Wednesday, February 26, 2014

Two historical questions about incoherent excitations

I believe that one of the most important concepts in quantum many-body physics is that of quasi-particles and the associated incoherent excitations. Often the one particle spectral function can be written in the form.

where the first term is a well-defined peak associated with quasi-particles and total spectral weight Z_k. 
The second term describes incoherent excitations, i.e., it has a weak dependence on the momentum k and as a function of omega is a broad distribution, in contrast to the sharp quasi-particle peak.
Futhermore, due to a sum rule [conservation of particle number] the total spectral weight of the incoherent part is 1-Z_k.

I think this equation is one of the most profound and important results in quantum many-body theory.

Some of this is illustrated in the figure below taken from a Nature Physics commentary by Nandini Trivedi.
The above equation and concepts have come to the fore over the past two decades due to wide studies of strongly correlated electron materials, particularly via dynamical mean-field theory and experimental ARPES [Angle-Resolved PhotoEmission Spectroscopy] studies.

I have two historical questions I am struggling to find answers for:

1. When and by whom was the equation above first clearly written down and elucidated?

I suspect sometime in the 1950-60s by Landau, Pines, Nozieres, Kadanoff, Baym, or Hubbard?
I looked in AGD and several other old books but could not find it.

2. When was the first time that the incoherent part of the spectral function was definitively observed in 
an experiment?

By this I don't mean just seeing some background [that could be noise], but actually showing that the incoherent background has the weight 1-Z_k. I presume an ARPES experiment in the past two decades.

I should know this, but just can't quickly find the answer.


  1. Regarding the first question, I would say that this equation was introduced, for the first time, in the work by Nozieres and Luttinger, Phys Rev. 127, 1423 (1962) (eq. 2.14), although a very detailed discussion about quasi-particles can be found in the book of Nozieres "Theory of interacting Fermi Systems "(Ch 4 Sec Ib), published that same year in its French version.

    1. Hi Luis,

      Thanks for the very helpful comment.
      The Phys. Rev. is very cursory. They have the equation [with Gamma-> 0] but do not actually call the second term the incoherent part, and never really discuss it.

      On the other hand, the book does have the equation and talk about the incoherent part. It is actually introduced in chapter 3, section 1d [page 71 in the 1997 reprint in Advanced Book Classics]. It even has a sketch.

  2. There is a more recent discussion in arxiv:0103393 where they reference Nozieres' book but also Abrikosov (same year, see page 12 just above Eq.5.

    1. Thanks.
      I am a bit confused. I could not find any discussion of the incoherent part above eqn. 5.
      Rather, I found the relevant discussion and equation above eqn. 9. No reference is given.

  3. Regarding question 2. This is difficult to see definitively in photoemission because there is a unknown matrix element for optical excitation that is light polarization and energy dependent, electron momentum dependent, and energy dependent. This generally musses any normalization that might be exist.

    1. One experimental ARPES paper that claims to measure the incoherent part is

  4. This equation was also noted as Eq. (10.2) in the book 'Course of Theoretical Physics Vol 9: Statistical Physics part 2', by Lifshitz and Pitaevskii (Pergamon Press 1980). But again no detailed discussion of the incoherent part.