The Spectral Function in Many-Body Physics and its Relation to Quasiparticles

Dear Robert, the answer to your question is trivial and your statement holds pretty much by definition.

You know, the Green's functions contain terms such as $$G(\omega) = \frac{K}{\omega-\omega_0+i\epsilon}$$ where $\epsilon$ is an infinitesimal real positive number. The imaginary part of it is $$-2\Im(G) = 2\pi \delta(\omega-\omega_0)$$ So it's the Dirac delta-function located at the same point $\omega$ which determines the frequency or energy of the particle species. At $\omega_0$, that's where the spectrum is localized in my case. If there are many possible objects, the $G$ and its imaginary part will be sums of many terms.

This delta-function was for a particle of a well-defined mass (or frequency - I omitted the momenta). If the particle is unstable, or otherwise quasi-, the sharp delta-function peak will become a smoother bump, but there's still a bump.

Because you didn't describe what you mean by "peak" more accurately, I can't do it, either. It's a qualitative question and I gave you a qualitative answer.

Cheers LM

Spectral function gives the number of state(or density of state if you divide volume,...etc), The peak means there's a state or there're several degenerate states there. In single particle system, spectral function are only delta function sets at where eigenstates are. Considering the many-body interaction (for ex: electron-electron interaction, electron-phonon interaction...etc in Condensed Matter) into hamiltonian as a perturb term and calculating the approximate solution in some degree, the new eigenstates ket could be called quasiparticle. Sometimes we called this particle as "dressed electron". It's just a approximation which merge those complicated interaction and electron into a "quasiparticle". Thus, the spectral function couldn't be so simple as a set of delta function in single electron system, but relates with the interaction, which add a so-called "self energy" term in spectral function. The real part of self-energy changes the peak position, the imaginary part changes the life time of the state.

you can see the ch1 & ch2 in this book: Green's Functions and Condensed Matter by G.Rickayzen.

Hope this message will help you.