## Saturday, April 17, 2010

### 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.

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

Steven Colyer said...

From the last paragraph of the last link in your first sentence:

One important general result of the research, according to Morris, is that the spin ice monopoles are one of the first examples of "fractionalization" – whereby a spin is split into two separate entities – in a 3D system. A familiar 2D example of fractionalization is the fractional quantum Hall effect, the discovery of which resulted in Robert Laughlin, Horst StÃ¶rmer and Daniel Tsu winning the 1998 Nobel Prize for Physics. Because this and other properties of spin ices should be shared by similar magnetic materials, it could lead to the development of new materials for making spintronics devices, such as magnetic memories.

What do these gentlemen have to say about these results? What does Seth Lloyd at MIT Mech Engg Quantum Computing Labs have to say about it? Or anyone involved in spintronics?

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

Well, what isn't these days, honestly? :-)

Phil Warnell said...

Hi Bee,

A terrific post which even in 2-d has one required to burn the wood a bit, as to be able to smell smoke. There wouldn’t be o 1-d way to have this understood by any chance? :-)

Best,

Phil

Phil Warnell said...

Hi Steven,

It appears that these scientists are the equivalent to what their counterparts being marketers are in business; that is in being spin doctors ;-)

Best,

Phil

Neil B said...

I'm sure there's some clever dodges, but I keep making this gripe about literal free monopoles (and recognized for decades): If you believe the "A field" (vector potential) is "real", then monopoles are very ugly since they have to wrap up A field lines in a sort of tail that goes on forever (or at least, has to end in another monopole.) Maybe Bee will explain it can be handled OK, but any realist must consider it ugly.

And, if E and B are really equivalent, there would be an electrical-A field analogy - and the same problem (to the extent you worry about model issues as "problems" ...)

BTW, we see that processes mimicking particles can occur in condensed matter - those weird pseudo-particles in superconducting materials etc. Of course, one can imagine "real particles" in "empty space" as being processes of the underlying background structure propagating along. (Yeah, I indulge that philosopher's habit of using lots of quote marks about conceptually tricky stuff ...) (& I assume "entropic force" was sarcastic.)

Plato said...

Steven:Or anyone involved in spintronics?

That thought of spintronics crossed my mind as well.

In a "parabox situation" it is important to understand that quantum gravity history as it might be used, might be used to define some "emergent principle as a algorithm written" may also be written as "quantum gravity signal" for computerized situations in numerical standardization relations?

I mean you have to have some format in which to translate the theoretical toward the truest versions of a "vision of the math." What ever that may be.

Best,

Neil B said...

Plato, did you mean "parabox" literally? All I can find of conceivable relevance is, on Wi-pee:
"The Farnsworth Parabox" is the fifteenth episode of the fourth production season of Futurama.

In any case, following up on my previous: AFAICT we need not even consider the "B" field to be fundamental and separately real. We write it to show effects, such as
F = q(E + v cross B),
but we could just consider it a velocity-dependent E field (unless there really are true B charges.) I suspect there's even a math structure to show that. This seems an obvious insight or challenge to me, but belief in monopoles and the validity of B as such is widespread. (Even as I realize that many scientists don't like scolding from those acting as philosophers.)

Uncle Al said...

Maxwell's equations would be wonderfully symmetrized by a single magnetic monopole.

Consider binding energy and annihalation emissions of monopolium orbital decay then annihalation (or zoom in and pop). No monopoles appear in particle accelerators. If the electron is remarkably light, must the monopole be remarkably heavy, [(m_p)^2]/(m_e), about 1.72 TeV? If so... why doesn't it decay?

Talk, talk, talk. Loft a satellite (well outside the magnetosphere) containing three orthogonal large area superconducting loops each with its own SQUID detector, and look.

Neil B said...

Al - don't you think I at least have a point about the dubious nature of an intrinsic B field (instead, it can be looked at as transformable E field.) - ?

BTW, since gravity too has a magnetic analogy - "gravimagnetism" (I prefer shorter gravitism) - then shouldn't there be a "gravitational monopole" as well? What a mess ....

Plato said...

Hi Neil,

No not as such in the idea of a scolding, but of a recognition of the attempts to walk a line, and realize that a mass exodus regardless of the ilk of one's disposition toward QG, there is hope "that some measure" might have a consequence in the way that has been set before scientists as a means to continue QG "in determination."

Discretized, but in a much different way?

It's not just with men that such bold adventures are made, as we know well the history inclusive of those woman who study, saids nothing about gender bias as a solution. Only those who would make it so.

Geometrical frustration

Some monte carlo operation perhaps or some "sphere packing problem?"

Viscosity perhaps, and the understanding where the vortices are, are even more appealing to me. Entanglement.

Sort of like understanding that satellites can move through space most easily projected through the tunnels in space understanding Lagrangian?

So here
an understanding of the dilemma set before one as a "paradox" places one in a position to ask "which direction" they will go.

But most surely the method by careful consideration there is no doubt:)

Best,

Plato said...

Hi Uncle,

Maybe in comparison?

4-Dimensional Quantum Gravity

Best,

Bee said...

Neil B: A point mass is a gravitational monopole.

Phil: You can do it in 1-d but you'll miss an essential feature. In 1-d, draw a line of dots. The "lattice" rule is one arrow in, one arrow out, ie the arrows all point in the same direction. To make a pair of defects, switch one arrow. Then pull them apart. It will look somewhat like this: -> -> -> <- <- <- -> -> ->. The one "monopole" is -><-, the other <-->. Now make a couple of them. What happens in 1-d is that they have a tendency to annihilate and it doesn't really become apparent that you can treat them as independent degrees of freedom, since there's only 1 dimension in which you can connect them. Best,

B.

Phil Warnell said...

Hi Bee,

Thanks, I thought you could manage to do it in 1-d and yes it’s true it doesn’t appear as legitimate, since restricted to only one degree of freedom (spatially). Then again is it really as time would still be required for anything to happen at all. The question of course then being, as what it would mean to switch one of times arrows and then have it stretch? ;-)

Best,

Phil

Neil B said...

Actually Bee, a point mass is the analogy to an electric charge "monopole" but not also to a magnetic charge - the latter being my original query. Gravi-magnetism is the additional field from moving matter, so there would be a gravimagnetic field around a pipe with flowing matter (Lense-Thirring effect is an example, see Frame-dragging.) In a gravimagnetic field, moving particles behave differently than in a regular g-field. (And I add background for readers in general, as usual.)

If there is a special particle that carries gravimagnetic charge, it would (on analogy with B-monopole) not attract masses but it would affect masses moving near it. Which makes we wonder, that isn't possible by itself since all particles should also have a simple gravitational field due to their mass-energy.

Phil Warnell said...

Hi Niel,

”Which makes we wonder, that isn't possible by itself since all particles should also have a simple gravitational field due to their mass-energy”

I have no idea how Bee would respond to this, yet I would say it relates to what you would find as being a particle at the most fundamental level, as to being a string, a loop or something else altogether? As for charge in my mind it relates to being a measure of potential, which is simply expressed through the presence and actions of the field.

Best,

Phil

Bee said...

You don't need a particle to have a gravitational field, you just need something that carries energy. (Note: in a classical theory, a field is not necessarily identified with particles.)

Sam & Mike said...

This has just been realised in 2D. In many ways, it is nicer since the defects can be imaged directly:

http://www.nature.com/nphys/journal/vaop/ncurrent/full/nphys1628.html

Bee said...

Hi Sam,

Yes, I had seen the paper. The pictures from the measurement are quite nice. There is also a video on the Imperial College website. Best,

B.

Neil B said...

Phil, B is right that you don't need a particle to have a g-field, distributions of "energy" (like, electric fields) would exert gravitation. However, I am correct as a subset: all particles should have g-fields. BTW, I wonder what kind of g-field a pulse of light would have: it is moving at c, so I can't imagine a solution analogous to classical E-fields. (We can't logically construct AFAIK an E field for a charge moving at c, since the relativistic distortion would make it a planar pulse of infinite amplitude (similar to a Dirac delta function) which is presumably physically absurd. Maybe that's one of the reasons gravity has to be inherently different from EM?

Zephir said...

Two magnets attached by their poles still doesn't make magnetic monopole.

http://www.energeticforum.com/renewable-energy/3826-magnetic-monopole-arises-when-magnets-repell-each-other.html

These monopoles have nothing to do with Coulomb's needles, which are just another (and conceptually quite different) way, how to fake monopoles. Of course, the existence of repelling magnetic domains violates nothing from fundamental laws of electrodynamics.

Bee said...

Neil: Fields of point-like particles, whether that's their electric or gravitational field, tend to diverge. That is of course "physically absurd" but it's a well-known absurdity. (And one of the reasons why you might want to consider 1-dimensional strings instead...) Best,

B.