Grand Canonical Ensemble: N operator problem

[atyy]

The Scholarpedia Higgs article by Kibble http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism does agree that it is misleading but conventional to talk about "spontaneous gauge symmetry breaking" when he refers to an Abelian model, and that it is better to say that there is an explicit breaking of the gauge symmetry by some gauge choices, but the state is gauge invariant. However, he also does say "the resulting theory does retain a global phase symmetry that is broken spontaneously by the choice of the phase of $\langle\Phi\rangle$."

I guess the interesting point about Kibble's remark here is that if the Abelian Higgs model does break a "global symmetry" because there is a phase with non-zero order parameter, why then are there no Goldstone bosons? The commentary given by Hansson et al is that the gauge invariant order parameter in this type of "symmetry breaking" is nonlocal, and doesn't give rise to Goldstone bosons. Hansson et al say that ground state degeneracy in the Abelian Higgs case depends on the topology of the manifold.

http://arxiv.org/abs/cond-mat/0404327 (p5) "With a local order parameter this would be a signature of Goldstone bosons. In fact, the Anderson-Higgs mechanism forbids any such bosons in the actual spectrum, which shows that a description based on $\phi_{D}$ does not have the character of the standard sigma model."

 

[DrDu]

Why is Galilei invariance (or Poincare invariance in the relativistic case) broken for a BEC? That doesn't make sense to me. How can a Galilei- (Poincare-) covariant theory break Galilei (Poincare) invariance?

At the moment, I find no better source than wikipedia http://en.wikipedia.org/wiki/Goldstone_boson
There is quite simple an argument working backward: In a BEC $\langle a_0\rangle\neq 0$. Hence particle number is unsharp. However, in a nonrelativistic context particle number conservation is a superselection rule following from the invariance under boosts from Galilei group and time reversal. So invariance under the subgroup generated by the boosts must be broken.

 

[DrDu]

I guess the interesting point about Kibble's remark here is that if the Abelian Higgs model does break a "global symmetry" because there is a phase with non-zero order parameter, why then are there no Goldstone bosons? The commentary given by Hansson et al is that the gauge invariant order parameter in this type of "symmetry breaking" is nonlocal, and doesn't give rise to Goldstone bosons. Hansson et al say that ground state degeneracy in the Abelian Higgs case depends on the topology of the manifold.

http://arxiv.org/abs/cond-mat/0404327 (p5)

The following article by Kennedy and King is interesting (Hansson cites the PRL short version of it)
http://projecteuclid.org/download/pdf_1/euclid.cmp/1104115008
They show that there is a Goldstone boson in Landau gauge and suspect that it doesn't couple to physical degrees of freedom.
For the relativistic Higgs models it is known that the Goldstone theorem holds in Lorentz gauge, but that the Goldstone bosons aren't in the physical sector and hence unobservable.

 

[vanhees71]

Haag's model is the BCS model, which I think this is usually considered gauge invariant, but the electromagnetic field is not dynamical.

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. As far as I can see all this is solved for a long time. The paper by Nambu has been already cited. A very clear description can also be found in the book by Schrieffer:

J. R. Schrieffer, Theory of Superconductivity, Westview Press (1999)

 

[DrDu]

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. As far as I can see all this is solved for a long time. The paper by Nambu has been already cited. A very clear description can also be found in the book by Schrieffer:

J. R. Schrieffer, Theory of Superconductivity, Westview Press (1999)

 

[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.)