Finding computationally effective and reliable ways to model interacting electron systems for crystalline materials is one of the pressing problems in condensed matter physics.
Researchers have found a quick but very precise way to achieve this in a new study. Physical Review B has published the research headed by Zheting Jin, a doctoral student in Applied Physics at Yale, and Sohrab Ismail-Beigi, who served as his thesis advisor.
Researchers in the areas have long been interested in developing techniques to precisely describe interacting quantum electrons because it can offer insightful information about numerous significant features of materials. However, it is challenging to describe the electrons at this level for a number of reasons. One is that they travel wavy because they are quantum mechanical, making it harder to track them. The other is how they communicate with one another.
According to Ismail-Beigi, Strathcona Professor of Applied Physics, Physics, and Mechanical Engineering & Materials Science, each element of this issue is “OK to deal with separately.” However, when waviness and interactions are present, the issue becomes so complicated that no one is able to effectively resolve it.
Similar to many challenging physics and math issues, the issue might theoretically be solved numerically by applying brute force, but the processing and storage requirements would be exponential in the number of electrons. with instance, the size of the computer required grows by a factor of two (usually even a greater number) with every extra electron added to the system. This means that even with the biggest supercomputers available today, researching a system with only roughly 50 electrons is impractical. A tiny nanoparticle possesses more than 1,000 electrons, whereas an individual iodine atom has 53.
“On the one hand, the electrons want to move around—that’s to take advantage of the kinetic energy,” Ismail-Beigi said. “On the other, they repel each other—’don’t come next to me if I’m here already.’ Both effects are captured in the well-known Hubbard model for interacting electrons. Basically, it has these two key ingredients, and it’s a very hard problem to solve. No one knows how to solve it exactly, and high-quality approximate and efficient solutions are not easy to come by.”
The Ismail-Beigi team has created a technique that belongs to a group of methods that make use of an auxiliary or subsidiary boson. As they work with one atom at a time, these methods typically use much less computational power but are only moderately accurate. The team of Ismail-Beigi tried a different strategy. Instead of focusing on one atom at a time, the researchers treat two or three bound atoms (referred to as a cluster) simultaneously.
“Electrons can hop between the atoms in the cluster: we solve the cluster problem directly, and then we connect the clusters together in a novel way to describe the entire system,” Ismail-Beigi said. “In principle, the larger the cluster, the more accurate the approach, so the question is how large a cluster does one need to get a desired accuracy?”
Cluster techniques have been tried before by researchers, however the computational costs were too high and the accuracy was lacking given the additional computational cost.
“Zheting and I found a clever way of matching different clusters together so that the quantities calculated between the different clusters agree across their boundaries,” he said. “The good news is that this method then gives a very highly accurate description with even a relatively small cluster of three atoms. Because of the smooth way one glues the clusters together, one describes the long-range motion of the electrons well in addition to the localized interactions with each other. Going into this project, we didn’t expect it to be this accurate.”
Computations using the new method are three to four orders of magnitude quicker than computations using published benchmarks.
“All the calculations in the paper were run on Zheting’s student laptop, and each one completes within a few minutes,” Ismail-Beigi said. “Whereas for the corresponding benchmark calculations, we have to run them on a computer cluster, and that takes a few days.”
The researchers expressed excitement about using this approach in the near future to solve increasingly challenging and real-world material challenges.
