega _{n}-\varepsilon }}{\frac {1}{i\omega _{n}-\Omega }}=-{\frac {n_{\rm {B}}(\varepsilon )+n_{\rm {F}}(\Omega )}{i\omega _{m}-\varepsilon +\Omega }}.}$

${\displaystyle \Pi (i\omega _{n})={\frac {1}{\beta }}\sum _{i\omega _{m}}{\frac {1}{i\omega _{m}+i\omega _{n}-\varepsilon }}{\frac {1}{i\omega _{m}-\varepsilon '}}={\frac {n_{\rm {B}}(\varepsilon )+n_{\rm {F}}\left(\varepsilon '\right)}{i\omega _{n}-\varepsilon +\varepsilon '}}.}$

${\displaystyle \Pi (i\omega _{n})=-{\frac {1}{\beta }}\sum _{i\omega _{m}}{\frac {1}{i\omega _{m}+i\omega _{n}-\varepsilon }}{\frac {1}{-i\omega _{m}-\varepsilon '}}={\frac {1-n_{\rm {F}}\left(\varepsilon '\right)+n_{\rm {B}}(\varepsilon )}{i\omega _{n}-\varepsilon -\varepsilon '}}.}$

The general notation ${\displaystyle n_{\eta }}$ stands for either Bose (η = +1) or Fermi (η = −1) distribution function

${\displaystyle n_{\eta }(\xi )={\frac {1}{e^{\beta \xi }-\eta }}.}$

If necessary, the specific notations n_B and n_F are used to indicate Bose and Fermi distribution functions respectively

${\displaystyle n_{\eta }(\xi )={\begin{cases}n_{\rm {B}}(\xi ),&{\text{if }}\eta =+1,\\n_{\rm {F}}(\xi ),&{\text{if }}\eta =-1.\end{cases}}}$

The Bose distribution function is related to hyperbolic cotangent function by

${\displaystyle n_{\rm {B}}(\xi )={\frac {1}{2}}\left(\operatorname {coth} {\frac {\beta \xi }{2}}-1\right).}$

The Fermi distribution function is related to hyperbolic tangent function by

${\displaystyle n_{\rm {F}}(\xi )={\frac {1}{2}}\left(1-\operatorname {tanh} {\frac {\beta \xi }{2}}\right).}$

Both distribution functions do not have definite parity,

${\displaystyle n_{\eta }(-\xi )=-\eta -n_{\eta }(\xi ).}$

Another formula is in terms of the ${\displaystyle c_{\eta }}$ function

${\displaystyle n_{\eta }(-\xi )=n_{\eta }(\xi )+2\xi c_{\eta }(0,\xi ).}$

However their derivatives have definite parity.

Bose and Fermi distribution functions transmute under a shift of the variable by the fermionic frequency,

${\displaystyle n_{\eta }(i\omega _{m}+\xi )=-n_{-\eta }(\xi ).}$

However shifting by bosonic frequencies does not make any difference.

${\displaystyle n_{\rm {B}}^{\prime }(\xi )=-{\frac {\beta }{4}}\mathrm {csch} ^{2}{\frac {\beta \xi }{2}},}$

${\displaystyle n_{\rm {F}}^{\prime }(\xi )=-{\frac {\beta }{4}}\mathrm {sech} ^{2}{\frac {\beta \xi }{2}}.}$

In terms of product:

${\displaystyle n_{\eta }^{\prime }(\xi )=-\beta n_{\eta }(\xi )(1+\eta n_{\eta }(\xi )).}$

In the zero temperature limit:

${\displaystyle n_{\eta }^{\prime }(\xi )=\eta \delta (\xi ){\text{ as }}\beta \rightarrow \infty .}$

${\displaystyle n_{\rm {B}}^{\prime \prime }(\xi )={\frac {\beta ^{2}}{4}}\operatorname {csch} ^{2}{\frac {\beta \xi }{2}}\operatorname {coth} {\frac {\beta \xi }{2}},}$

${\displaystyle n_{\rm {F}}^{\prime \prime }(\xi )={\frac {\beta ^{2}}{4}}\operatorname {sech} ^{2}{\frac {\beta \xi }{2}}\operatorname {tanh} {\frac {\beta \xi }{2}}.}$

${\displaystyle n_{\eta }(a+b)-n_{\eta }(a-b)=-{\frac {\mathrm {sinh} \beta b}{\mathrm {cosh} \beta a-\eta \,\mathrm {cosh} \beta b}}.}$

${\displaystyle n_{\rm {B}}(b)-n_{\rm {B}}(-b)=\mathrm {coth} {\frac {\beta b}{2}},}$

${\displaystyle n_{\rm {F}}(b)-n_{\rm {F}}(-b)=-\mathrm {tanh} {\frac {\beta b}{2}}.}$

${\displaystyle n_{\rm {B}}(a+b)-n_{\rm {B}}(a-b)=\operatorname {coth} {\frac {\beta b}{2}}+n_{\rm {B}}^{\prime \prime }(b)a^{2}+\cdots ,}$

${\displaystyle n_{\rm {F}}(a+b)-n_{\rm {F}}(a-b)=-\operatorname {tanh} {\frac {\beta b}{2}}+n_{\rm {F}}^{\prime \prime }(b)a^{2}+\cdots .}$

${\displaystyle n_{\rm {B}}(a+b)-n_{\rm {B}}(a-b)=2n_{\rm {B}}^{\prime }(a)b+\cdots ,}$

${\displaystyle n_{\rm {F}}(a+b)-n_{\rm {F}}(a-b)=2n_{\rm {F}}^{\prime }(a)b+\cdots .}$

Definition:

${\displaystyle c_{\eta }(a,b)\equiv -{\frac {n_{\eta }(a+b)-n_{\eta }(a-b)}{2b}}.}$

For Bose and Fermi type:

${\displaystyle c_{\rm {B}}(a,b)\equiv c_{+}(a,b),}$

${\displaystyle c_{\rm {F}}(a,b)\equiv c_{-}(a,b).}$

${\displaystyle c_{\eta }(a,b)={\frac {\sinh \beta b}{2b(\cosh \beta a-\eta \cosh \beta b)}}.}$

It is obvious that ${\displaystyle c_{\rm {F}}(a,b)}$ is positive definite.

To avoid overflow in the numerical calculation, the tanh and coth functions are used

${\displaystyle c_{\rm {B}}(a,b)={\frac {1}{4b}}\left(\operatorname {coth} {\frac {\beta (a-b)}{2}}-\operatorname {coth} {\frac {\beta (a+b)}{2}}\right),}$

${\displaystyle c_{\rm {F}}(a,b)={\frac {1}{4b}}\left(\operatorname {tanh} {\frac {\beta (a+b)}{2}}-\operatorname {tanh} {\frac {\beta (a-b)}{2}}\right).}$

${\displaystyle c_{\rm {B}}(0,b)=-{\frac {1}{2b}}\operatorname {coth} {\frac {\beta b}{2}},}$

${\displaystyle c_{\rm {F}}(0,b)={\frac {1}{2b}}\operatorname {tanh} {\frac {\beta b}{2}}.}$

${\displaystyle c_{\rm {B}}(a,0)={\frac {\beta }{4}}\operatorname {csch} ^{2}{\frac {\beta a}{2}},}$

${\displaystyle c_{\rm {F}}(a,0)={\frac {\beta }{4}}\operatorname {sech} ^{2}{\frac {\beta a}{2}}.}$

For a = 0: ${\displaystyle c_{\rm {F}}(0,b)={\frac {1}{2|b|}}.}$

For b = 0: ${\displaystyle c_{\rm {F}}(a,0)=\delta (a).}$

In general,

${\displaystyle c_{\rm {F}}(a,b)={\begin{cases}{\frac {1}{2|b|}},&{\text{if }}|a|<|b|\\0,&{\text{if }}|a|>|b|\end{cases}}}$