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

recently, I stumbled accross a concept which might be very helpful understanding quasiparticles and effective theories (and might shed light on an the question How to calculate the properties of Photon-Quasiparticles): the spectral function $$A\left(\mathbf{k},\omega \right) \equiv -2\Im G\left(\mathbf{k},\omega \right)$$

as given e.g. in Quasiparticle spectral function in doped graphene (on arXiv).

It is widely used in many-body physics of interacting systems and contains the information equivalent to the Greens function $G$. For free particles, $A$ has a $\delta$-peaked form and gets broader in the case of interactions.

The physical interesting thing is, as I read, quasiparticles of interacting systems can be found if $A$ is also somehow peaked in this case. I don't understand this relationship, hence my question:

### What is the relation of the spectral function's peak to the existence of quasiparticles in interacting systems?

Thank you in advance
Sincerely

Robert

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You'll find the term "spectral function" in nuclear physics as well, were it plays the role of a energy and momentum distributions (as measured in some interaction) for nuclear components. Without being familiar with the application you cite, it appears to have a similar meaning. –  dmckee Jan 22 '11 at 21:06
@dmckee: Thank you for this connection. If I can get my hands on $A$ in this meaning I will have to look if I in turn can understand something from nuclear physics as well :) Greets –  Robert Filter Jan 23 '11 at 11:55

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

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Thank you for your response. Could you please comment on why quasiparticles in interacting systems correspond to peaks of $A$ - can one e.g. see cooper pairs and derive BCS theory (for which I am obviously no expert) from this viewpoint? Furthermore, can one apply this to composed systems with different $G$ for different domains along with appropriate boundary conditions? I am sorry for the lengthy comment and the maybe not so sharp question. Greets –  Robert Filter Jan 23 '11 at 11:51
There is a little more to this. The spectral function can be almost directly measured (e.g. with ARPES or surface tunnelling experiments), and so is a massively important link between experiment and theory. The real fun begins when the spectral function doesn't show a dominant peak... –  genneth Jan 30 '11 at 10:29
@genneth: Thank you for the further info! You may want to consider making another nice answer out of your comment with some references and further explanations :) - I assume that most people do not read the comments. Greets –  Robert Filter Jan 30 '11 at 12:34

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.