Hello,
How did we end up with bosonic Matsubara Frequency for dressed phonon propagator ? Is it due to phonon being a boson ?
Regards,
Krishnendu
Hello Krishnendu,
Yes. The phonon propagator has bosonic Matsubara frequencies, as phonons are bosons, so its propagator is periodic in imaginary time:
D(\tau+\beta)=D(\tau),
which gives \Omega_l=2\pi lT.
Second, in the electron self-energy, the phonon carries the frequency difference between two fermionic frequencies:
\Omega_l=\omega_j-\omega_{j’}.
Since fermionic matsubara frequency omega_j=(2j+1)\pi T,
their difference is always \Omega_l=2(j-j’)\pi T,
which is bosonic.
So the bosonic frequency is not an extra assumption; it follows both from the statistics of the phonon and from energy conservation at the electron-phonon vertex.
Dressing the phonon does not change this. It modifies the phonon spectral function, but the Matsubara frequencies remain bosonic.
In fact, we can also express it in another way, in terms of fermionic Matsubara frequencies. It depends on convience. In the current EPW, the Coulomb term (one term) is written in Bosonic notation, while the rest of the Eliashberg part is expressed in fermionic notation.
Hope this helps.
Best,
Shashi