# When do gauge theories have protected gapless excitations?

Goldstone's theorem states that a system in which a continuous symmetry is spontaneously broken necessarily has gapless excitations. (A hand-waving "proof" of Goldstone's theorem can be given by noting that very large wavelength spatial variations in the order parameter should have very small energy cost).

On the other hand, there appears to be another way to obtain protected gapless excitations, namely by having a (possibly emergent) local gauge invariance where the gauge group is continuous, so that there is a gapless gauge boson. What is the most general statement that one can make in this case, analogous to Goldstone's theorem? Clearly, the gauge charges must remain deconfined, since we know that confined/Higgs phases of gauge theories are gapped. Is the existence of deconfined gauge charges (for a continuous gauge group) a sufficient condition to ensure gaplessness, and if so, how would one prove this?

Is the existence of deconfined gauge charges a sufficient condition to ensure gaplessness?

I think the answer is NO, such as the $Z_2$ gauge theory in 2+1D and 3+1D.

I believe that the existence of deconfined gauge charges of a continuous gauge group is a sufficient condition to ensure gaplessness?

Hastings and I have a paper (http://arxiv.org/abs/cond-mat/0503554 ) arguing for a "Goldstone's theorem" for gauge theory: Gapless gauge boson is robust agianst any local perturbations. In other words, the gaplessness of gauge boson is topological, no local perturbation can gap them.

But the gaplessness is not protected by gauge symmetry. In a lattice gauge theory, with gapless gauge boson, even a local perturbation that break the lattice gauge symmetry cannot give the gauge boson a mass (or a gap).

Both gauge symmetry and gaplessness are results of long range many-body entanglement.

• Dear Prof. Wen, thanks for your answer. Indeed I should have said "continuous gauge group". It was reading your paper with Hastings that inspired me to ask this question, but in that paper you just argue that the gapless phase is stable to small perturbations. My question is, does there exist ANY phase in which there are deconfined gauge charges (for continuous gauge group) but not gapless excitations, and if not, why not? Aug 19, 2015 at 16:24

Here is a partial answer that depends on a particular choice of local gauge constraint. In a U(1) gauge theory, the usual gauge constraint is just Gauss' Law, $$\nabla \cdot \mathbf{E} = \rho.$$ This in turn implies Coulomb's Law $\mathbf{E} \sim 1/r$ for the electric field surrounding a deconfined point charge. Such a long-range interaction ought to be mediated by a gapless photon. Hence, we conclude that deconfined gauge charges implies gaplessness in U(1) gauge theory, at least for this choice of gauge constraint.