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March 28, 20255 min readPhysicists Catch a Quantum Butterfly Spreading Its WingsIn a first, physicists have directly seen Hofstadters butterflya long-sought-after fractal in the quantum realmBy Gayoung Lee edited by Lee BillingsA team of scientists from Princeton University has measured the energies of electrons in a new class of quantum materials and has found them to follow a fractal pattern known as Hofstadters butterfly, which has long been predicted, but the new study marks the first time it has been directly observed experimentally in a real material. Yazdani GroupAlmost 50 years ago, computer scientist Douglas Hofstadter predicted that a butterfly would spread its wings in the quantum world. Under the right conditions, tiny electrons in a quantum system could produce an energy spectrum composed of fractals, intricate self-repeating structures that would form a very striking pattern somewhat resembling a butterfly, he wrote in a seminal 1976 paper.Many physicists have attempted to create Hofstadters butterfly in different formats, with varying degrees of success; the first such spectra emerged about 25 years ago. The difficulty in observing the effect was, in part, because Hofstadters initial prediction posited that it would require colossal magnetic fields beyond the reach of any laboratory. Most experimental efforts consequently sought to summon the butterfly in silico, within the confines of computer simulations, and those reliant on physical quantum systems studied its properties using largely indirect measurements.Now, however, what may be the first-ever direct, real-world observation of the butterfly has emerged from the complex quantum dance of electrons sandwiched between two microscopic layers of graphene. The results, published recently in Nature, are all the more remarkable because they were unexpectedthe researchers involved werent even trying to hatch Hofstadters butterfly from its quantum chrysalis.On supporting science journalismIf you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.I think it was happy accident, says study co-author Kevin Nuckolls, a physicist at the Massachusetts Institute of Technology. I think this is common for physics experiments [in which] you see something weird. You spend a couple hours on it and decidelike, Ill give it a couple more days.At the time of their experiment, Nuckolls and his co-authors were all part of the same Princeton University lab, studying how superconductivitythe resistance-free flow of electricitymanifests in graphene, a two-dimensional crystal formed by a single layer of carbon atoms arranged in a hexagonal pattern. When two sheets of graphene are stacked one atop the other, with a slight rotational offset of about 1.1 degrees so that the hexagons dont exactly overlap, a so-called magic-angle configuration is formed. When subjected to a magnetic field, the electrons within each sheet zip back and forth between the carbon atoms, exhibiting superconductivity and other bizarre properties.Manufacturing such twisted bilayer graphene is as much art as science. It often yields off-kilter duds that dont have the proper angle. So for each attempt, the researchers checked their work by directly probing the sheets with a scanning tunneling microscope (STM). The resulting images show the flow of electrons through the material and can indicate whether any given assemblage has hit the magic angle.Generally when were making these devices, we dont know what angle this twisted bilayer graphene comes out at until we put it into our microscope, explains Dillon Wong, a co-author of the study and formerly a researcher at Princeton. And most of the time its at a completely wrong angle, and were just disappointed.As expected, things went wrong this time around, tooa first look showed that the graphene was undershooting the intended angle of 1.1 degrees. But because this particular graphene bilayer was closer to another known but slightly smaller magic angle, Nuckolls decided to keep imaging it anyway with the STM.This plot shows how the energies (vertical axis) of electrons change as a function of magnetic field (horizontal axis), and cluster into separated Hofstadter electronic bands (multi-colored shaded regions). Nuckolls and his colleagues were able to identify how the energy levels of electrons were self-repeating on different scales, as was predicted would be the case in the formation of "Hofstadter's butterfly," a type of quantum fractal.Yazdani groupThe first images werent that impressive, Nuckolls admits, but once the researchers zoomed out to see a fuller picture of the system, they became more intrigued. They only realized several days later, however, that the sandwiched electrons appeared to be fulfilling Hofstadters half-century-old prediction. Their delay isnt so surprising, given that they werent looking for the pattern in the first placeand that it only became apparent through careful tracking of the electrons collective behavior.The idea behind Hofstadters butterfly is that youre looking at how the [band structure of electrons] moves when you have the magnetic field on one axis and the electrons energies on the other, and plotted on that diagram, the band forms a fractal structure that looks like a butterfly, explains Myungchul Oh, study co-author and now a physics professor at Pohang University of Science and Technology in South Korea. Past experiments were indirect, Oh says, in the sense that they werent looking at the actual energy transformations but rather proxy measurements, such as the spatial distributions of electrons.Once Nuckolls, Wong and Oh decided this particular system was worth deeper scrutiny, they tasked Michael Scheer, a graduate student in theoretical physics at Princeton, to come up with more robust models of the notional interactions at work to better understand exactly what was occurring and how.Hofstadters butterfly is kind of like a fingerprint, Scheer says. Its really detailed, informational and very sensitive to the model that you have and, on the other hand, to the material that youre measuring and its physical parameters. That interplay between theory and experiment can reveal an enormous amount of information that researchers can use to learn about the materials properties, Nuckolls adds. In other words, studying Hofstadters butterfly in twisted graphene bilayers could be of broader utility, opening the way for further enlightening investigations of the phenomenon in other systems and materials.One of the biggest merits of this work is that its ... really managed to go in a very special regime of parameters to be able to see new physics, says Cristiane Morais Smith, a condensed matter physicist at Utrecht University in the Netherlands, who wasnt involved in the new work. What was very special was that [they] could go to a situation in which these small magnetic fields [like those of an STM] were enough to probe what [they] wanted to probe, she says, which should allow other groups to easily replicate and elaborate on the experiment.Hofstadter, now age 80, politely declined Scientific Americans request for comment about the new result, noting that he had only rarely revisited his prediction ever since making it about half a century ago and would be unlikely to properly comprehend the paperwhich, he added, he had no plans to read. Over the years I have seen many claims of experimental replication of the [predicted] recursiveness, he says. But they are all extremely coarse-grained, and none of them has come close to detecting a genuine recursively nested structure. That will perhaps happen in another few decadesif humanity still exists at that point.Even so, the new work takes humanity at least several footsteps (or wing flaps?) toward realizing Hofstadters predictions. This initial result is ripe for follow-up studies, Oh says, such as examining whether Hofstadters butterfly will still take flight in graphene sandwiches subjected to far stronger magnetic fields. Id love to see how and whether the Hofstadter pattern would be emulated on higher-scale magnetic fields, he says.Theres something very satisfying about us working on this problem 50 years after Hofstadters calculation, Nuckolls says. In Hofstadters original paper, he basically concludes that what Ive calculated and predicted is really awesome, but no ones ever going to see it because the necessary magnetic fields are never going to be achieved. Yet 25 years after that, researchers started seeing the first evidence supporting his calculations. Now our work is able to probe exactly what he had predicted.
