Jun 7, 2010

Spin Liquids

[Simulation of a quantum spin-liquid performed on a flat honeycomb structure - by The University of Stuttgart]

As I have been reading many posts, specially via Condensed Concepts, about spin liquids, I decided to learn a bit about that. So, following the new plan for this blog of using it to help me understand things, I am writing what I have found in the following arXiv article, which is a compilation of lectures given at the famous Les Houches Conference in France:
There are some minor typos, so just be sensible when reading. Nothing serious. It seems that although the definition of a spin liquid may make sense in the classical setup, it would not be realisable there, so the concept is usually only applied to quantum spin models. Anyway, let me describe what I understood from the above paper. If anyone have more interesting things to add or corrections to make, I would be very happy to hear them and learn more about the subject!

The term (quantum) spin liquid refers to the ground state of a spin model where no symmetry of the model Hamiltonian is broken.

Let me include at this point two paragraphs for the less technical audience that has been brave enough to continue reading up to here. Physicists may feel free to skip the next two paragraphs as I am going to explain the basic concepts for a broader audience. I will get back to a more technical description after them. First, a spin model is a mathematical model describing particles with spin (the quantum version of magnetic moment) usually, but not necessarily, on a lattice (a collection of points linked by lines). These models are defined through a so-called Hamiltonian function, which is just a formula that gives the energy of the model for each configuration of the spins in the lattice. The ground state is the minimum energy configuration, that should be favoured at zero temperature as all physical systems like to minimise the energy and there are no thermal fluctuations at T=0. You can think of the energy as a cost function that you always try to keep at a minimum. Finally, a symmetry of the Hamiltonian is some kind of modification that you do to the spins or any other variable in the Hamiltonian such that when you put this modified variable back into the formula, the Hamiltonian does not change.

The symmetry part requires some more explanation, I know. Consider an Euclidean vector with two coordinates  v=(x,y) and let us assume that in some system there is a Hamiltonian depending on it given by H=xy, i.e., the product of its two coordinates. If we multiply the vector v by -1, then each coordinate is multiplied by -1 and the Hamiltonian becomes H=(-x)(-y)=xy. It doesn't change. Therefore, the multiplication of v by -1 is a symmetry of the Hamiltonian. Of course physically meaningful symmetries are more interesting, although the one I have just gave you may be considered as a very special case of a more general local gauge symmetry, but we are not going to talk about that now. The important is the idea.

Consider now, for instance, the ground state of the (ferromagnetic) Heisenberg model. This is the classical example of symmetry breaking. The Hamiltonian is rotationally invariant as it is given by the scalar product of the spin vectors, but the ground state has magnetic order with all spins pointing in the same direction and obviously changes if they are rotated (although a rotation takes it to another ground state). The word liquid in spin liquid is an analogy with the transition from the liquid to the solid state. The liquid is homogeneous and looks the same everywhere, so it has a continuous translational symmetry of the liquid. On the other hand, a crystalline solid (let us not talk about glasses at this point...) breaks that symmetry in the sense that it is not invariant by a continuous translation, but by very specifically ones. The same works for rotations, but the basic idea is what matters. This can also be stated as the fact that the spins do not develop any long range order (LRO) at zero temperature, they are completely disordered even then.

That is the reason why these ground states can be realised only in the quantum setup. Classically, there are no fluctuations at zero temperature to destroy an ordered state. However, at the quantum level there exist quantum zero point fluctuations that can do job. They can disorder the spins from their ordered states guaranteeing that they will not break any symmetry.

An interesting characteristic of ground states with broken symmetries is that they are degenerate. By applying the symmetry operation that is itself broken by these states, you get another ground state. This kind of degeneracy should not happen in the spin liquids, but their ground state can still be degenerate, the degeneracy coming from another kind of order called topological order, about which I am certainly going to write a more detailed post in the future.

It seems that there was no observation up to date of a spin liquid phase in a real system, although many simmulations on almost realistic models seem to observe it. For instance, the article Exotic Quantum Spin-Liquid Simulated: A Starting Point for Superconductivity?, from where I took the picture for this post, describe one of them. As I said, Ross McKenzie from the blog Condensed Concepts have been writing a lot about that recently, so let me just list some of his posts
Just to summarise things: Spin liquids are ground states that break no symmetry from the original Hamiltonian, they have no long range order an can only appear in quantum systems because these have fluctuations even at zero temperature and these quantum fluctuations can destroy the order. Although from the experimental side these states have not been realised so far, it seems that theoretically they are reasonably understood, although through the article in the beginning of this post it seems to me that this understanding is only at the mean field level. The detailed description of these states, even theoretically, is still lacking and seems to be an interesting topic of research.

As a finishing note, Misguich at the end of the paper gives an interesting connection of spin liquids with Kitaev's toric code (which the reader may already know from previous posts). The ground state of this topological quantum code is a spin liquid. Although the ground state has a degeneracy, this degeneracy is topological and has nothing to do with the breaking of a symmetry by the ground state, so we are still fine. This analogy is only a final observation in the article, but given that the search for an experimental realisation of a topological code is also a hot topic, this gives another path through which spin liquids may be observable in real systems, although in this case they would be engineered instead of natural.

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