## Friday, August 09, 2013

### Skyrmions could make hard disks 100 times smaller

Remotely related: sci-fi gets real: tech junkies should look at 27 science-fiction concepts that morphed into reality in 2012.
Nature's Ron Cowen reviewed a technical paper in Nature that is one month old,
Writing and Deleting Single Magnetic Skyrmions (Niklas Romming and 7 co-authors from Hamburg).

Skyrmions, some topologically non-trivial solutions of non-linear sigma-models first described by Tony Skyrme in the 1960s, may be thought of as tiny vortices of atoms. Because in this very recent breakthrough, Romming et al. became able to create and destroy them at will, it's plausible that they may be used in future magnetic information storage technologies.

I've been in love with skyrmions decades before I knew their name.

It really began when I was 15. I was obsessively reading Albert Einstein's book "My World View" ("Mein Weltbild", in a Czech translation) that I had found somewhere in the bookshelves (I guess that it would originally belong to my paternal grandfather, a professional painter/artist and geometry teacher).

In this book, one that probably overlaps with "Ideas and Opinions" heavily, Einstein popularly presents his views and insights about relativity, religion, socialism, Jewish questions, Nazism, meanders, Max Planck, alleged incompleteness of quantum mechanics, and other things.

Einstein wrote many inspiring things, many things that looked deeply ethical, many political ideas I would later find myself in disagreement with, many ideas about physics that were right, and some ideas about physics that were wrong.

Those nearly 25 years ago, I was only beginning to be exposed to quantum mechanics and for a year, I was an employee of Einstein's dream to construct the unified field theory de facto as a classical field theory, if you allow me to use the standard terminology. After some months, I had to begin to steal ideas from proper quantum mechanics to explain the hydrogen atom, before I was forced to steal all of quantum mechanics, of course, but let me avoid the hydrogen atom here.

While its quantum dynamics implies that some quantum numbers are discrete, there are also other observables that have to be discrete in the real world (because they were observed as discrete!) although such a quantization rule seems hard to get in a classical field theory. In one of the essays, Einstein wrote something like (using a modernized terminology):
Quantum mechanics is probably incomplete and a complete description should still be looked for. There is no proof that an old-fashioned, realist, classical theory may not account for the quantum phenomena. For example, the quantization of the electric charge could follow from a classical field theory. There could be a classical field theory that allows us to derive that whenever the charge density vanishes on the boundary of a region, the region contains a charge that is an integral multiple of the elementary charge.
I took that as a homework and apparently found a solution. Imagine that in each point of the spacetime, there is a field that takes values on a three-sphere. If $$\vartheta(x,y,z,t)=0$$ corresponds to a conventionally preferred point of the sphere (the North Pole) in the same way that we know from the two-sphere, we may add a potential energy term to our action such as$S_{\rm pot}\sim - C \int \dd^4 x\,\vartheta^2$ that will place the value of $$\vartheta$$ in the majority of the spacetime close to the value zero. However, in a limited three-dimensional region, the field $$\vartheta$$ may probe all points of the target three-sphere. We may figure out that in those regions, the real space may be "wrapped" on the target space three-sphere.

The charge density may be calculated as the "solid angle" spanned by the infinitesimal region of space in the three-sphere. That means that the charge current is proportional to the Hodge dual of a Jacobian of a sort,$j^\kappa = \frac{e}{6}\cdot\frac{1}{2\pi^2} \varepsilon^{\kappa\lambda\mu\nu} \partial_\lambda V^a \partial_\mu V^b \partial_\nu V^c \varepsilon_{abcd} V^d$ where $$V^a$$ is the four-vector embedding the three-sphere pointer into a four-dimensional Euclidean space of a sort; we always have $$V^a V_a=1$$. I hope that I inserted the right normalization factor above; $$1/6$$ avoids the multiple counting over the permutations of $$a,b,c$$ while the other factor divides by the "full solid hyperangle" i.e. the surface/volume of the unit three-sphere $$2\pi^2$$ for the integral of $$j^0$$ over the regions where something happens to be an integer multiple of $$e$$.

This seemed like a cute idea. Later, I learned that the magnetic (monopole) charge density actually is represented by a similar topological trick. However, the electric charge is quantized for purely quantum mechanical reasons. Due to the quantization of energy in the quantum harmonic oscillators, one may only add energy to charged fields by creation operators whose electric charges are quantized. There's nothing wrong about this intrinsically quantum explanation for the electric charge.

In the 1990s, the discovery of dualities (and S-duality in particular) showed that these two constructions or explanations for the charge quantization are equivalent although the proof is in no way obvious.

In 1998, I still didn't know the word "skyrmion" although my adviser Tom Banks was telling me that I should have found out what the word meant. ;-) But when I and Ori Ganor asked how the cylindrical M2-branes stretched between pairs of M5-branes are represented at the Coulomb branch of the 6-dimensional (2,0) theory, they are represented by skyrmions, too. The fivebranes become knitted.

This 6-dimensional construction differs from the 4-dimensional construction above by some changes to the dimension only. First, the pointer field isn't labeling a three-sphere but a four-sphere. You may obtain the corresponding vector $$V^a$$ from the 5-dimensional transverse Euclidean space as the separation of the corresponding points of the two M5-branes normalized so that it is a unit vector, i.e. as$V^a = \frac{\Phi^a_M - \Phi^a_N}{|\Phi_M -\Phi_N|}$ where the index $$a=1,2,3,4,5$$ labels the transverse dimensions to the M5-branes and $$M,N$$ label the M5-branes themselves (Chan-Paton indices of a sort).

There's one more difference between the six-dimensional and four-dimensional case. The six-dimensional theory has five and not just four spatial dimensions. So the four-sphere may only be wrapped by four spatial dimensions and the solution remains constant in 1 remaining spatial dimension (plus 1 temporal dimension). That's why the resulting skyrmionic objects are strings rather than point-like objects. They become tensionless strings ("the" tensionless strings known in this theory) in the $$\Phi_M-\Phi_N\to 0$$ limit where the Coulomb-branch-based description of the theory breaks down.

In 2000, Ken Intriligator used some nice anomaly considerations to derive structurally similar terms in the six-dimensional theory. I've tried to see that the equations are equivalent to the skyrmion-based ones but the two papers always seemed slightly inequivalent at the end.

In various effective descriptions of nuclear physics, one encounters nonlinear sigma-models and the baryon number seems to be exactly given by the skyrmionic wrapping number. I guess that the detailed implementation of the nonlinear sigma-models is inequivalent in the condensed-matter-physics setup by Romming et al. but the mathematics is going to be analogous. In a foreseeable future, this 50-year-old piece of mathematical physics that has appeared at various places of real physics may dramatically improved magnetic information storage systems.

1. Dear Lubos,

2. Ha ha,

this is an additional cute point against sourpusses who loudly demand that all legitimate physics has to have direct every-day implications for their close minded view of the world, high energy theoretical physics is therefor no good, etc ... :-D.

This is really a cute application of some serious physics, thanks for explaining this even though I do not get all the fine grained details of course :-)

3. kashyap vasavadaAug 9, 2013, 6:14:00 PM

I agree with Dilaton and "and?" that mathematical content of Lubos' blog is superior to any other physics blog around. So Lubos, keep it up!

4. I saw also you cite some very good arxiv papers. I worked through some algebraic topology, mainly Hatcher and for the geometry part Griffiths. Of course, I still have some parts to read from the huge book about "mirror symmetry" and Baker & Baker on string theory but I would like to know if you know a shorter introduction to the way in which topology and group theory connects to string theory... That could help me now. Thanks in advance and keep the good work :)

5. *Becker&Becker ... sorry

7. Why the inertia? ;)

8. I first silently lurked around here too, before I dared to post my first TRF comment, believe it or not ... ;-) !

9. I have an obsession with 6d (2,0) SCFT. This paper was on my radar for quite some time and I think now is the time to read it.
BTW I still can't use the Disqus with Chrome to log on to TRF (the Disqus log on window disappears). Each time I want to log on to TRF I must use IE. Maybe because I have Windows 8? Does anyone else have a similar problem with Chrome and Windows 8?

10. What is the setting in your Chrome for "Allow third-party cookies to be set"? If the setting is "No" then you must exempt disqus.com explicitly. Try it and let us know if it works.

11. It worked!

Thanks Eugene!

12. Well, you will laugh but I found this blog searching for "Sean Carroll, nonsense" so I cannot really start posting here in a less anonymous way :)) I nevertheless share many ideas I found here and I like string theory although I might have things to learn about it before reaching Lubos's skills of argument. All the posts on this blog about quantum mechanics are right and it brings some sort of relief to see that not everyone is into "many worlds interpretation of Quantum Mechanics"... it is funny thinking that everyone used ghosts in doing some QFT calculations but nobody ever went to the public screaming "Ghosts are real!!!" well, anyhow, great blog!

13. In SQCD, for example Seiberg duality, can one still see the Baryon number as being given by the skyrmion wrapping number?

14. Dear any, SQCD - theories subject to Seiberg dualities - are gauge theories with polynomial Lagrangians.

The Skyrmions are solitions in nonlinear sigma-models, e.g. in effective field ttheory descriptions that involve pion and rho-meson fields.

15. right, like the magnetic description of SQCD. which too is an effective field theory after adding an irrelevant operator of mass dimension. Give the magnetic squarks a vev and the magnetic gauge bosons become your massive rho-meson (and mesino). Still not sure how to see the skyrmions :) best, any.

16. I may be ignorant of something about the Lagrangians you mention but just to be sure that we're on the same frequency, do you understand that the skyrmions only appear in non-linear - and therefore non-polynomial and non-renormalizable - sigma-models? See

http://www.scholarpedia.org/article/Nonlinear_Sigma_model

for some explanation of the NLSMs. The kinetic terms have field-dependent "metric tensor" prefactors in NLSMs. You may have pions and rho but the configuration space has to be compact - a sphere of a sort.

17. i agree with what you are saying, and i will have to think about your comments on config. space and field dependent metric tensor, although if i understand it correctly the NLSM may be related (is gauge equivalent) to a linear one. i.e. there is a G/H lagrangian that is relatable to a linear one with gauged H. hence the rho mesons....may i mention this http://arxiv.org/abs/1202.2863v3 and the introduction.

My feeling is that some sort of gauged bosonic skyrmion should be present by using certain field configurations of the magnetic description.

18. Hi Lubos,

I was thinking how the free SCFT theory in 6d may arise in this picture. Since we must retain CF invariance the M5 branes should coincide of course; so my intuition says that the way to go from a strongly coupled (2,0) SCFT to a free SCFT (i.e. to switch off the interactions of the tensionless strings) is the width of the M2 brane (i.e. the M2 dimension in the worldvolume of the M5 brane) to collapse and thus M2 to become a thin stringy like membrane. This way the Wilson surface on M5 would collapse to a Wilson loop.

But I’m not sure if I got it right. Do you know if there are
any papers on this issue?

19. Dear Giotis, I am not sure I understand what you want the papers to be about.

First, yes, the conformal symmetry is broken by the separation of the M5-branes. In the (2,0) theory, the separation scalars have dimension mass^2 - it's from the kinetic terms being mass^6 - so one can see that the transverse thickness in the 4 dimensions that belong to the 5+1 dimensions in which the (2,0) theory is defined is given by Thickness = scalarvev^{-1/2} by dimensional analysis.

The cylindrical M2-branes considered here are strings simply because 1 of the dimensions is in the transverse dimensions to the M5-branes, so it's "internal" from the (2,0) viewpoint, while the remaining 1 spatial and 1 temporal dimension is along the M5-branes. There is never any Wilson surface inside the (2,0) or M5 world volume theory. When M2-branes are fully parallel to M5-branes, they dissolve into the electromagnetic flux of the self-dual 3-form field strength H fields in the (2,0) theory.

A way to get a free theory limit is to have the large scalarvevs in the (2,0) theory. In that limit, the transverse (4D) thickness of the strings goes to zero, mass goes to infinity, and they don't affect low-energy physics. The interactions in this low-energy theory are weak in this limit and may be viewed as effects of integrating out the heavy thin strings.

20. Thanks Lubos,

There was a misunderstanding then; I was talking about the Wilson surfaces which appear in the 6d theory (see e.g. http://arxiv.org/abs/hep-th/9809188).
Probably you have some other configuration that I didn't quite understand.

The question was generally, how can I get a free SCFT in the M5/M2 picture i.e. without breaking CF invariance.

21. I see. Just be careful, the Wilson surfaces from the 1998 paper aren't really "object" that last in time. They're localized in time, i.e. generalized instantons.

The same is true for the Wilson loops, of course, but I didn't understand how you reduced the surfaces to the loops. In general, the reduction from (2,0) to N=4 Super Yang-Mills proceeds by compactification of 2 dimensions on a small torus.

22. My aim is to switch off the interactions of the tensionless
strings while retaining conformal invariance (in order to get a free SCFT in 6d). So I was thinking if I could make the membranes thin (stringy like with zero width) then in 6d the tensionless strings will become point like and thus their paths won’t be able to intersect.

But I’m not sure if these things have already been solved. That’s why I was asking if there are any papers on this issue i.e. how to get a
free SCFT in 6d.

23. Just a correction:

What I wrote above for Wilson surfaces turning into Wilson loops is complete nonsense of course. Sorry for that; I got confused due to AdS7/CFT6 duality. The Wilson surface of 6d (2,0) *is not* the world sheet of the tensionless string.

Thanks for the correction Lubos.