Deceptive advertising

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

Researchers provide new evidence for the existence of type-II Weyl semimetals, which would be both conducting and insulating in different spatial directions. ” (my emphasis)

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.


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.

You can read about this in our arXiv preprint.

What is a panoply?

My paper with Oleg Starykh, “Quantum Lifshitz field theory of a frustrated ferromagnet” was just published in Physical Review Letters.  Our original title, which is in the version we posted on the arXiv back in October 2015, was “A panoply of orders from a quantum Lifshitz field theory”.  We had quite some argument with a referee, who objected to the title, and despite my fondness for getting obscure words into physics publications, we adopted the new title.   A panoply is “a complete or impressive collection of things.”  This was inspired by the remarkably rich phase diagram of a very simple model: the spin-1/2 1d Heisenberg chain with ferromagnetic nearest-neighbor interaction and antiferromagnetic second-neighbor interaction.  In an applied field, a variety of interesting phases emerge.  Individually, these have been understood, but our goal was to synthesize all the different phases together to find a unified picture.


Schematic phase diagram, incorporating the main features predicted by the quantum Lifshitz field theory

The referee objected because we were not successful at a complete and rigorous synthesis. But I think we got pretty far.  What we uncovered was a simple and novel quantum critical theory, which captures the universal aspects of the system, and for which many (but not all) properties could be obtained exactly.  Moreover, we could show that it explained a first order metamagnetic transition, a vector chiral phase, and a spin nematic state.  Maybe not enough to be a panoply in terms of completeness, but I at least was impressed by how far this simple field theory could go.   You can read the paper and decide for yourself.

Icing model

Lucile Savary and I just posted a paper on the arXiv on “Disorder induced entanglement” in spin ice systems.  This was stimulated by experiments by Satoru Nakatsuji, Collin Broholm, and their collaborators on Pr2Zr2O7, which was supposed to be a type of “quantum spin ice”.   The idea was that the exchange couplings between Pr spins would involve a lot of spin-flip interactions, which induce quantum dynamics.  Their experiments revealed that actually disorder was more important than the quantum exchanges.

This turns out to be related to a bit of very common atomic physics.  Pr3+ is what is called a non-Kramers ion, which means that it has an even number of electrons.  It forms a two-level system which we can describe using a spin-1/2 operator like a Pauli matrix, but which is not quite a usual spin.  In particular, for such a non-Kramers ion, the spin operator is not odd under time-reversal.  Actually in this case the z component is odd, but the x and y components are even (this is equivalent to the condition T^2 = +1 that theorists like).  As a consequence, disorder in the material, for example misplaced ions in the Zr sites, or missing/extra O, exert electric fields inside the sample that induce local random “transverse fields”, i.e. terms like h_x S^x +h_y S^y, with different h_x,h_y for each spin.  This effect is electrostatic in origin, so it should be a robust and dominant one.


Schematic phase diagram of the random transverse field Icing model.  See the preprint for an explanation.

A transverse field is a textbook way to induce quantum dynamics in a classical system.  For example, the transverse field Ising ferromagnet is the paradigm for quantum criticality.  What we learned is that in non-Kramers systems, a transverse field can be induced without any real magnetic field, even without breaking time-reversal symmetry.  So we thought: let’s put this to work for us!  One should be able to controllably induce quantum dynamics by introducing disorder in an otherwise classical magnet.  The spin ice pyrochlores are a natural place to look.  The two most studied materials are Dy2Ti2O7 and Ho2Ti2O7.  Both seem to be modeled extremely well by a classical Ising model.  Dy3+ is a Kramers ion, so it does not work for us, but Ho3+ is a non-Kramers ion, so we’re in business!  A good model for disordered Ho2Ti2O7 is thus the classical spin ice Ising model plus a local random transverse field.  Inspired by the textbooks, we called this the random field Icing model.

Yes, the entire post has been a set up for the name.  I’m really proud of it.  The physics is pretty interesting too.  You can read about in the arXiv article.  That’s the original text – we’ve had to cut it since to fit journal length constraints, in our rather long and painful referee process which is still on-going (I don’t understand why so many referees are so bitter – lighten up!).  We rather like the original text which is more pedagogical.

Kitaev, perturbed

No, Alexei is not upset.  Only his model is being perturbed.  In 2006, Kitaev introduced his “honeycomb model” – an exactly soluble Hamiltonian describing spin-1/2 spins interacting via anisotropic exchange on the nearest neighbor bonds of a honeycomb lattice.  He showed that it is a beautiful example of a quantum spin liquid: a highly entangled very non-trivial zero temperature phase of matter.  Most remarkably, its elementary excitations are not spin waves as in a usual magnet, but instead they are Majorana fermions and some exotic “vortices”!  In the past few years, following a beautiful proposal by Jackeli and Khaliullin, it has been recognized that despite the apparently artificial appearance of the model, it might be a good first approximation to a number of real materials.  So one can hope maybe to find Kitaev’s quantum spin liquid in the laboratory!

This hope is reasonable because of stability: Kitaev showed that any small perturbation to his model which preserves time-reversal symmetry leaves the system in the same spin liquid phase.  So an experimental material doesn’t need to be extremely finely tuned to land in the state.  How would one look for it?  The most powerful probe in quantum magnetism is inelastic neutron scattering, which measures a scattering amplitude proportional to the number of excitations of a given momentum and energy created in response to flipping a spin.   It can be calculated exactly for Kitaev’s soluble model.

That is a pretty straightforward calculation, which yields a surprise: there are no excitations created below some minimum energy, or “gap”.  This appears to be a “spin gap”.  It is surprising because Kitaev’s solution shows that there is no true gap: the Majorana fermions actually have a massless relativistic dispersion, like light, so exist at arbitrarily small energies when the wavelength is arbitrarily long.


Here is the calculated structure factor – actually just the new gapless contribution.  The “obvious” part obtained from the Kitaev point adds to this, and contributes above the dashed line.

It’s a bit of a weird result, but it is so straightforward it can’t be wrong…or can it?  In fact, in a paper that just appeared on the arXiv, Xue-Yang Song, Yi-Zhuang You and I showed that the “spin gap” is only a feature of the exactly soluble model.  For any generic Hamiltonian in Kitaev’s spin liquid phase, there is not even an apparent spin gap.  By combining quantum mechanical perturbation theory and field-theoretic arguments, we were able to work out precisely how the gap fills in, and what the low energy structure of spin excitations looks like in a generic system.

This work was spearheaded by Xue-Yang Song, who is a third year undergraduate at Peking University!  Well done Xue-Yang!  You can read about it in the preprint.

Kagomé encore: kapellasite

A few days ago I (finally!) finished a paper with several friends – let me single out Oleg Starykh, Donna Sheng, and Shoushu Gong (how cool a name is that??!!) – on a combined analytical and computational study of a spin-1/2 kagomé antiferromagnet.  It wasn’t the usual nearest-neighbor one, which seems to manage to stay controvbcversial forever, but a less-studied variant, where a longer-distance exchange coupling is dominant.  This is believed to be a good model for the mineral kapellasite.  It turned out that this leads to a natural way to think about the lattice as decomposed into many constituent one dimensional spin chains.  Just that insight is enough to understand nearly everything, and with a little work, make a remarkable number of very detailed – and successful – comparisons between analy
tics and DMRG computations.

I like this picture of a valence bond solid order that occurs in this model, which comes completely from analytic predictions, and matches the DMRG results extremely well.

The paper is on the arXiv: arXiv link