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Hofstadter Butterfly

Observing quantum physics with Big Red 2 and 3

At Indiana University, physicists are taking a closer look at how electrons behave, with help from hundreds of millions of calculations powered by HPC clusters Big Red 2 and 3. Their work aims to classify the types of electrons present in a magnetic field. 

The experiments expand upon recent discoveries that proved the existence of the Hofstadter Butterfly. First described in 1976 by Douglas Hofstadter (who was later an IU professor), the plot -- one of the earliest examples of computer graphics -- described the spectral properties of electrons moving in a crystal lattice under the influence of a magnetic field.

Daniel Ariad, Johns Hopkins University
Daniel Ariad, Johns Hopkins University

Once a theoretical concept, the Hofstadter Butterfly was quantifiably proven through experiments using graphene in 2013. The observable effect of electrons moving through magnetic fields also reveals a series of quantized plateaus in Hall resistance, in the perpendicular direction to the flow of electrons. In order to quantify these plateaus, known as the Quantum Hall effect, the team runs calculations using Big Red 2 and 3. “It’s a classification problem,” said Daniel Ariad. “Knowing this number not only tells you the orbiting radius of the electrons but it also tells you many other properties of the electrons, like the different classes of the electron's orbits. So now you can classify the electrons,” he said.

Babak Seradjeh, Indiana University
Babak Seradjeh, Indiana University

The Hofstadter Butterfly assumes that electrons are moving through a perfect network of graphene. Ariad and his team use calculations that assume flaws in the system as a way to define the variable orbiting radius of electrons. Each point on their version of a quantized plateau is made up of hundreds of millions of data points, making the team one of Big Red’s top users for 2019. “In these types of classifications, the discovery is made possible by supercomputers like Big Red 3. The intricacies of what is happening in a physical system are too great for us to have any equation we can solve on paper. We need to use the computers to solve and find these values,” said Babak Seradjeh, a collaborator on the project.

Herbert Fertig, Indiana University
Herbert Fertig, Indiana University

Like many research projects involving supercomputing, this one is interdisciplinary. “It brings many disciplines of physics together. There are classifications, phases of matter, phase transitions, metrology, quantum theory, computational aspects,” said Herbert Fertig. “Because the plateaus are very flat, the effect used to define the resistance standard, by the National Institute of standards in technology,” said Fertig. “Having something so accurately and precisely quantized, that effect is used to say what one Ohm of resistance is, through the height of the plateaus,” he said. The quantized plateau also informs science’s understanding of measurement. Together with another quantum phenomenon in superconductors called the Josephson effect, the quantized Hall plateaux are used to measure the value of Planck's constant with great accuracy. As of 2019, the Planck’s constant has been fixed to define the exact weight of a kilogram.

Similar to how computers enabled the discovery of the Hofstadter butterfly, the team using Big Red hopes to uncover new patterns in the electron's dance under the magnetic field.