Thursday, May 26, 2011

The complex nature of the self energy

I am trying to understand under exactly what conditions it is (or is not) meaningful to use a self energy to describe and understand experiments on a strongly correlated metal which may be (at least in some sense) a non-Fermi liquid. This is particularly motivated by a recent paper on the overdoped cuprates.

Below are some statements which I am trying to ascertain the truth of and relationship between them. I believe
1. and 2. are always true.
3. and 4. are equivalent but are not always true.
5. is true.
I am not sure about 6.

1. The one-electron Greens function G(k,E) is an analytic function of energy E.

2. One can always define a self energy by Dyson's equation
where G0 is the non-interacting Greens function. This self energy will be an analytical function of E.

3. If E is treated as a complex variable G(k,E) has isolated simple poles in the complex plane. These poles correspond to quasiparticles. One can then write down a Boltzmann transport equation for these quasiparticles.

4. The self energy can be written as a convergent perturbation expansion. This ensures adiabatic continuity and the existence of quasi-particles.

5. If G(k,E) has a non-integer power law dependence on E there will be a branch cut in the complex plane. This means a description in terms of quasi-particle poles is inadequate.
[This is what happens in one dimension with Luttinger liquids].

6. Branch cuts in the plane may mess up the Kramers-Kronig relation which relates the real and imaginary parts of the self energy.

So, I welcome thoughts about the above claims.
Is there somewhere that this is all written down and discussed clearly?
I have gleaned the above from my subconscious memory of a diverse range of sources.
The figure is from Piers Coleman.


  1. If you have a system for which k is not a good variable (i.e. finite systems) the analog of the Greens function would be an expectation value of a Greens operator. Is the Dyson equation true in the general case (i.e. can you always use this to define an operator-valued self energy)?

  2. I would say that as a function of complex E, the Green's function always has discontinuities on the real axis, either in the form of poles or branch cuts. This is clear from the Lehmann representation, discussed for example in Mahan's book. Evaluated right above the real axis it is equal to the retarded Green function and it is equal to the advanced one when evaluated right below the real axis. The discontinuity is the single particle spectral weight.
    Finally, the analytical properties of the self-energy are the same as those of the Green's function. That can be deduced for example from Appendix A of M. Potthoff, Eur. Phys. J. B 36, 335 (2003). Hence, for the self-energy the Kramers-Kronig relations follow from causality and are completely general I believe.

  3. Re statement 3, a quasiparticle pole is not a pole of the Green's function (as given by the Lehmann representation) but of an analytic continuation of the Green's function. This also explains why the existence of such poles located a finite distance from the real axis does not contradict the Lehmann representation. Both Nozieres and Schrieffer discuss this analytic continuation in their books. Nozieres' discussion (in an Appendix) is particularly thorough.