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In solid state physics emergent particles are very common.

  1. How one determines if they are gap-less excitations?

  2. Do the defects in spin ice called magnetic monopoles have mass?

  3. What is the mass of emergent particles in the fractional quantum hall effect (FQHE)?

I am not looking for a deep theory, just the general picture.

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  1. I do not think there is anything one can say in full generality about determining whether "emergent" particles. Goldstone's theorem guarantees gapless excitations in some cases, but really you just gotta talk about the specfic problems.
  2. The spin-ice monopoles (which must be produced in pairs) should have a mass gap since introducing a new pair requires you to violate the spin-ice rules at a new location which has a finite energy cost. See Equation (2) of the original paper by Castelnovo, Moessner and Sondhi.
  3. The excitation spectrum of FQHE has a thirty year history of experimental and theoretical study of an enormous number of distinct states, so I don't know that a short answer can be given.
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