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
where formula_2 is the inverse temperature and the frequencies formula_3 are usually taken from either of the following two sets (with formula_4):
The summation will converge if formula_7 tends to 0 in formula_8 limit in a manner faster than formula_9. The summation over bosonic frequencies is denoted as formula_10 (with formula_11), while that over fermionic frequencies is denoted as formula_12 (with formula_13). formula_14 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 formula_15, a certain Feynman diagram is represented by an integral formula_16, at finite temperature it is given by the sum formula_17.
Summation formalism.
General formalism.
The trick to evaluate Matsubara frequency summation is to use a Matsubara weighting function "h""η"("z") that has simple poles located exactly at formula_18. 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.
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")"h""η"("z") over all poles of "g"("z"),
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.
Choice of Matsubara weighting function.
To produce simple poles on boson frequencies formula_21, either of the following two types of Matsubara weighting functions can be chosen
depending on which half plane the convergence is to be controlled in. formula_24 controls the convergence in the left half plane (Re "z" < 0), while formula_25 controls the convergence in the right half plane (Re "z" > 0). Here formula_26 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 formula_27
formula_30 controls the convergence in the left half plane (Re "z" < 0), while formula_31 controls the convergence in the right half plane (Re "z" > 0). Here formula_32 is the Fermi–Dirac distribution function.
In the application to Green's function calculation, "g"("z") always have the structure
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 formula_34. 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.
Table of Matsubara frequency summations.
The following table contains formula_35
for some simple rational functions "g"("z"). The symbol "η" = ±1 is the statistical sign.
[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.
Applications in physics.
Zero temperature limit.
In this limit formula_36, the Matsubara frequency summation is equivalent to the integration of imaginary frequency over imaginary axis.
Some of the integrals do not converge. They should be regularized by introducing the frequency cutoff formula_38, and then subtracting the divergent part (formula_38-dependent) from the integral before taking the limit of formula_40. For example, the free energy is obtained by the integral of logarithm,
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
which shows step function behavior at zero temperature.
Green's function related.
Time domain.
Consider a function "G"("τ") defined on the imaginary time interval (0,"β"). It can be given in terms of Fourier series,
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
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
with
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.
Operator switching effect.
The small imaginary time plays a critical role here. The order of the operators will change if the small imaginary time changes sign.
Distribution function.
The evaluation of distribution function becomes tricky because of the discontinuity of Green's function "G"("τ") at "τ" = 0. To evaluate the summation
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 formula_51 as the weighting function for formula_52, and formula_53 for formula_54.
Bosons
Fermions
Free energy.
Bosons
Fermions
Diagram evaluations.
Frequently encountered diagrams are evaluated here with the single mode setting. Multiple mode problems can be approached by spectral function integral.
Appendix: Properties of distribution functions.
Distribution functions.
The general notation formula_64 stands for either Bose ("η" = +1) or Fermi ("η" = −1) distribution function
If necessary, the specific notations "n"B and "n"F are used to indicate Bose and Fermi distribution functions respectively
Relation to hyperbolic functions.
The Bose distribution function is related to hyperbolic cotangent function by
The Fermi distribution function is related to hyperbolic tangent function by
Parity.
Both distribution functions do not have definite parity,
Another formula is in terms of the formula_70 function
However their derivatives have definite parity.
Bose–Fermi transmutation.
Bose and Fermi distribution functions transmute under a shift of the variable by the fermionic frequency,
However shifting by bosonic frequencies does not make any difference.
Derivatives.
First order.
In terms of product:
In the zero temperature limit:
The function "c""η".
Definition:
For Bose and Fermi type:
Relation to hyperbolic functions.
It is obvious that formula_90 is positive definite.
To avoid overflow in the numerical calculation, the tanh and coth functions are used
Low temperature limit.
For "a" = 0: formula_97
For "b" = 0: formula_98
In general,