Landau fans in TBG

There’s a lot of excitement over twisted bilayer graphene (TBG), and increasingly over other sorts of moiré heterostructures from 2d materials. In TBG, most of the interest is due to the discovery of many correlation phenomena that occur at partial filling of the mini-bands nearest the charge neutrality point. Theory says that they are very flat, and have novel topological aspects to them. The flatness explains their tendency to be unstable to interactions that induce the correlated states. But the details of that flatness still matter to any theory. There are lots of models for this – how do we know what is right? It would be great to be able to actually measure the bands in experiment.

Example of a “fan diagram” obtained for TBG

The most sensitive way to measure bands experimentally is via quantum oscillations. Semiclassically, these are oscillations as a function of doping/field/etc due to interference of electron trajectories that go in closed orbits in momentum space. So one can learn about these orbits. Quantum mechanically, this is due to Landau level formation. Anyway, many groups have seen quantum oscillations in TBG in the regime where correlated states form. But guess what? They do not really agree with the naïve explanations based on most band theories. There are many discrepancies, but the most obvious one is that even close to charge neutrality, the “magic angle” samples do not show the oscillations that would be expected from Dirac cones at the corners of the moiré Brillouin zone. The expected behavior is actually seen at larger angles. So something different happens near the magic angles that needs to be explained.

My students Kasra Hejazi and Chunxiao Liu and myself wondered if the differences might be related to topological changes that happen near the magic angles, in the “standard model” of these systems due mainly to Bistritzer and Macdonald. So these guys set out to do a fully honest calculation of the quantum oscillations. This is a variant of the Hofstadter “butterfly” problem, and the true electron spectrum is actually fractal. But they managed to subdue the beast of a challenge and work things out rather completely. Short answer: for certain angles in the “magic range” one indeed gets anomalous quantum oscillations consistent with experiment just from band structure. This is associated with additional topological transitions, or semi-classically to orbits that are not near the moiré zone corners.

This could be an explanation of the experiments! But maybe it is just a piece of the puzzle. The electron-electron interactions can and probably do modify the quantum oscillations. Maybe they could even “fix” the one dis-satisfying feature of our calculations: the anomalous oscillations only occur for a rather narrow range of angles. Maybe interactions make them more robust?

Anyway, the paper is out there – have a look:

On the challenges of reading papers

This post is really just for fun. Everyone knows that reading a professional level paper at the forefront of research is often challenging. This morning I printed out a copy of a paper I wanted to read, from a pdf in my browser. The result was even more difficult to understand than usual:

Quite remarkable! Now it’s not impossible that the author of this paper is reading this. If so, can you identify it? A hint to that person is that I wanted to read it based on a recent visit I made to the author’s university.

Moiré is better

I’ve taken too long of a hiatus from writing here.  Now seems like a good time to get back on the horse. Here at the KITP we just finished a two week Rapid Response program on “Correlations in Moiré Flat Bands” (follow the link to see the talks). I’ve blogged last year about this field, which was driven by two remarkable experimental papers from Pablo Jarillo-Herrero’s group at MIT that came out around March. They found that two sheets of graphene, twisted relative to one another by about 1 degree, showed signed of strong electron correlation and superconductivity as a result of an effective long-wavelength superlattice created by the moiré interference pattern of the two slightly mismatched atomic potentials. The results were very impressive, but would this be a one-time success, or would it signal a nascent area of research?

Now 9 months later, it’s clear that the initial discovery has indeed given birth to a vibrant and promising field. At least 6 more leading experimental groups have observed correlation physics in similar moiré structures, and their progress is both impressive and inspiring. There are many observations of superconductivity, and now correlated insulators have been discovered at every integer “filling” of the moiré superlattice. Several groups showed measurements indicating apparent ferromagnetism or metamagnetism in the vicinity of the “1/4 filling” insulators, and furthermore signs that these states may be topologically non-trivial Chern insulators — i.e. they may possess a quantized Hall effect and edge states. This might even coexist with superconductivity.

Other measurements explored the electrical transport at temperatures above these ordering instabilities, finding evidence for a large T-linear component of the resistivity: with a slope 100 times larger than in untwisted graphene. The origin and significance of this was robustly debated. I was struck by the wealth of data from some of these experiments: a full measurement of the resistivity over a range of electron densities spanning two entire bands, and over a range of temperatures covering the full bandwidth (and more)! I can’t recall seeing anything quite like it. It reminds me of the idea, which arose in ultra-cold atomic physics, or doing “precision many body theory”. Understanding these volumes of data in quantitative detail may be a similar challenge in the solid state.

Being a theory institute, we saw of course a wide range of theoretical contributions presented and discussed. It’s clear that in this short time the field is already passing into a more refined stage. The initial furor of discovery is past and the opportunity for “quick and dirty” theory is passing with it. Theorists now must get serious and contend with the realities of this fascinating class of structures. There are many questions. A few that come to mind: What exactly are the implications of “fragile topology” here? How important are asymmetries induced by the realities of the devices? Which phenomena are due to phonons, and which phonons exactly? How much can the models for these systems really be simplified?

For definitive answers, I expect theory to need to get quantitative, and to confront directly the measured quantities like resistivity and its temperature, field, density dependence. A lot of the attendees are working towards this, and I am optimistic we will see major progress in the near future. More moiré measurements and theory is indeed better!

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.