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## Collective modes and restoration of gauge invariance in superconductivity

After the first explanation of superconductivity by Bardeen, Cooper and Schrieffer, it was for several years a matter of concern to render the theory charge conserving and gauge invariant. I have been reading the article by Y. Nambu, Phys. Rev. Vol. 117, p. 648 (1960) who uses Ward identities to establish gauge invariance and the book by Schrieffer, "Theory of superconductivity" from 1964.
While I can follow the steps of the calculation, the physical content is not quite clear to me.
While the difference between a free and a "dressed" Greens function is quite clear to me, the
concept of a dressed vertex is much less. I only see it as a formal device to calculate the current-current correlation functions. The collective modes somehow fall out as homogeneous solutions of a Bethe Salpeter type equation.
Schrieffer stresses that the mechanism in fact is not peculiar to superconductivity but holds also in normal metals. I know of some discussions of the "backflow" which is also not too clear to me. I suppose that these matters are better understood now half a century later. Maybe someone knows a more pedagogical reference?

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I'm not really qualified to answer your questions but maybe some references might help you on your way.  If you are quite familiar with the functional integral approach then I think this is the more modern approach to dealing with superconductors in a gauge invariant manner as well as identifying the collective modes. See for example this article by van Otterlo, Golubev, Zaikin and Blatter, where they also discuss ward identities. Another pedagogical description can be found in Altland and Simons book Condensed Matter Field Theory.  I think the connection to the dressed vertices might be found from the vertex generating functional (or effective potential) $\Gamma[\Psi]$ ($\Psi$ being the pairing field) that one can obtain with the functional integral technique (for definition and introduction see for example sec. 2.4 in Negele and Orland's book Quantum Many-Particle Systems). In Weinberg's book The quantum theory of fields Vol II you can find a treatment of superconductors in terms of the effective potential. Also see this article by Weinberg.  I hope this helps in some way. Let me know if it was useful or if you find any more pedagogical references.

Recognitions:        Science  Advisor  Dear Jensa,   thank you very much for your answer. I am having a look at the article by van Otterlo. I knew about the work by Weinberg, but I am not sure if it is what I am looking for.  There seems to be a highly relevant article written by Martin in the book "superconductivity" ed. by Parks. However, I only have partial access to it at the moment via google books. Maybe I come back to you after having read the articles.

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## Collective modes and restoration of gauge invariance in superconductivity

Dear Jensa,

I was reading some articles in the last time and I think I at least understand the problem:
Immediately after the paper by BCS, people became worried because the calculation of the Meissner effect was not Gauge invariant and it was shown that at least for the response to longitudinal fields violation of gauge invariance could lead to any result. It became clear that to restore gauge invariance one has to consider collective excitations which are now known as Nambu Goldstone bosons and correspond to variations of the phase of the gap parameter.
The whole history is quite fascinating as it lead to the discovery of the Higgs mechanism in field theory, but that's another story (told e.g. in the Noble lecture of Nambu).
In a superconductor, current and charge can be carried by both particle like excitations resulting from acting with the Valatin Bogoliubov operators on the vacuum and by the collective Goldstone mode. The currents carried by both modes are conserved separately, if calculated correctly. I was trying to understand this from the following two articles:

1. S. Cremer, M. Sapir and D. Lurie, Collective Modes, Coupling Constants and Dynamical-Symmetry Rearrangement in Superconductivity, Nuovo Cim., Vol 6, pp. 179, 1971

2. L. Leplae, H. Umezawa and F. Mancini, Derivation and Application of the Boson Method in Superconductivity, Phys. Rep. Vol 10, pp. 151-272, (1974)

I liked the exposition in 1. but I do not understand exactly what he is doing in section 4. Especially he sais:
"We shall exhibit here a simple
alternative technique which yields considerable physical insight into the dy-
namical mechanism involved (*). Our method is based on the well-known
fact that the matrix elements of an operator density between physical one-
particle states can be computed by coupling the operator in question to an
external c-number field and evaluating the lowest-order transition amplitude
induced by the external field. Applied to superconductivity, this technique
will be shown to lead very rapidly to the results of LEPLAE and UMEZAWA (5)
i.e. to the explicit expression of the current operator in terms of the physical
quasi-particle and Goldstone field operators."
He then directly discusses some Feynman diagramms the relevance of which is not directly clear to me from the aforesaid.

The article 2. uses some apparently non-standard techniques but adresses the interesting point more directly which I never exactly understood and which seems to be at the heart of the problem: How exactly do we separate particle like modes from collective modes?

Although I like in principle the "modern" treatment as e.g. in the article by Otterlo you gave me, I don't see how to identify or treat single particle like excitations after having performed the hubbard stratonovich transformation.

Maybe you have some fresh ideas.

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