Usually, the absence of Goldstone modes in a superconductor is seen as an example of the Anderson-Higgs mechanism, related to the fact that there is gauge invariance due to the electromagnetic gauge field coupled to the charged electrons.

However, this is puzzling in light of the fact in (3d) neutral atomic gas the fermionic degrees of freedom also undergo BCS transition, and there will be Goldstone modes since Goldstone theorem applies.

Mathematically the same model with an attractive $ g\bar{\psi}\bar{\psi} {\psi} {\psi}$ term is used for both superconductor and atomic gas BCS transtion. Suppose no external electromagnetic field is applied (so the Anderson-Higgs mechanism does not apply), then how come in one case there are Goldstones, while in a superconductor there are only gapped plasmons?

Usually, the absence of Goldstone modes in a superconductor is seen as an example of the Anderson-Higgs mechanism, related to the fact that there is gauge invariance due to the electromagnetic gauge field coupled to the charged electrons.

However, this is puzzling in light of the fact in (3d) neutral atomic gas the fermionic degrees of freedom also undergo BCS transition, and there will be Goldstone modes since Goldstone theorem applies.

Mathematically the same model with an attractive $ g\bar{\psi}\bar{\psi} {\psi} {\psi}$ term is used for both superconductor and atomic gas BCS transtion. Suppose no external electromagnetic field is applied (so the Anderson-Higgs mechanism does not apply), then how come in one case there are Goldstones, while in a superconductor there are only gapped plasmons?

I guess another way to put the question is, when no external EM field applied, are there Goldstone modes in a superconductor, and why?

Usually the fact that the absence of Goldstone modes in superconductor is seen as an example of Anderson-Higgs mechanism, related to the fact that there is gauge invariance due to the electromagnetic gauge field coupled to the charged electrons.

However this is puzzlling in light of the fact in (3d) neutral atomic gas the fermionic degrees of freedom also undergoes BCS transition, and there will be Goldstone modes since Goldstone theorem applies.

Mathematically the same model with an attractive $ g\bar{\psi}\bar{\psi} {\psi} {\psi}$ term is used for both superconductor and atomic gas BCS transtion. Suppose no external electromagnetic fied is applied (so the Anderson-Higgs mechanism does not apply), then how come in one case there are Goldsteones, while in superconductor there are only gapped plasmons?

Usually, the absence of Goldstone modes in a superconductor is seen as an example of the Anderson-Higgs mechanism, related to the fact that there is gauge invariance due to the electromagnetic gauge field coupled to the charged electrons.

However, this is puzzling in light of the fact in (3d) neutral atomic gas the fermionic degrees of freedom also undergo BCS transition, and there will be Goldstone modes since Goldstone theorem applies.

Mathematically the same model with an attractive $ g\bar{\psi}\bar{\psi} {\psi} {\psi}$ term is used for both superconductor and atomic gas BCS transtion. Suppose no external electromagnetic field is applied (so the Anderson-Higgs mechanism does not apply), then how come in one case there are Goldstones, while in a superconductor there are only gapped plasmons?