A Matsubara sum is a summation of the following form,
which notably appears as the inverse
Fourier transform of the
Matsubara Green’s function:
is a meromorphic function on the complex frequency plane,
i.e. it is holomorphic
except for a known set of simple poles,
and is a real parameter.
The Matsubara frequencies are defined as follows
for bosons (subscript ) or fermions (subscript ):
How do we evaluate Matsubara sums?
Given a counter-clockwise closed contour ,
recall that the residue theorem
turns an integral over into a sum of the residues
of all the integrand’s simple poles that are enclosed by :
Now, the trick is to manipulate this relation
until a Matsubara sum appears on the right.
Let us introduce a (for now) unspecified weight function ,
which crucially does not share any simple poles with ,
This constraint allows us to split the sum:
Here, we could make the rightmost term look like a Matsubara sum
if we choose such that it has poles at .
We make the following choice,
where is the Bose-Einstein distribution for bosons,
and is the Fermi-Dirac distribution for fermions:
The distinction between the signs of is necessary
to ensure that for all when
(take a moment to convince yourself of this).
The sign flip for is also needed,
as negating the argument negates the residues
Indeed, this choice of has poles at the respective
Matsubara frequencies of bosons and fermions,
and the residues are given by:
With this, our contour integral can now be rewritten as follows:
Where the top sign () is for bosons,
and the bottom sign () is for fermions.
Here, we recognize the last term as the Matsubara sum .
Isolating for that yields:
Now we must choose .
Earlier, we took care that for ,
so a good choice would be a circle of radius .
If , then encloses the whole complex plane,
including all of the integrand’s poles.
However, because the integrand decays for ,
we conclude that the contour integral must vanish
(also for other choices of ):
We thus arrive at the following results
for bosonic and fermionic Matsubara sums :
- H. Bruus, K. Flensberg,
Many-body quantum theory in condensed matter physics,