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In usual three-dimensional superconductors the would-be Goldstone mode is eaten up by the electromagnetic gauge field via the standard Higgs mechanism.

I am thinking now about a problem of a two-dimensional superconductor, i.e., a paired fermions living in two spatial dimensions that are electrically charged and thus couple to dynamical electromagnetic field that lives in three spatial dimensions. For high-energy physicist I am thinking about a 2+1 dimensional brane where a complex scalar lives that minimally couples to a U(1) gauge field that lives in 3+1 dimensions.

I am wondering now what happens to a 2+1 dimensional Goldstone mode in this kind of system? Is it eaten up or remains gapless in this case?

I am sure the answer should be well-known because people in high energy physics must have thought about this kind of problem maybe in the context of string theory or brane models.

Thank you in advance for the answer!

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This is not a complete answer, rather I am very interested in someone answering this question so I wanted to do my part.

What I do know is that the Goldstone mode gets promoted to the plasma frequency in 3D, however in 2D the plasma mode is no longer gapped and the mode remains gapless (https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.65.1482). However how this resonates with the the usual prescription of Higgs mechanism in field theory I do not know.

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