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: https://arxiv.org/abs/1903.11563

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