Skin depth and Mermin-Wagner theorem

Question

I recently became aware of the Coleman-Mermin-Wagner theorem presented in [1802.07747] for higher-form symmetries and I have a question about how it might be applied to electromagnetism.

The theorem states: continuous $p$-form symmetries in $D$ spacetime dimensions are never broken if $p ≥ D − 2$.

In 3+1 spacetime dimensions, we consider the photon to be the Goldstone associated with a continuous $U(1)$ one-form symmetry. However, as stated in [1802.07747], in 2+1 dimensions, this interpretation as a Goldstone is no longer allowed.

Questions:

1. Suppose I have a dielectric material at finite temperature in 3+1 spacetime dimensions. Effectively, I now have a 3 dimensional system. Does this mean that the $U(1)$ one-form symmetry cannot be spontaneously broken?

2. If the answer to the above question is "yes," then the photon field can no longer be viewed as a Goldstone and thus need not satisfy Goldstone's theorem. As a result, I would expect the low-frequency dispersion relation $\omega \propto k$ should not hold (in the absence of fine-tuning). Are there any circumstances in which this dispersion relation holds exactly at arbitrarily low frequency (again in the absence of fine-tuning)?

3. If the answer to the above question in "no," then can we interpret the Coleman-Mermin-Wagner theorem as mandating that all finite-temperature materials have a finite skin-depth beyond which electromagnetic radiation gets exponentially damped?

Answer 1

Great question! However, the question isn't really specific to 1-form symmetries. You can ask the same question about a system in (2+1)-D that spontaneously breaks a 0-form $\mathrm{U}(1)$ symmetry. At nonzero temperature the Coleman-Mermin-Wagner theorem forbids the $\mathrm{U}(1)$ symmetry to be spontaneously broken, so you can ask what happens to the Goldstone modes. The answer is, they are still present, with no frequency gap opening up, as long as you are below the Berezinskii–Kosterlitz–Thouless (BKT) transition temperature.

A nice explanation for this is provided in the following paper:

https://arxiv.org/abs/1908.06977

where it is argued that a sufficient requirement for Goldstone modes in $d$ spatial dimensions is the existence of an emergent $d$-form symmetry that has a mixed anomaly with the 0-form $\mathrm{U}(1)$ symmetry. Such an emergent symmetry is basically equivalent to the absence of vortices, and is present in the (2+1)-D system below the BKT temperature.

I imagine there is probably a similar statement that applies to (3+1)-D electromagnetism. It has two emergent $\mathrm{U}(1)$ 1-form symmetries that have mixed anomalies with each other. That statement is presumably robust even at nonzero temperature, even though the spontaneous symmetry breaking is not, and that would be sufficient to imply a gapless photon.