*Updated 2020-03-22
*

One problem with scientific publishing is that the most up-to-date information about a topic is spread out across numerous extremely technical journal articles, none of which explains the concept from scratch. In response to a request from a friend (see previous post), I thought I would take a little time to try to answer the question: “what is a spinon?”

There two answers this question, (1) what are the spinons themselves and (2) what role do spinons play in the theory. I will try to answer both of these questions here for an audience that is fairly knowledgeable about physics, but not necessarily experts in condensed matter physics.

# What are spinons?

Spinons are **quasiparticles**: excitations of some underlying system. They don’t really make sense outside the context of that system, so we first need to explain our system.

In our paper we are interested in the transition between the Néel state (an alternating arrangement of up and down spins) and the valence-bond solid (VBS). We start with a square lattice of spins; at each vertex on the lattice there is a local S=1/2 degree of freedom (a spin). In a material, these spins might be localized electrons on a lattice of atoms. The VBS is formed when these spins all pair up with one of their neighbors to form singlets as in the figure below (blue ovals). Crucially, they pair up in ordered rows or columns to minimize the energy. There are four of these possible ordered states.

Within the VBS, you can make an excitation by breaking one of those singlet bonds to make a triplet (indicated by two up arrows below). This is called a triplon, a gapped bosonic spin-one excitation. Triplons are the conventional bosonic excitations that you expect in this type of system.

Once we have this triplon, we can imagine moving the two halves of it around independently. Each of these halves corresponds to a **spinon**. Each spinon carries half the spin of the original triplon, so they are spin-half. Pulling the spinons apart increases the energy because it introduces ‘unhappy’ bonds (indicated by the red haze). The further apart we pull them, the more energy is required and therefore spinons are *confined* (just like quarks within a proton).

**Aside:** if we take the spinons very far apart, then we can think of them a ‘vortices of the VBS order parameter’, a complicated way of saying that each spinon would be at the intersection of four domains representing each possible VBS order. I wanted to point this out to justify showing this excellent figure. There are also some great illustrations in this preprint.

Near the phase transition, the spinons become deconfined and behave as independently-propagating spin-half

*bosons*. This is unusual because spin-half particles are usually

*fermions*. Fermions cannot share a quantum state with other fermions. Electrons are fermions, which is why each atomic orbital can only accept two electrons (one up and one down). Bosons, on the other hand, are happy to share the same state, which is related to the phenomena of Bose-Einstein condensation and superconductivity. In fact, in our paper we demonstrate that the spinons are bosons by showing evidence of them forming a Bose-Einstein condensate.

## What’s this about spin-half bosons?

The fun headline about spinons is that they are spin-half bosons, but what does that even mean? Broadly speaking, there are two types of particles: bosons and fermions, which refer to the type of particle statistics that they obey. Fermions refuse to occupy the same state as other fermions, and bosons are happy to share. Most matter particles (electrons, protons, neutrons) are fermions. Bosons are often force-carrying particles, like photons or the Higgs boson. Particles can also have spin: intrinsic angular momentum. Spin behaves a lot like the particle is spinning, but it’s a property of the particle itself, not something it does.

In your first quantum mechanics class you learn that bosons always have integer spin (S=0,1,2…) and fermions always have half-integer spin (S=1/2,3/2….). Why? Because of the spin-statistics theorem. So then how can spinons carry S=1/2 *and* be bosons? An important caveat to the spin-statistics theorem is that it only applies to particles in 3D space, so 2D particles, like spinons, don’t have to obey the rule.

# What are spinons for?

The theoretical purpose of spinons is rather technical, but very roughly, they are needed to explain an unusual phase transition. The system we study (the J-Q model) hosts a transition from a Néel state (breaking 3D rotational symmetry) to a VBS (breaking a discrete four-fold symmetry). The conventional theory of phase transitions, known as Landau Theory, predicts that a transition between phases breaking unrelated symmetries, such as this one, should be first order (and therefore associated with the release of latent heat energy). After very extensive study, however, there is no evidence of this latent heat, nor any other hallmarks of a first order phase transition. It would appear that Landau theory does not describe this phase transition. If we instead use the theory of deconfined quantum criticality, where there are deconfined “fractional” particles (spinons) at the critical point, then it is possible for the phase transition to be continuous and we can explain the lack of latent heat.

# Some caveats

Here I’ve only discussed a pretty narrow view of spinons, focusing on how they work in the system I’ve studied. There are many things called spinons, and perhaps the only definition that would apply to all of them is “some kind of collective excitation that carries S=1/2” (credit: Anders). The picture of what a spinon looks like varies depending on what the background state is. These diagrams would look very different if we were picturing spinons on a background of the Néel state.

## Further reading

This is complex topic, but there are a few places you can go to read more depending you your level of interest. There is a more detailed version of this description in the introduction of my dissertation (free copies available). The theory of deconfined quantum criticality was first described in this Senthil, et al. Science paper. Some of the best numerical evidence is in this Science article by Shao, Guo and Sandvik. A great discussion of the details of the spinons themselves is in this (very) recent preprint.

*Thanks to Anders Sandvik for providing feedback on an early draft of this post.*