What is the difference between a magnon and a spinon? For a long time, I thought the terms "magnon" and "spinon" were equivalent, describing the collective spin excitation in a system. Lately, I have seen remarks in the literature that they indeed do differ, however I don't know in what sense. Can somebody, please, explain, how exactly do these technical terms differ?
 A: I think the magnon is a special case of the spin wave.  Whereas spinon refers to the general quasiparticle that carries all spin of an electron, magnon refers to the limiting case of spin wave quantized in such a manner that it becomes part of an anti-magnetic cloud of quasiparticles.  However, this may not be the complete story!
Some usage of the terms:
"In contrast to magnons, the spinon excitation amplitude is identical for all three orthogonal directions, α = x , y , z". ("Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain", Mourigal, Enderle, et.al., p.4, Nature Physics 16 June, 2013)
"It is necessary to clarify our terminology of spinon and magnon in our spin S = 1 systems. We call a spinon (a Gutzwiller projected state of) an unpaired fermion cm or more generally a Bogoliubov quasi-particle γm, which carries spin-1 (it is quite different from the slave particle approach for spin-1/2 systems where a spinon carries spin- 1/2), whereas a magnon is a physical spin-1 excitation in the spin system. In our Gutzwiller projected wavefunction approach, a magnon in the Haldane phase is a combination of a spinon plus a global Z2 flux (for spin-1/2 systems, a magnon is a combined state of two spinons)."  ("Gutzwiller approach for elementary excitations in S = 1 antiferromagnetic chains", Footnote 3, Liu, Zhou, Ng, New Journal of Physics, 14 August 2014)
A: The two answers given so far are wrong.
A magnon is an excitation carrying spin-$1$. A spinon is an excitation carryong spin-$\frac{1}{2}$. This has nothing to do with it being an excitation above a ferro- or antiferromagnet. The difference is much more dramatic, such that magnons are 'normal'/standard, yet spinons are very special.
Suppose you have some $SU(2)$-symmetric Hamiltonian (in more than one spatial dimension) made out of spin-$\frac{1}{2}$ particles which is in a ground state that spontaneously breaks the $SU(2)$. If you flip a single spin you've created a magnon, not a spinon. Intuitively you might think a spin-flip in a spin-$\frac{1}{2}$ system carries a half-integer spin, but that's not true: while it is true that $|\downarrow\rangle$ and $|\uparrow\rangle$ each carry a half-integer spin, their difference is an integer spin. In other words: magnetic phases (in more than one dimension) have magnon quasi-particles.
Spinons are much weirder. In fact since any local spin operator changes an integer amount of spin, you cannot create a single spinon with a local operator! Hence spinons are examples of fractionalized particles: they can only arise as part of a physical disturbance. For example spin liquids can give rise to spinons. A less exotic but still nice example is the spin-$\frac{1}{2}$ Heisenberg chain in one dimension where a single spin flip actually creates two emergent quasi-particles (two spinons). In a way a similar thing already happens in the transverse-field Ising chain $H =  -\sum S^x_n S^x_{n+1} + g \; S^z_n$: imagine going to the ordered phase and applying a single spin-flip operator -- can you see how this actually creates two quasi-particles?
A: What I can give you is the difference in spin's chain. The two, magnons and spinons, are excitations around different background states. The magnons are excitations around the ferromagnetic vacuum and the spinons are excitations around the antiferromagnetic vacuum. Because of that, a lot of properties are different between this two: the magnons are bosons and the spinons are fermions; magnons have spin 1 and spinons 1/2. The dispersion relation of a magnon in a spin chain is:
$$
\varepsilon (p)=4J\sin^2(\frac{p}{2})
$$
and the dispersion relation of the spinon is:
$$
\varepsilon (p)= \frac {\pi}{2} \cos (p)
$$
The S-matrix are different as well. All this follows because excitation are a state-dependent concept. So yes, this two are collective excitation in a "sea" of spins, but they are excitations around different states.
