Research at the boundaries


Spectral intensity at the Fermi energy for a domain wall in a model of Mn3Sn.   Fermi arcs arc visible connecting projections of Weyl points (marked as dots).

Spontaneous symmetry breaking is one of the fundamental ideas in physics.  In condensed matter, the paradigm is the Ising model of magnetism.  In the Ising model, spins spontaneously orient below the critical temperature, in one of two possible time-reversed orientations, say “up” or “down”.  Without special preparation, when this happens, a system will form domains or both orientations, separated by domain boundaries.  These boundaries are a simple example of a topological defect.  Other examples are vortices in superfluids, and dislocations  and disclinations in crystals.  In general, domain walls form whenever there is a discrete broken symmetry, and so are not limited to Ising models.

These defects, which live in real space, are “old” examples of topology in physics.  The more modern versions are defects in momentum space.  Notable are Weyl points, which are linear crossings of a pair of non-degenerate bands, that can be viewed as monopoles of the Berry curvature.  They are related to a variety of cool phenomena, like chiral edge states and Hall effects, and “teleportation” of electrons from one Weyl point to another.

What happens when we bring the two types of topology together?  The magnetic order which forms a spontaneously broken symmetry state directly influences the formation and locations of Weyl points.  This is because magnetic order locally acts like a Zeeman field on electrons, and modifies their band structure.  So if Weyl points are present in a magnetic system, they generally re-orient in momentum space as one crosses a magnetic domain boundary.  We can expect intimate coupling of the extended electronic states and domain walls.  For example, chiral edge states may live on the latter, and the walls themselves may have distinct transport properties.  Conversely, when one drives a current in the system, carried by the conduction electrons, it may “push” the domain boundaries in unusual ways.

Dr. Jianpeng Liu, a postdoc here, recently studied this with me in the context of Mn3Sn, an exciting material which shows a classic indication of band topology, the anomalous Hall effect, up to room temperature.  It has interesting magnetic order, whose domains and domain boundaries we described theoretically, including their coupling to Weyl fermions.  This material is being studied by a number of experimental groups – we learned about it thanks to Professors Satoru Nakatsuji and Yoshi-Chika Otani at the ISSP in Japan.  We expect there will be a lot more to see experimentally in this interesting material!  You can read our paper in PRL.

Spin liquid versus spin orbit


The interest in quantum spin liquids seems to be growing in physics.  Part of this is due to a flood of experiments on new materials.  Many of these draw their spins from a different part of the periodic table than did the previously studied spin liquids: Ru, Ir, Yb, Pr instead of Cu.  In these heavier elements, spin-orbit coupling (SOC) is much stronger, and may even play a dominant role in the local physics of the magnetic ion.  Some hotly studied materials like this are α-RuCl3, Na2IrO3, YbMgGaO4, Yb2Ti2O7, Pr2Zr2O7.  The upshot of the SOC is that the spins in these materials have very anisotropic interactions, that generally depend on the direction of the bond.  Theory of quantum spin liquids, however, has been largely restricted to Heisenberg models with isotropic SU(2) spin-rotation symmetry.

A notable exception is Kitaev’s honeycomb model, which is extremely anisotropic, but as he showed, exactly soluble.  Unfortunately, exact solutions are rare, and more general approaches are needed.  Even for extensions of Kitaev’s model, which have been heavily studied in the context of materials like α-RuCl3 and Na2IrO3, the vast majority of theoretical studies have employed methods that cannot actually diagnose spin liquids, like classical energy minimization, or exact diagonalization on very small systems.

We decided to try to apply a relatively well-established approach to SU(2) invariant quantum spin liquids, the Gutzwiller construction, to a general anisotropic problem.  The Gutzwiller construction works by starting with a ground state wave function of a “fake” system of free spin-1/2 fermions.  A quadratic Hamiltonian for these fermions determines this wave function, generally constructed by filling momentum states.  Then one “Gutzwiller” projects this fermion wavefunction onto the physical spin subspace in which there is exactly one fermion per site.  Since the fermions were placed into extended momentum states, the result is highly entangled in real space.  It is a pretty established method, and the Gutzwiller projected states are amenable to computationally tractable Variational Monte Carlo (VMC) simulations.  They give some interpretation and classification of spin liquids, and there is a physical interpretation of the Gutzwiller fermions as “spinons”: exotic neutral excitations of the quantum spin liquid.

We took as our test problem a microscopic model of strongly anisotropic spins on the triangular lattice, which may be relevant to YbMgGaO4, and is also relatively convenient as the triangular lattice has only one spin per unit cell.  Still, you pay a price for giving up SU(2) symmetry: even the nearest-neighbor model has 4 independent exchange parameters, so there is a large phase space to study.  Jason Iaconis, the student who spearheaded this project, also added 2nd and 3rd neighbor (isotropic) exchange which makes the phase space even bigger.  To compensate, we opted to restrict the consideration of spin liquid states to just one variety, known as U(1) spin liquids, a choice motivated by some interpretations of YbMgGaO4.  With this assumption, there are 6 possible spin liquid states, and we tried to judge the competitiveness in terms of energy, amongst each other, as well as with magnetically ordered states.  This was still a pretty ambitious problem, and Jason did a remarkable job, helped by another student, Chunxiao Liu, and a postdoc, Gábor Halász, who have become experts in projective symmetry groups, a tool important for understanding these wave functions.

I won’t try to summarize the outcome very much, except to say that a number of issues arose in the process.  We found that for some purposes the Gutzwiller projected states are inadequate, and needed to be improved.  Jason came up with a novel way to do that.  While we hoped to find a state with a fermi sea of spinons, which had been suggested in experiment, this seemed hard to stabilize.  Maybe it could be stabilized beyond Gutzwiller, but it is not clear.  We did argue that this could be tested directly in experiment by measuring the thermal Hall conductivity, which we showed should be substantial in such a spinon fermi sea.

If you are interested, please have a look at the paper.

Crossed-chains/planar pyrochlore


The crossed-chains lattice is a useful one to consider for the study of frustrated magnetism.  It mimics the local structure of the three-dimensional pyrochlore lattice, with corner-sharing tetrahedra, as if the tetrahedra were squashed into a plane, so is sometimes called the planar pyrochlore.  My group worked on the antiferromagnetic Heisenberg model on this lattice (check here).  But unlike its more famous cousin the kagomé lattice, it is not as common to find visuals of it.  The kagomé lattice is named after a type of Japanese fishing basket.  I’m wrapping up a two week vacation in Bali, Indonesia (if you can, go!), where they also have a lot of kagomé baskets — mostly containing chickens.  But I spotted some crossed-chains baskets in the market today, so here we are.  Enjoy!

Classic Rock Revival

I’ve written a commentary for June’s version of the online journal club for condensed matter physics.  Get it here in advance!

Paper 1 (arXiv:1705.00570) : High pressure floating-zone growth of perovskite nickelate LaNiO3 single crystals, by Junjie Zhang, Hong Zheng, Yang Renb and J. F. Mitchell.

Paper 2 (arXiv:1705.02589) : LaNiO3 – a highly metallic and antiferromagnetic strongly correlated transition metal oxide, by Z. W. Li, H. Guo, Z. Hu, L. Zhao, C.-Y. Kuo, W. Schmidt, A. Piovano, T. W. Pi, D. I. Khomskii, L. H. Tjeng and A. C. Komarek.

The rare earth nickelates, with the chemical formula RNiO3 and the perovskite structure based on the cubic lattice, are one of the paradigmatic families of materials undergoing metal-insulator transitions (MITs). They are prominently featured in the authoritative reivew from 1998 on Mott MITs by Imada, Fujimori, and Tokura. With increasing ionic radius of the rare earth element R, the nickelates become increasingly metallic, and in the standard phase diagram, a line of finite temperature MITs terminates at a zero temperature MIT lying between R=Pr, with an antiferromagnetic insulating ground state and the end member, R=La, with a paramagnetic metallic ground state. This phase diagram has probably been reproduced hundreds of times in the literature. In the intervening decades the nickelates have re-risen periodically as prominent research subjects for diverse reasons.

It might be surprising to learn that, despite the fact that the nickelates were an established topic already in the 1990s, this classic rock (mineral) LaNiO3 (LNO) did not actually exist, at least in single crystal form. All the work on LNO was carried out on polycrystalline powders and thin films. Single crystals are not necessarily better than the former, but they are often different, and usually studying them is clarifying. Rather remarkably, as explained in the two featured papers above, after all this time, this year two groups have managed to produce single crystals of LNO, using the floating zone technique with high oxygen pressure.

Both group’s crystals are rhombohedral, with R-3c symmetry, as expected from polycrystal studies. They display a broad maximum of magnetic susceptibility somewhat above 200K, and show metallic behavior down to the lowest measured temperatures, both features consistent with the standard phase diagram. However, a major surprise occurred in the study by Li et al (paper 2) who observed an antiferromagnetic ordering transition at 135K. Neutron diffraction showed clear peaks at the wavector (1/4, 1/4, 1/4) in pseudocubic notation, which is the same type of antiferromagnetic order found in the low temperature phase of all the other nickelates (R=Pr,Nd etc.). Thus according to this study the ground state of LNO is not a paramagnetic but an antiferromagnetic metal! A transition is not observed in paper 1 above, but I would note that the residual resistivity (zero temperature limit of the resistivity) in paper 1 is about 14μΩ-cm, about 36 times larger than the 0.38μΩ-cm found in paper 2, which may indicate that extreme purity is required to realize the antiferromagnetic state.

Added comment 6/26:  My colleague Susanne Stemmer pointed out to me that paper 2’s resistivity may be too extremely low.  The value they show is as low as that of copper, even up to room temperature – this seems particularly surprising and perhaps indicates some issue with the resistivity measurement.  It may be that the difference between the two crystals is something other than purity…Anyway, everyone should be aware that any time two experiments on the same system disagree, there is uncertainty and further work is needed to resolve what is different, and why.  So…please read on, but keep this in mind: be careful, and remember that what follows discusses the implications assuming the other results are solid (the resistivity is not crucial for the conclusions about magnetism).

Assuming the result holds up, what do we learn from this revision of the nickelate phase diagram? It has bearing on a long-standing debate in the literature on the driving force behind the MIT and antiferromagnetism in these compounds. For the more insulating nickelates, R=Sm and smaller rare earths, two transitions occur on lowering temperature. The first transition coming from high temperature is the MIT one, and occurs without magnetic ordering. It is characterized by a symmetry lowering from orthorhombic to monoclinic, driven by an alternating contraction and expansion of NiO6 octahedra around every other Ni site. At lower temperature, antiferromagnetic order at the (1/4, 1/4, 1/4)  wavevector develops. Partly based on these observations, a view that the non-magnetic structural changes drive the MIT and the antiferromagnetism was proposed. Initially thought of as “charge ordering” of differently charged nickel ions, this has also been deemed “charge disproportionation” (I will use this name for concreteness) and most recently a “site selective Mott transition”, viewed through the lens of Dynamical Mean Field Theory (DMFT). In the Nd and Pr compounds, the structural and magnetic transitions occur together, leaving open the chicken and egg question of whether charge disproportionation or magnetism are the driving force here. DMFT studies seem to support the continuing importance of charge disproportionation, while other work advocated for a magnetic mechanism.

At least for LNO, the new work points to a clear answer, since antiferromagnetism occurs in the metallic state. A preliminary search by the authors did not reveal any structural or electronic changes associated with charge disproportionation, though it remains possible that the effect is simply very small. Antiferromagnetism seems to be in the drivers seat in LNO. It would be natural to think then that in the nearby materials PrNiO3 and NdNiO3, antiferromagnetism might again also be a driving factor. There one does observe charge disproportionation, but in fact it follows on symmetry grounds: in an underlying orthorhombic crystal (which these are), charge disproportionation is always induced by (1/4, 1/4, 1/4) antiferromagnetism as a secondary order parameter. This is not true for rhombohedral symmetry, consistent with the absence of charge disproportionation in antiferromagnetic LNO.

Many prior experimental and theoretical results should be revisited in light of these findings. Very narrow atomic scale superlattices of LNO, found to display an antiferromagnetic metallic state, might be stabilizing the inherent three-dimensional order of the crystals. A pseudogap observed in tunneling measurements of LNO films might be indicative of local formation of the antiferromagnetic state. More generally it is inspiring to see previously “impossible” crystals grown using modern machines, and we may hope to enjoy the fruits of these labors more broadly across interesting electronic materials.

SYKness makes a metal incoherent


Sometimes ideas come from surprising places.  Recently Alexei Kitaev made a splash by pointing out that a problem studied decades ago by Subir Sachdev and Jinwu Ye, who were studying quantum spin glasses, has bearing on the problem of quantum gravity.  What is now called the Sachdev-Ye-Kitaev (SYK) model became the subject of intense study as a toy model in which gravity and black hole physics emerges from totally different microphysics: quantum mechanics of many fermions.  It has a very simple Hamiltonian:

H_{SYK} = \sum_{ijkl} U_{ijkl} c_i^\dagger c_j^\dagger c_k^{\vphantom\dagger} c_l^{\vphantom\dagger} ,

where the coefficients U_{ijkl} are totally random.  Here I’ve written what is called a “complex” SYK model, slightly different from what Kitaev studied, which was written for Majorana fermions.  The difference is not crucial for the gravity connection, but it is helpful for us.  The key feature is that according to this Hamiltonian electrons move collectively, transitioning from two “orbitals” (kl) to another two (ij) in pairs, never alone.  You could visualize it like this:


This is a “toy model” because what emerges (in the limit of a large number of orbitals N) is not the gravity of our world, but a slightly different, and lower-dimensional version.  But still it is remarkable and beautiful.

The SYK model also describes the physics of strongly interacting electrons, forming an unconventional metallic state.  This metal is what condensed matter physicists refer to as a “non-Fermi liquid”, which behaves differently from typical metals like copper, in which electrons act almost independently.  In the SYK metal, you cannot separate motions of single electrons, but instead only see a collective dance of the whole electron fluid. The problem of understanding non-Fermi liquids is a forefront one in the field of quantum materials, but there are very few solvable theoretical models for them.  As such, the SYK model is quite valuable.

The drawback of the SYK model is that it is “zero dimensional” — every electron in it interacts with every other one, and so there is no built-in locality to the problem.  Because of this we cannot imagine driving a flow from one side of the system to another, and hence cannot discuss conductivity.  We should view the SYK model as describing a strongly-interacting “quantum dot”: a little chunk of non-Fermi liquid metal.

A natural thing to do is to try to string together these quantum dots to build up a higher dimensional system.  Already a couple of very nice papers along these lines have appeared: this paper, and this one which is a generalization.  In both these works, coupling between the SYK dots is chosen to preserve the scaling of the SYK model: electrons must hop in pairs between the dots.  Consequently the whole system remains a non-Fermi liquid and no new energy scale is introduced by the coupling.

We just posted a paper in which we make a different choice — just ordinary one-electron hopping between the dots:

H = \sum_x H_{SYK}(x) - \sum_{\langle x,x'\rangle} t_{ij}(x,x') c_i^\dagger(x) c_j^{\vphantom\dagger}(x') + {\rm h.c.} .

In this model, there is a competition between the SYK interaction U and the hopping t.  For t<<U, we have a strongly correlated system.  The header image is suppose to show this: electrons zipping between the correlated fluid dots.  Fortunately, it remains soluble in the large N limit and one can obtain a rich structure.  Briefly, at low temperatures and low energy, the system is a Fermi liquid.  We can extract the parameters of the Fermi liquid theory: effective mass, Fermi liquid interactions, quasiparticle residue, etc.  However, at higher temperatures it is an incoherent metal, and displays properties that are a mixture of those of the pure SYK model and a regular metal.  Together, it is striking how many of the properties are similar to those of experimental correlated metals:

  • Small coherence scale Ec = t2/U
  • Large effective mass (Sommerfeld coefficient) γ ~ m*/m ~ U/t
  • Small quasiparticle weight Z ~ t/U
  • T2 low-temperature resistivity with Kadowaki-Woods ratio A/γ2 = constant
  • linear in temperature resistivity at high temperature
  • linear in temperature thermal “resistivity” T/κ at high temperature
  • Fermi liquid Lorenz number L = κ/(T σ) = π2/3 at low T
  • non-Fermi liquid Lorenz number L = π2/8 at high T

Anyway, I’m pretty pleased with how it all works out.  Maybe the most remarkable part of this story is that the person who really spearheaded all the calculations, Xue-Yang Song, is still an undergraduate.  Pretty impressive.  You can read the preprint here.

March for Science

marchI went down to Los Angeles yesterday for the March for Science.  Although there was also a march in Santa Barbara, I wanted to see a larger scale event.  In LA there were 50,000 people registered, and I could believe there were that many there!  It was packed, and the atmosphere was great.  Amazing to see so many people care about science!  It was a lot of fun to see the creative signs people had put together.

My impression was that actual scientists were a pretty small fraction of the participants. Not being a scientist has some advantages.  As a non-scientist, you can feel comfortable with the sign on the left, which is a good sign, but I can’t help but be bothered by the fact that there are in reality no elements “T”, “M”, or “Es”.  The sign on the far right is definitely from a scientist: it takes a physicist to recognize a Hamiltonian!  Wish I had thought of that myself.

The beat of the drum

The hottest off-shoot of topological insulators these days is the study of nodal electronic structures.  By this I mean the situation in which two different bands touch along some locus in momentum space.  This started a few years ago with the re-discovery of Weyl points – I wrote a commentary some time ago with the poetic title, “Weyl electrons kiss”, using kissing as a metaphor for band touching.  The actual contact for Weyl fermions is linear in energy versus momentum, so a bit angular compared to lips.  The kissing image seems more apt for quadratic band touching, which also occurs and is interesting in its own right.

The contact locus between bands can be extended, and form for example a loop in momentum space.  This is often called a “nodal loop”.  An exotic phenomena that happens in such cases is that there can be a branch of surface bound states which exists only for surface momenta which project inside the nodal loop.  If one plots the energy of the surface state versus the two dimensional momentum of the surface, it forms a “drumhead” that stretches across the projection of the nodal loop into the 2d surface Brillouin zone.  I think this drumhead surface state was first pointed out by Volovik, though I do not recall the reference at the moment.

To a good approximation such a drumhead state is “flat”, i.e. its energy is nearly constant. A flat band has no kinetic energy, so one might think it should be susceptible to interactions.  With Jianpeng Liu, a postdoc at KITP, we studied this and indeed found various instabilities due to electron-electron forces.   Here is a phase diagram from our paper, for a simple limit of our simple model:


You can see in the inset the charge density wave pattern that forms in our simple model – it looks a bit like the drumhead surface state is “ringing”, though this is really oscillation of the charge density as a function of distance into the material.

This sort of tendency to form ordered states means that nodal loop semimetals should be an interesting place to study two-dimensional electronic surface phase transitions.    A neat feature of this situation is that the nodal loop means there are gapless states in the bulk, and that the surface bound states become bulk-like as they approach the momenta of the loop.  These attributes mean that surface quantum criticality in this situation is different from the criticality of a purely 2d system, and is instead inextricably tied to the 3d nature of the system.

You can read our paper if you want to know the details.

Strange days

I feel a certain sense of unreality lately.  Science in general, and my own scientific situation, seems in a strange state.  The administration just released its proposed budget, and it is awfully bad news for science.  The huge cuts to the EPA were expected, and get the most press, but the situation is uniformly grim for basic science.  A 20 percent cut to the NIH and 17 percent to the DOE office of science are also massive, and the latter, if it happens, is likely to be devastating to physics.  The National Science Foundation (NSF) is not even mentioned in the budget document, but probably it is considered one of a set of “other agencies” slated for a 10 percent cut.  Those percentages might not seem large, but they are bound to be distributed unevenly: these agencies have various commitments that have already been made, and need to be prioritized.  This level of cut could easily translate into many programs not being renewed, complete cancellation of any new grants, etc.. I wonder how many jobs will be lost at national labs and universities, and how many students and postdocs will lose their support.   I can only hope that congress does not follow these recommendations.

It is interesting to make some comparisons internationally, but hard to find very recent data.  According to wikipedia as of 2014, the US ranked 11th in the world in its research and development spending as a fraction of GDP.   Anyone in my field has had the experience of traveling to Germany, Switzerland, South Korea, Japan, etc., and being overwhelmed by the availability and level of research funding there.  American science is still strong on any scale, but for how long can it remain so under these terms?  It is hard to imagine a great country making these decisions.

There’s a personal side to it for me.  Our KITP grant is up for renewal this year from the NSF.  The KITP is without a doubt the most successful visiting center for theoretical physics in the world.  Among other things, the KITP’s postdoctoral program has trained some huge fraction of the leaders in theoretical physics, who have gone on to become faculty at all the top institutions.  The KITP’s scientific programs chart the future of physics.  It has become a model for other institutes worldwide, though the competition doesn’t come close to its level of productivity despite in many cases significantly more financial support in these imitators.  The KITP’s budget has been stagnant for many years, even before the change of administration.  I’ll be attending a “reverse site visit” in D.C. to present the case for renewal of the KITP’s grant to the NSF in May.

The KITP’s NSF grant does not support my own group’s research.  For that I rely on my own grants.  This year my DOE grant from the office of Basic Energy Sciences is up for renewal.  A 17% cut  does not bode well for that.  This DOE grant supported nearly all of our group’s work on quantum spin ice and pyrochlores, which has been probably the most influential part of my research in the past decade.  It supported my (former) graduate student Dr. Lucile Savary, who received a permanent CNRS research position in France, and has been winning a lot of prizes lately.

In mid-April, I’ll be traveling to New York City to present a proposal to the Simons Foundation, a private philanthropic organization that funds mathematics and science.  My team is competing for what they call a “Simons Collaboration”: the goal being to bring a group of researchers together to attack some major scientific challenge that needs the combined efforts of a team, and which cannot be supported by governmental means.  With the proposed budget changes, the latter criteria should not be too hard to satisfy.  It seems that private support for science may become especially crucial given the governmental trends.  Already nearly all of our postdocs in condensed matter at the KITP are funded by the Betty and Gordon Moore Foundation, another fantastic philanthropic foundation.

In order not to end on a low note, let me say that these thoughts about funding are the backdrop for my main concern, which is the science itself.  That is flourishing!  It’s an exciting time to do physics.  This has been a great year for the recognition of the topological revolution.  I’m amazed by the talent of young minds in the field: I’ve been working with an undergraduate on non-Fermi liquids, hydrodynamics, and disorder – research that crosses the boundaries between condensed matter physics and quantum gravity, which I myself would not have been close to understanding at that age!  At the Boulder summer school for condensed matter and materials physics last July, the students were simply incredible.  Our KITP postdocs are doing breakthrough work on their own at the cutting edge of topological matter. Experiments on quantum materials are rich and varied (though increasingly the ones I care about are done outside of the US), and it seems likely we will see increasing technological impacts of these basic research areas soon.  I don’t think my own research has ever been as diverse as it is now.  So it is a great time to do physics, even though I worry about being able to afford it.






Recently I’ve spent some time thinking about domains in some magnetic materials, trying to understand how they influence and interact with topological electronic structure and Hall transport.  In the process, I played around a bit with simple models of the domain formation and dynamics.  This is a kind of classic problem in statistical mechanics.  You have a system which in equilibrium spontaneously orders below some temperature, for example a ferromagnet.  There are several possible directions to the magnetization, all with the same energy.  If you suddenly lower the temperature from above the ordering temperature, where there is no magnetization, to below it, magnetization starts to form.  This is called a “quench”.  But at different points in space, the orientation is different, and random.  Then over time after the quench, nearby regions begin to align with one another, and larger regions, called domains, develop, within which the magnetization is uniform.  These grow with time, and this process is called “coarsening”.

The simplest example is an Ising magnet, in which there are only two types of domains.  But there are more complex situations.  Here I was considering a case in which the magnetization orients within a plane, and has 6 preferred directions, say multiples of 60 degrees.  This is modeled by a simple Langevin dynamics

\gamma \partial_t \theta_i = -\frac{\partial H}{\partial \theta_i} + \eta_i(t)

where \gamma>0 is a damping constant, \eta_i(t) is a stochastic Gaussian delta-function correlated “white” noise, and H is a Hamilonian

H = -J \sum_{\langle ij\rangle} \cos(\theta_i-\theta_j) - \lambda \sum_i \cos 6\theta_i,

where J gives an energy for neighboring spins to align, and \lambda >0 is a six-fold anisotropy that favors the 6 domains.  This is what would be called “model A” dynamics of an XY model with 6-fold anisotropy.

It is surprisingly easy to simulate this thing – I did it with Julia’s DifferentialEquations package. The animation shows the evolution with time starting from a completely random initial condition, on a 90×90 square lattice.  It is beautiful how the domains develop.  You also see some “vortices”, which are points around which the angle winds by \pm 2\pi, which due to the anisotropy become the intersection point of 6 domain walls.  Over time the domains grow – you can see the “walls” between domains moving – and the vortices slowly meander and eventually will find a way to annihilate one another.  The size of the domains, the distance between vortices, etc., all show power-law dependence on time in this type of problem.  Can you spot the different kinds of vortices (clockwise versus counter-clockwise winding), and see some of the annihilation events?

In a real material eventually the domain motions will be affected by disorder, which slows and finally stops them in some frozen or nearly stationary state. Lots of other effects can also come in, like extra forces at the boundaries of the sample, applied fields, strains, etc., that can make the eventual structure quite complex and hard to predict.  But still it is cool.