# Coarsening

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.

# A new quantum spin liquid?

I’ve been feeling I’ve not been posting enough.  I’ll try to get back into it…

Inelastic neutron scattering intensity of YbMgGaO4, from two recent papers:  arXiv:1607.02615 (left) and arXiv:1607.03231 (right).

I’ve been studying quantum spin liquids for many years now.  Or maybe I should say looking for them?  Quantum spin liquids have both theoretical and experimental definitions.  In theory, they are exotic phases of quantum matter whose ground states are intrinsically and inextricably in quantum superposition.  From this quantum entanglement of the ground state they obtain the ability to support exotic excitations that behave like particles different from the elementary ones in our world: emergent non-local “quasiparticles”.  In experiment, they are materials containing spins that avoid order at the lowest temperature, for which the conventional magnetic excitations – magnons – are absent, and which show some anomalous properties, e.g. unusual thermodynamics.  Somehow it has been hard to bring these two meanings together.  Either this is a failing of theory, a lack of the right measurements, or maybe we just haven’t found the right material yet.

The last option would make things easy – maybe we will just find a new system where the agreement is obvious.  So I get excited when a new material with some promise appears.  About a year ago, I heard from Gang Chen, my former student, now a professor at Fudan university in China, of YbMgGaO4.  This is a material with a triangular lattice of Yb ions, each of which hosts an effective S=1/2 quantum spin.  Typical of 4f rare earth ion spins, it has very short-range but highly anisotropic interactions with its neighbors.  Guided by Gang, the experimentalists identified it as a spin liquid: at least the spins remain correlated but disordered down to millikelvin temperatures.  How and why, remains to be understood.

One intriguing thing is some similarity to the thermodynamic properties observed in the more famous organic spin liquid materials, which have been studied for many years.  And both share the triangular lattice structure.  The reason the new YbMgGaO4 material is of particular interest, though, is a practical one.  The Yb spins have large magnetic moments, and they are dense, so it is a good material for inelastic neutron scattering studies.   This is the most powerful (i.e. data-rich) technique to study magnetism we know today, so this is a big deal.  Moreover, single crystals big enough for such measurements are now available.  The above images are from papers this year by two different groups, arXiv:1607.02615 (left), and arXiv:1607.03231 (right), which carried out the first such measurements, and show a smooth continuum of excitations, i.e. an apparent absence of conventional magnons.  Nice pictures.

For now I won’t speculate (at least in writing!) about what’s going on, but it is certainly interesting and I’m sure it will be an active research subject for the near future.  Let me also mention that in a material like this, the chemistry is largely independent of the rare earth element, so Yb can be substituted for almost any other 4f atom.  This means a wide variety of magnetic systems can be found with identical geometry.  So YbMgGaO4 is just the first of many new materials we can probably expect to see studied soon.  Something to look forward to!

# Congratulations!

Congratulations to David Thouless, Duncan Haldane, and Mike Kosterlitz for the 2016 Nobel prize in physics!  They pioneered applications of topology to condensed matter physics.  It took a while, but now the use of topology in condensed matter is commonplace.

These guys did foundational work, and the field has developed a lot since their seminal contributions.  Given this prize, it will be interesting to see how the work of Charlie Kane and others on topological insulators fares with the Nobel committee in the future!

# Passion for Physics

I hold a weekly group meeting, Wednesday afternoons, with an extended group including some students, visiting KITP grad fellows, postdocs, and sometimes other visitors.  It runs an hour, and generally unstructured.  Yesterday our new postdoc, Xiao Chen, who just arrived from UIUC, talked about his work on “out of time ordered” correlations, which is a new-ish way to study dynamics of quantum systems, the transition to chaos and thermalization, and other related phenomena.  There are connections of this subject to condensed matter physics, ultra-cold atoms, quantum information science, and even black hole information (which is where this originated!).

It’s a pretty stimulating subject, and after an hour when the meeting ended, four of us stayed there and kept talking for about another hour – I didn’t keep track – chatting about all these different connected ideas, and what we did and didn’t understand.  None of us are really expert in this subject, and only Xiao has worked on it very seriously.  If you know me, you know my interests tend to lean towards “real world” topics – i.e. specific to some material or experiment.  This is pretty far from that!  But how can you not be fascinated by these concepts that are so far ranging that they touch both the foundations of statistical mechanics and the origins of gravity?  I don’t know how, but there were still only four of us that stayed.

In most professions, free-ranging discussions don’t seem like a part of “work”.  With all the many responsibilities of an academic position, it can be hard to take time for them even for us.   Yet I firmly believe that such dialogues are critical to learning, developing new ideas, and doing good research.    More than anything else, if you have a passion for physics, you should love this stuff!   It made my day yesterday – thanks people!

I just saw the summary for a feature story on “Physics”, the online journal highlighting what the APS considers exciting developments

Weyl semimetals are definitely a hot subject lately.  These are electronic materials in which non-degenerate (i.e. spin-split) conduction and valence bands touch at a single point (called a Weyl point) in three dimensions.  Anton Burkov and I actually were one of the first groups to predict these things some years ago (see this paper), hot on the heels of Ashvin Vishwanath’s group (here).  Well full disclosure requires me to point out that Herring discovered them in 1937!  Anyway, like the related topological insulators, these things are not actually all that rare, and are pretty easy to find using very standard DFT methods, so the subject is booming.

One of the latest twists is the discovery of “type II” Weyl semimetals (if you buy into this, you now call the original ones type I).  I’m of II minds myself about this.  On the one hand, the type II Weyl point is no different from a type I Weyl point, the only distinction is that in the type II case,  the dispersion away from the point goes only in one direction of energy, in some directions in k space.  On the other hand, because of this type of slope, such a type II Weyl point cannot be isolated: there are always states at the same energy elsewhere in k space.  And those other states form a Fermi surface, which has much more density of states than the vicinity of the point itself.  So I would have thought type II Weyl points are mainly special in that they by construction are less able to influence electronic properties than the type I Weyl point (which can be isolated in energy and in that case would control all the physics).

But I’ve been wrong before about what becomes big.  Type II Weyl points…sure!

Nevertheless, I found the summary of this Physics piece crazy.  Type II Weyl semimetals would be (what does “would be” mean here, anyway?  In some hypothetical world?) both conducting and insulating in different directions?  Certainly not!  As cool as Weyl fermions are, whatever type they are, they are extremely well-described by non-interacting quasiparticle theory.  According to this cornerstone of the theory of solids, electrons travel in directions determined by their group velocity, and because they certainly disperse in three dimensions, they also move in all directions.  So the statement, at face value, is pretty much absurd.

When I read the actual feature, a few paragraphs down there is some discussion where a magnetic field is mentioned in this context, which might allow for some speck of truth in the summary.  Actually it has been known for decades that any low density electron system becomes, in high magnetic field, much more conducting along the field than normal to it.  This is a called the ultra-quantum limit. If conditions are right, ultra-quantum electrons can be conducting along the field and insulating normal to it.  This does not depend much upon the nature of the band dispersion.

So, well, I don’t know what the writer had in mind, and maybe there is some way it could make sense here. Yet…I doubt it.

I should say that there certainly are materials that are conducting in some directions and insulating in others.  Usually this is related to structure: they are built from linear chains or planes, that are weakly coupled.  The disparity between conducting in different directions can be magnified by magnetic fields, by strong interactions, by disorder, or all of the above.  I don’t think this is particularly rare.

# Strange Stuff

I gave a public lecture in Boulder, CO, in conjunction with the Boulder Summer School for Condensed Matter Physics, entitled Strange Stuff: A Second Quantum Revolution.  You can watch it on youtube.  Or look at a pdf of the slides.

# An interesting article

I occasionally write articles for the online journal club in condensed matter physics.  I’ve written one that should appear in next month’s listing.  Thought I would share it early here.  It highlights a recent article by a few of my colleagues at UCSB, who are over in Microsoft’s Station Q research center.  They present some new ideas of time translation symmetry breaking in quantum systems.

Spontaneous symmetry breaking is a fundamental paradigm in physics, from the Higgs field of the standard model to its many manifestations in condensed matter and materials physics. It occurs whenever a symmetry of the equations of motion, or the Hamiltonian for Hamiltonian dynamics, is not fully preserved by physical quantities. The broken symmetry can be internal or global, such as spin-rotation symmetry which is broken across the Curie point of a ferromagnet, or a spacial symmetry such as translations which are broken from their continuous form in a fluid or gas to a discrete subgroup in the transition to a crystalline solid.

A seemingly more exotic idea is time-translation symmetry breaking (TTSB).  By virtue of the analogy with spatial symmetry breaking, such a situation was deemed a “time crystal” by Wilczek.  In this language it appears very exotic, but it can be recast in more familiar form.  Any oscillator is in a sense an example of TTSB: a translation by less than the period of the oscillator alters the configuration of the oscillator.  Oscillation obviously occurs easily in finite classical and quantum systems — c.f. the simple harmonic oscillator.  The trickiness comes if you want the oscillation to be a robust, universal feature of a system.

To wit, most of our understanding of universality rests on equilibrium statistical mechanics.  Unlike the more conventional forms of symmetry breaking, persistent oscillations are not present in equilibrium, almost by definition: in equilibrium all observables settle down to average values determined by the rules of statistical mechanics.  A recent cogent discussion is in this article.  Hence, a system with TTSB must be out of equilibrium.  This in ensured by driving with external forces, influx of energy, etc.  Then spontaneous oscillations can certainly arise by various mechanisms.  For example in the AC Josephson effect, driving a Josephson junction above its critical current leads to oscillations of voltage.  There are well-known oscillating chemical reactions.

Such mechanisms explain spontaneous oscillations at short times, but not their coherence.  Naively small perturbations or noise can induce phase shifts that build up over long times, spoiling the perfect phase coherence.  The formal question, analogous to that in ordinary spontaneous symmetry breaking, is whether the oscillations remain synchronized over long time and space separations, i.e. is there “long range order”?  Again, in classical systems there is a long history of asking this question, from influential work by Winfree on biological rhythms to studies of narrow band noise in charge density waves.

In the highlighted paper, Else et al address the existence of TTSB in quantum systems, with driving perturbations periodic in time.  Since the underlying symmetry of the dynamics in this case is already discrete, TTSB must also be discrete — it manifests if physical quantities oscillate with a period larger than that of the drive.  Any finite system of this type has eigenstates of Floquet type: states where $|\psi(t+T)\rangle = e^{i\phi} |\psi(t)\rangle$, where T is the period of the drive.  This is the analog of a stationary state in Hamiltonian mechanics, and obviously in such a state expectation values are invariant under translations by T.  So there is no TTSB in a Floquet eigenstate.

However, it is not obvious that the dynamics of a generic state behaves the same way, and in fact Else et al construct an example where they do not.  Specifically, they present a simple model of spins in which the unitary evolution over a period T consists of two parts, $U(T) = U_2 U_1$.  They take $U_1 = \prod_i \sigma_i^x$, which flips every spin in the z basis, and $U_2 = \exp [i H(\{ \sigma_i^z \})]$, where H is a local Hamiltonian-like function, so that evolution by $U_2$ assigns a state-dependent phase to each product state in the z basis.  It is straightforward to show that consequently all the Floquet eigenstates are Schrödinger cat states, i.e. they are a superposition of two macroscopically distinct components with all flipped spins in one component relative to the other.  Because such a cat state is exponentially difficult to construct from a product state, they argue that a generic initial state never relaxes to a Floquet-like state, and instead undergoes what is basically a persistent Bloch oscillation living for a time that grows exponentially with system size.  The situation has close analogies to the usual symmetry breaking in an Ising ferromagnet, in which the true finite system ground states are cat states, but are never reached in physically relevant times.

This establishes a very simple example of TTSB in a periodiocally driven quantum system, and a nice connection of TTSB to non-local entanglement. The authors further focus on the case in which the phase factor derives from a strongly disordered Hamiltonian, in which the undriven Hamiltonian would exhibit many body localization. In that context, they argue that many body localization lends stability to TTSB. Like most results for many body localization, the stability argument is not rigorous, but it is reasonable, and indeed they present numerical results consistent with this claim. Moreover, the new work shows that prior theoretical studies giving instances of MBL phases which symmetry protected topological order or discrete symmetry breaking also exhibit TTSB (see the highlighted articles’ Refs.[24-30]).

One may wonder whether this strong disorder regime is the only situation where TTSB is stable, or whether there may be other examples. Are there examples of TTSB beyond the simplest discrete multiplication of the drive period? Assuming TTSB indeed exists, then there should be dynamical phase transitions into or out of the oscillating situation, which would also be interesting to study. On a complementary front, the simplicity of discrete TTSB seems a practical target for experiments with ultra-cold atoms or other driven quantum systems. Despite the current obsession of theory with topology, it seems that the old fashioned notion of symmetry breaking still has some surprises left for the community.

You can read the article here: arXiv

# A picture is worth ~4000 words

from arXiv:1605.04199.  On the left is experiment, in the right is theory.

Theory isn’t always right, and even when it is right, it isn’t always relevant. Sometimes it is hard to tell, in both respects.  Occasionally, though, confirmation of a theory is strikingly clear.  Humans are visual creatures, so we really like to “see” this in a picture.

It’s been about 9 years since my group wrote a paper  of about 4000 words presenting a theory of frustrated antiferromagnets based on the special geometry of the diamond lattice.  We predicted a unique “spiral surface” of strong spin fluctuations in momentum space, which we proposed could be observed by neutron scattering in the material MnSc2S4.  At the time, the experiment was impossible because there were no single crystals available.  A pity.   9 years is a while.  Jason Alicea and Emmanuel Gull, both graduate students at the time, are now successful professors at Caltech and Michigan, respectively, and Simon Trebst, who at the time was a researcher at Microsoft, now is a Professor Doctor in Cologne.   Doron Bergman is making loads of money in the SF Bay area.

But in the last month intrepid experimenters in Europe presented new results, after preparing large enough single crystals to do the measurements.  In this paper, you can actually see the spiral surface – or rather a cut through it, which makes a kind of rounded square, in the raw data.  Pretty cool!

# He looks pretty young to me

Tonight at 6pm there is a “Café KITP” talk by Tim Hsieh, one of our esteemed postdocs at the KITP, entitled “The age of entanglement”.  These talks are held at the Soho bar and lounge in Santa Barbara, and aimed at the general public.  It’s certainly true that quantum entanglement is everywhere in theoretical physics these days.  I’m sure Tim will give a great presentation.

# Quantum skyrmions

A skyrmion is a topological defect in a ferromagnet.  In a two dimensional system, it consists of a configuration in which every possible orientation of the magnetization occurs, “wrapping” the full sphere once.  One can visualize it by putting the magnetization down at the origin and up at infinity, and rotating smoothly in a plane containing the radial and vertical directions, between these two orientations.  At some radius the magnetization is in the plane, and rotates by 360 degrees as one moves around the origin.

This is a classical topological defect.  What happens with quantum mechanics?  The skyrmion is finite object, so one might imagine that it could behave as a quantum particle.  If it is large, it involves many spins and will be heavy and classical.  But if it is small, what happens?  Rina Takashima, a graduate student from Kyoto University and a KITP graduate fellow, worked with me on this question and we found that indeed a skyrmion becomes a “quasiparticle”.  It has some unusual dynamics and this can lead to interesting physics, for example a sort of “Bose condensation” of skyrmions. So far our work is really focused on chiral magnets, for which the in-plane spin component is fixed by the material.  In the future we hope to look at non-chiral ferromagnets, where skyrmions can also occur but where the “chirality” of the in-plane twist is arbitrary and can be spontaneous.