Dirac points: the movie

There’s a lot of excitement about twisted bilayer graphene, which I wrote a bit about earlier.  We have been going back to basics and studying the band structure of these systems, using some of the models which have been largely accepted, at least in broad terms, to describe them.  These models take the form of two continuum coupled Dirac equations with a spatially periodic hopping term connecting them.  The first thorough early study was by Bistritzer and MacDonald back in 2011.  They discovered that (1) for nearly all angles the original Dirac points of persist, (2) at certain “magic” angles the Dirac velocity vanishes, and (3) near these magic angles the low energy bands become exceptionally narrow over the entire Brillouin zone.

What we showed in our recent preprint is that actually the vanishing Dirac velocity is just one of several topological phase transitions associated with merging/splitting/annihilation of Dirac points as the angle is varied.  The above video shows how these points move around the moiré Brillouin zone as the angle is varied, in a range of angles close to the first magic angle (about 1 degree).  The parameter \alpha shown at the upper right of the video is inversely proportional to the angle.  Please see our preprint for more details.

Topological superconductivity in twisted bilayer graphene?


If you went to the American Physical Society March meeting this year, you probably heard about exciting experiments discovering correlated insulators and superconductivity in twisted bilayer graphene.   This is exciting because it seems like the first real observation of strong correlation physics in graphene outside the strong field quantum Hall regime.  The moiré patterns generated by twisting these layers are fascinating and is an intriguing “twist” on correlated electron theory as well.  You can gauge the fascination of theorists by this by checking out how many papers appeared rapidly on the topic since on the arXiv.

Thanks to my colleague Cenke Xu, who works very quickly, he and I were the first ones in this wave of theories.  Our work was very simple, and perhaps naïve in some ways, but it does give an idea of the richness that might occur in these systems.  We found that even a relatively simple model leads to topological superconductivity, which would be quite exciting if true.  Our paper was recently published in PRL, and you can find a Physics Viewpoint that discusses it as well.  I made the image in this post due to a request from the editors for this Viewpoint, but apparently they didn’t like it.  So instead I’m sharing it here.

My group is continuing to work on the subject, and I plan to post something on our more recent work soon.


Counter current heat exchanger


Counter current heat exchange in a gull’s leg.  Art by Michael McNelly (after Ricklefs. 1990. Ecology. W.H. Freeman, New York).  Taken from this page

In biology class in high school, I learned that birds have a clever mechanism to help keep warm in winter.  They need to supply blood all the way down their bare legs to their feet.  This is potentially a way to lose a lot of heat, as it would be lost readily to the environment without any insulation.  However, in a bird’s leg, the artery in which blood flows down is positioned in contact with the vein that carries the blood back up, and heat is exchanged across the contact.  In that way most of the heat does not flow down the leg, and the lower portion of the leg maintains a much lower temperature where it does not lose heat.  This is called counter current heat exchange.

My group recently stumbled across an analog of this phenomena in a very different venue – a type of Hall effect in a quantum spin liquid.  This was prompted by a beautiful experiment by Kasahara et al, just published in Nature , reporting the discovery of a quantized thermal Hall effect in the material \alpha-RuCl3.  A thermal Hall effect means that some heat moves perpendicular to an applied temperature gradient, or conversely, that a temperature gradient appears perpendicular to the flow of heat.  You can imagine there is some relation to the bird’s leg, in which the main heat flow is vertically up and down the leg, but heat also passes horizontally from the artery to the vein.  In the experiment, a heat current is applied along the x direction of a sample, and a temperature gradient develops along the y axis.  The remarkable thing – not present in the bird – is that the magnitude of the temperature difference divided by the heat current is quantized.

Quantization of the thermal Hall effect was predicted a long time ago for systems that possess chiral edge states, like those in the quantum Hall effect (which is a similar but much easier to measure effect involving electrical current and voltage rather than heat current and temperature).  However, the theory behind the thermal Hall effect presumed that these edge states are the only things carrying heat.  In the experiment by Kasahara et al, however, it is clear that most of the heat is actually being carried by motions of the atoms that make up the crystal, rather than chiral edge states.  So the theory needed to be reconsidered.   A few of us – Mengxing Ye, Gábor Halász, Lucile Savary, and I – developed a theory of the thermal Hall effect including the lattice.  What we found was that the approximate quantization of the thermal Hall effect could be explained if the lattice acts as a counter current heat exchanger between two edge states that play the role of the artery and the vein in the bird.  It was rather surprising to us, and indeed we found that actually the lattice’s involvement in heat transfer helps to observe the effect.


Schematic of the counter current heat exchange in the thermal Hall effect.  An edge state at the upper and lower edge (and the boundaries) is shown as a tube, with horizontal arrows showing the flow of heat in the edges.  Vertical arrows show the heat flow from the edge into the bulk lattice, which acts as the medium to exchange heat between the edges.  For an explanation of all the other labels, see our paper.

This is a rather quick summary.  I actually wrote a commentary on the experiment in the journal club for condensed matter physics, which you can read here.  This explains the quantum context more, and why it is such a cool discovery.  You can also read our paper on the arXiv.   The final version has appeared in Physical Review Letters.

HFM 2018


I’m on the program and organization committees for this summer’s Highly Frustrated Magnetism conference, which is going to be conveniently located at UC Davis, California, from July 9-14, 2018.

The conference is *very close* to the deadline to submit abstracts for talks or posters.  Please consider making a contribution and attending!  More information, and instructions on how to submit an abstract, are on the conference web page here:


APS elections

We’ve just had a round of local elections, and you may still be having a hard time getting over the national ones from a year ago.  There’s an easier one underway – elections at the American Physical Society.  I’m one of the candidates for “Member at Large” of the Division of Condensed Matter Physics (DCMP) Executive Committee. If you are a member of DCMP, you probably got an email explaining how to vote.  The executive committee mostly makes decisions related to the representation of condensed matter physics at the annual American Physical Society meeting.  But it also “has general charge of the affairs of the division.”

It seemed to me like a reasonable time to run for the committee.  There certainly are affairs that are of concern in DCMP and indeed to all of science and beyond.   For example, the proposed tax on tuition and fee remission for graduate students is a big one, that you can read about in the NY times.  That’s a nightmare for universities and something that would be extremely damaging to research and higher education in the US.

If you happen to be a member of the APS and the DCMP, I’d be happy to have your vote 🙂  I’ll try to be a responsible and responsive representative.

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.