Why are there no Goldstone modes in superconductor?

Question

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?

Answer 1

The Goldstone modes of a superconductor (considered as an electron gas with attractive potential) are physically the oscillations of electron density. As electrons are intrinsically charged, this will inevitably create the EM field. So we get plasma oscillations and nonzero mass of the Goldstone mode.

If we somehow turn off the Coulomb interaction between electrons, I guess that the Goldstone modes (density waves) will indeed become massless. But that's physically meaningless for electrons. So, answering your question, I would say that the external EM field does not play role here. Indeed, Goldstone modes acquire mass due to dynamic coupling with EM field.

Answer 2

A neutral fermionic superluid has gapless soundwaves that can be regarded as a Goldstone mode due to the spontaneous breakdown of translational symmetry, but the fluctuating quantity in the sound wave is not the order parameter $\langle \psi\psi\rangle$ but rather the combination
$$ \Phi(x,t)= \sqrt{\rho} e^{i\theta} $$ where $\rho$ is the fluid density and $\theta$ phase of $\langle \psi\psi\rangle$.

Further the non-relativistic system does not have separate radial and tangential modes. Instead the radial and tangential part are coupled with the $\rho-\rho_0$ being the density fluctuations and ${\bf v}\propto \nabla \theta$ being the back-and-forward motion of the fluid.

There is some experimental evidence for fluctuations in the magnitude of $\langle \psi\psi\rangle$ but this is usually strongly damped.