Magnetic Monopoles in Spin Ice

Last summer it went through the news: magnetic monopoles had been observed! First time I read this headline I was pissed off that nobody had told me. Upon reading the article it turned out however that the discovery was not one of elementary magnetic monopoles, but instead magnetic monopoles in condensed matter systems. It's okay nobody told me that because I'm not exactly known for my close ties to the condensed matter community. I thought at the time it might be worth mentioning on this blog, but changed my mind at the prospect of explaining what a corner-touching tetrahedral lattice is. Now last month I heard a seminar by Steven Bramwell, one of the experimentalists who made it into the headlines and one of the authors of the Nature article reporting on the discovery. He offered a simplified two-dimensional picture that I found very illuminating and totally blog-suitable.

Coulomb's Needle

Before we start, let me point out that while elementary magnetic monopoles, when discovered, would mean a change to the fundamental laws of electrodynamics, the magnetic monopoles in condensed matter systems are perfectly compatible with usual electrodynamics. We all learn in school that there are no magnetic monopoles. But there are configurations that appear like magnetic monopoles. In fact, I learned in Bramwell's talk that Coulomb did not only come up with a potential for electric point charges, but also with one for magnetic point charges. The latter didn't survive it into modern textbooks though. What Coulomb did was to measure the magnetic field around the end of a very thin needle. You can nicely measure this field, except of course exactly inside the needle. We know today that the magnetic field lines going through the needle into its tip do exactly balance the outgoing ones that Coulomb measured. There are thus no sources for the magnetic field. Whether or not the needle is straight doesn't matter, it just has to be something that's essentially one-dimensional. That the field around the end-tip looks like the one for electric charges is simply a consequence of the geometry*.

In any case, you all know what happens if you take a magnet and break it apart. Instead of getting two opposite magnetic charges, you just get two smaller magnets, so that's not useful. The way you “make” a magnetic monopole is to instead create something akin the needle that Coulomb was measuring the field of, deforming the needle, and then hiding it in a haystack. In the end, the only thing you can sensibly talk about are the endpoints of the needles which stick out if you measure the magnetic field.

Spin Ice

The system that the experimentalists used for their measurements is called a “spin ice.” The spin ice, named for its resemblance to water ice, features the above mentioned corner-touching tetrahedral lattice with little magnets on the corners of the tetrahedron that arise from the magnetic moments of the atoms at these locations. This molecular arrangement is akin to the H₂O ice. In ice, the larger oxygen atom sits in the middle of a tetrahedron, surrounded by 4 hydrogen atoms, two of which are nearby (the one "belonging" to the central oxygen atom), and two which are further away (belonging to neighboring oxygen atoms) and making up so-called "hydrogen bonds". If you draw an arrow towards the oxygen in the center when the hydrogen is close, and an arrow away when it's far, you have a corner-touching tetrahedral lattice with four arrows to or from the center of the tetrahedra (see picture).

Now think of a structure where instead of hydrogen atoms there are titanium atoms and rare-earth atoms such as holmium located at the corners of the tetrahedra. These metal atoms have magnetic moments which in this lattice configuration only can point either inwards or outwards of the tetrahedra, just as the arrows indicating the positions of hydrogen atoms in ice. That's the spin ice.

Magnetic Monopoles on a Sheet of Paper

Confused? We can vastly simplify that spin ice by looking at a 2-dimensional analogy. You can do it on a sheet of paper, as shown in the picture below. Use a pencil and have an eraser ready. The little arrows are the magnets. The lattice rule is that to each square, two arrows go in, and two go out, as with the spin ice. This is the preferred configuration.


[Click to enlarge]
Now let us create a defect in that lattice by switching one of the arrows. With this you now notice that, above the background of the lattice, there's a magnet sitting there. The one pole has three arrows in (red), the other one three arrows out (green). If you'd measure the magnetic field, you would find find these two poles.


[Click to enlarge]
Next step is to pull the ends of the magnet apart without creating further defects. The below picture shows a first step. If you draw it on a sheet of paper, you will easily see that you can pull the defects further apart, not necessarily in one straight line. The defects will be connected by a one-way oriented path of arrows along the path that you've pulled them apart. That's creating and deforming Coulomb's needle if you wish.

[Click to enlarge]

Final step is to add a few more such paired defects and and pulling them apart. Then sit back, look at your lattice and try to find Coulomb's needle, ie the connection between the oppositely charged defects. You'll notice the following. There is no unique connection between any two defects. They can be connected by many different lines, and they can no longer be considered paired either. The below picture shows an example with two defects and some of the possible connection lines. If you look closely, you will find more connections than I've drawn.


[Click to enlarge]

Thus, you have “hidden” the needle in the background lattice. The defects are no longer paired, but can move around independently as if they were, you get it, magnetic monopoles! You can then go and measure some of their properties, and these measurements confirming their existence is what made the headlines last summer.

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* No, wait, it's actually an entropic force!