Matsubara frequency
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.
Matsubara summation refers to the technique of expanding these fields in Fourier series
are called the Matsubara frequencies, taking values from either of the following sets (with
): which respectively enforce periodic and antiperiodic boundary conditions on the field
Once such substitutions have been made, certain diagrams contributing to the action take the form of a so-called Matsubara summation
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.
With the weighting function, the summation can be replaced by a contour integral surrounding the imaginary axis.
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.
[5] 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), 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.
, 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.
There are also two types of Matsubara weighting functions that produce simple poles at
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
In that case, any choice of the Matsubara weighting function will lead to identical results.
The symbol η = ±1 is the statistical sign, +1 for bosons and -1 for fermions.
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 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 It satisfies the periodic boundary condition G(τ+β)=G(τ) for a boson field.
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.
The small imaginary time plays a critical role here.
To evaluate the summation both choices of the weighting function are acceptable, but the results are different.
Multiple mode problems can be approached by a spectral function integral.
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.
In terms of product: In the zero temperature limit: Definition: For Bose and Fermi type: It is obvious that
To avoid overflow in the numerical calculation, the tanh and coth functions are used For a = 0:
