# Mermin Wagner theorem in superconductors, massive Goldstone mode

For a continuous symmetry breaking one receives a massless Goldstone mode which leads to divergent phase fluctuations in 2 and 1 dimension, thus we end up with a disordered state.

However in superconductors this Goldstone mode (phase mode) can be gauged away (Higgs effect). So why is Mermin Wagner theorem invoked for superconductors? There should be no gapless mode to disorder it? Or is it some subtlety with the Higgs effect in 2 dimensions?

(A small additional question: Anderson showed that this phase mode in fact gets promoted to the plasma frequency when including Coulomb-interaction. Is this equivalent with the usual approach of "gauging away" the phase?)

• I understand better the question. From what I gather, since the phase always appears with a gradient in the action, the inverse propagator is always proportional to $q^2$ (even though the there is a gap in the spectrum due to the Higgs mechanism). Now, the thermal physics is controlled by the zero frequency part of the propagator, which is still proportional to $1/q^2$. There is thus an infrared divergence in low dimensions, even with the gap. – Adam Jun 12 '17 at 14:33
• But the phase has redundancy, so isn't that divergence comming from the fact that we sum over physically equivalent configurations? By "gaugeing" away the residual freedom there is no phase mode left and the real freedom should be included in the gauge invariant, now massive, gauge field. Then I cannot find any infrared divergencies – JWDiddy Jun 12 '17 at 14:59
• Anderson mechanism is the same as Higgs mechanism: en.wikipedia.org/wiki/Higgs_mechanism second paragraph of the introduction. – FraSchelle Jun 13 '17 at 12:28
• 2D systems are more rich than Mermin-Wigner-Hohenberg theorem, see for instance Kosterlitz-Thouless-Berezinskii phase transition : see Wikipedia about Mermin-Wagner theorem (i can not copy-paste the link, sorry) – FraSchelle Jun 13 '17 at 12:30
• @FraSchelle Ok but it seems that Anderson used the explicit Coulomb interaction to show that the mode phase mode acquires mass. But maybe this is because he didn't regard the gauge field as a "dynamical field". I mean coulomb is mediated by the gauge field in the field theory context. Do you have any inputs on the main question? Any hints :) – JWDiddy Jun 13 '17 at 12:37