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What are phonons, rotons, and maxons, and what does their dispersion curve have to do with superfluidity? I understand that they are quasiparticles, but I'm not entirely sure what that implies. Are these concepts purely notational and/or mathematical? Or are these three quasiparticles fundamentally different from each other?

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From my personal nomenclature: notaton (not-a-ton) n. Any notional "object" in a physical theory with "particle-like" properties that is not believed to represent a real particle. Examples reggeons, instatons. –  dmckee Dec 7 '10 at 19:11
    
@dmckee: I like that. But it should be noted that instantons aren't quite particle-like. For me a particle is an object that arises from a particle approximation of some QFT (by the means of creation-annihilation operators). Instantons are exact, non-perturbative solutions. And even their physical interpretation isn't really particle-like. They are more appropriately described as mediators of tunneling. At least, that's my intuitive view; I'd be happy if someone disagrees with it to provide their own intuition about instantons. –  Marek Dec 7 '10 at 19:56
    
@Marek: I'll defer to you on the details. IN the model that was used to introduce the things to me they obeyed both number conservation and momentum conservation which is "particle-like" for my purposes. –  dmckee Dec 7 '10 at 20:07
    
@dmckee: momentum conservation is obeyed also by fields (or indeed any system that behaves under suitably symmetrical physical laws). On the other hand, number conservation isn't something that is usually obeyed by particles (just take any QFT). Number conservation actually hints that instanton is indeed a true field (e.g. in standard setting you usually have precisely one field for any particle type so that it is indeed conserved). –  Marek Dec 7 '10 at 20:23
    
Clearly somebody thought it was enough like a particle to give it a particle-like name ;-) –  David Z Dec 7 '10 at 22:03

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Superfluid Helium-4 has a very well studied excitation structure -- at very low momenta, there is a low energy excitation, the phonon, that corresponds to a periodic density fluctuation in the superfluid with well defined wave-number and an energy $E = c \hbar k$ (c being the speed of sound in the superfluid). Though others might quibble with me over vocabulary, I prefer to call phonons "collective excitations" and reserve "quasiparticle" for excitations that correspond to renormalized single-particle excitations (like an electron in a Fermi Liquid).

In either case though, what is meant is simply that the excitation is long-lived, or, by the uncertainty principle, that it has a sharply defined energy, and here the phonon does.

This linear relation breaks down at higher momenta, where the $E,k$ curve turns down, then turns back up. Near the local maximum, there are sharp excitations with an inverse-parabolic dispersion. These are the maxons. Near the local minimum we have a sharp excitation with a parabolic dispersion, and these are the rotons.

Historically rotons were introduced by Landau, with a guess for their dispersion ($E=\Delta+(p-p_0)^2/2m$), not merely as a mathematical device, but because a superfluid described only by the phonon dispersion fails to capture the actual thermodynamics observed in Helium-4. On the other hand, Landau wasn't concerned about a qualitative, microscopic picture of what rotons are, in the sense that we know what phonons mean for the local density of the system.

So, I am actually not too sure what rotons are in that same microscopic sense, and the same for maxons, although it is apparent that a gas of phonons and a gas of rotons are very different things. On the other hand, this is somewhat paradoxical since all three of these excitations are just different parts of the same dispersion curve, so to say they are fundamentally different is a phrase I wouldn't use without a great deal of caution...

Lastly, knowing all of the excitations present in superfluid helium is very important for calculating its thermodynamic properties, etc., however I think (and hopefully someone will correct me if I'm wrong!) the only excitation that is intrinsic to superfluidity is the phonon mode - this is because the superfluid state breaks a continuous gauge symmetry of the normal fluid, and thus phonons are the massless Goldstone modes of the superfluid state.

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Sometimes quibbling over vocabulary is all the fun we get ;-) +1 nice answer. –  David Z Dec 15 '10 at 19:06
    
Thanks for a very nice answer. I assume by "dispersion" you mean a distribution of spectroscopic peaks (or, in the theoretical sense, energy levels/states)? –  David Hollman Dec 15 '10 at 20:02
    
By dispersion I mean very precisely the relationship between energy and momentum $E = f(p)$, so rather it is where your spectroscopic peaks appear if you were to plot them as a function of the momentum you imparted to the system with your probe. –  wsc Dec 15 '10 at 21:07

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