Grand Canonical Ensemble: N operator problem

[DrDu]

Yes, Schrieffers book is nice. Also the following article by S. Cremer, M. Sapir and D. Lurie, Collective Modes Coupling Constants and Dynamical-Symmetry Rearrangement in Superconductivity, Il Nuovo Cimento, Vol 6(2), pp. 179 much more accessible than Nambu's paper. I found it especially interesting, how they derive the appearance of bound composite modes, something I find rather difficult to grasp in field theory.

Could you comment on my doubts about Greiters paper?

 

[vanhees71]

Hm, I get more and more confused. On the one hand I know for sure that there cannot be spontaneous symmetry breaking of a local gauge symmetry and Higgsing such a local gauge symmetry does not lead to Goldstone modes, which is very important for the electroweak standard model since there shouldn't be a massless scalar or pseudoscalar particle, because it's not observed after all.

On the other hand many people state that in superconductivity there is a Goldstonde mode present, including the just cited paper in #36. However, there the Hamiltonian is not gauge invariant, and they only consider the spontaneous breakdown of the usual U(1) symmetry of the effective Hamiltonian.

I always thought that superconductivity is just Higgsing the electromagnetic local U(1) symmetry due to a condensate of Cooper pairs. Then there shouldn't be any Goldstone modes in a superconductor.

Is there any experimental hint for gapless excitations in a superconductor?

 

[DrDu]

You mean the paper by Cremer et al? They show that the Goldstone mode is no longer massless once the electromagnetic field (i.e. local gauge invariance) is taken into account. Their hamiltonian is invariant wrt local gauge trafos.

 

[atyy]

Hm, but (3) is just the effective two-body potential of fermions, not the electromagnetic interaction. You get necessarily Cooper pairs as effective degrees of freedom, when this becomes attractive (in condensed matter physics via the electron-phonon interactions), as it is very nicely described in Haag's paper, but as I said, I'm not an expert in condensed matter physics, so that perhaps I misunderstand all this.

I'm hardly an expert either, but if I have correctly understood Greiter http://arxiv.org/abs/cond-mat/0503400, the BCS theory is gauge invariant even though it omits the electromagnetic gauge field, in the sense that since the gauge field is ignored, the theory only has to be invariant under the part of the gauge transformation in Eq 14, which is a rewrite of the gauge transformations in Eq 9 and 13. If the electromagnetic field is included, then one has to add Eq 11 to Eq 9 and 13 for the complete gauge transformation. (But I think DrDu doesn't agree that Greiter's gauge transformation is correct.)