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).See also reviews in Gizmodo and those via Google News. Thanks to Viktor K. for the link.
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.
0 comments:
Post a Comment