Annual Report 2020

A Collaborative Paradigm for Cracking the Code of Quantum Systems

Center for Computational Quantum Physics

Every object derives its properties from its electrons, which interact in a scaffold of atoms and whose motion is choreographed by the rules of quantum mechanics. Breakthroughs in areas from computing to energy storage rely in part on understanding these interactions well enough to create materials tailored to specific requirements.

Tracking quantum interactions in real materials is comically complex — a single gram of hydrogen, for example, contains about 1023 atoms. Equations with that many variables are intractable to even the most powerful supercomputers. So physicists must come up with ways to analyze the essential physics of these systems without considering every aspect of the motion of the underlying particles, predicting how electrons will dance about their atoms as circumstances change. From these predictions, physicists can deduce, for example, how a material’s electrical properties change under pressure. The trouble is that there are many prediction methods, and each approaches the problem differently and communicates the results in its own mathematical dialect. 

A visual representation of where electrons are most likely to be found around a chain of hydrogen atoms. Brighter colors denote higher probabilities, and dashed lines represent contours of constant probability. At this spacing between atoms, electrons try to link pairs of adjacent atoms to form dihydrogen molecules. Because the protons are fixed in place, these molecules can’t form. Instead, each electron ‘leans’ toward a neighboring atom. Click the plus button to toggle additional elements in this image. Credit: M. Motta et al./Physical Review X 2020

One method provides snapshots of electrons as they move about. Another calculates the probability of finding electrons in particular configurations. Some methods predict behavior well at high temperatures, some at low temperatures. Some are best suited to describing the evolution of materials over time, while others are better for a material that’s sitting still.

Stitching together, from all these techniques, a cohesive picture of how even one material behaves is a conceptual and logistical challenge. “We were sometimes getting consistent results, but more often than not, we were not getting consistent results, or we were computing things that can’t be directly compared,” says Antoine Georges, director of the Flatiron Institute’s Center for Computational Quantum Physics (CCQ). 

Georges and colleagues at the CCQ are now changing that with a “multimethod, multimessenger” approach: They take a simple mathematical model of a material and throw every computational method they have at it. By marrying these techniques and resolving their differences, CCQ researchers aim to kick-start a new era of materials design and take on grand challenges such as developing practical superconductors, which conduct electricity with zero resistance at reasonable temperatures.

But getting there requires a culture change, one that the CCQ is leading: Physicists must step out of the silos in which they work and join forces to surmount an overarching hurdle. The payoff is that “you get much more information when you combine different methods with different potential sources of error or systematic biases,” says CCQ co-director Andrew Millis.

To that end, CCQ researchers turned to one of the simplest models around: an infinitely long chain of hydrogen atoms. An endless queue of hydrogen may not seem realistic, but it’s a perfect theoretical playground for getting all these computational techniques to play well together. “We wanted a system that all the methods can actually handle that also brings out all the complexities of these problems,” says CCQ research scientist Miles Stoudenmire. 

The scenario goes like this: Line up simulated hydrogen atoms and space them a few tenths of a nanometer apart — just a few times the width of the atoms themselves — and then slowly decrease the distance. As the atoms bunch up, let a crew of computational techniques figure out how the electrons respond and how that response affects the lineup’s overall behavior.

A particle wave passes through a crystal structure containing electrons (arrows). The structure has antiferromagnetic order because its electrons have alternating up and down spins. Credit: Lucy Reading-Ikkanda/Simons Foundation

Even in this ‘simple’ situation, a diversity of behavior emerged. The hydrogen chain went through three phases: It started as an antiferromagnet, a state in which the intrinsic magnetic orientation of the electrons alternated direction. As the atoms crowded together, the electrons started spending more time between neighboring pairs of atoms —  a state reflective of hydrogen molecules wanting to form. As the spacing shrank further, the whole chain transitioned from being an electrical insulator to a metal.

“As we bring the atoms closer together, the whole electronic structure changes,” Millis says. Such insulator-to-metal transitions are intriguing for various applications, and it took many methods to reveal the underlying mechanism.

The lessons learned from this model system can be applied to other, more practical lineups. “Hydrogen is particularly squirrelly; we can’t really build this in the lab,” Stoudenmire says. “But it’s not pie in the sky. It’s close to solving other chainlike molecules, like a chain of DNA.”

Other well-established physics models have also been getting the multimethod, multimessenger treatment. The Hubbard model is a simple mathematical prescription for how electrons interact in a 2D array of atoms. But fleshing out its physics has remained a challenge. 

“The Hubbard model has been the Mount Everest of our field,” Millis says. “It’s beautiful, it’s impressive … it abstracts away all of the actual complexity of solids while leaving all the difficulty of quantum mechanics.”

Here again, CCQ researchers saw an opportunity to set a plethora of computational techniques working cooperatively on the problem. Specifically, they wanted to see how electrons in the Hubbard model responded to plummeting temperature. 

Credit: Lucy Reading-Ikkanda/Simons Foundation

As the temperature dropped, the material went through three distinct phases. At high temperatures, it was a soup of electrons dancing every which way. As the temperature decreased, the system became a metal, with electrons moving in a more orderly fashion. And as the temperature continued dropping toward absolute zero, it transitioned to an antiferromagnetic insulator.

Characterizing the various phases of the Hubbard model is a major achievement, setting the stage for further investigations of superconductivity.

“All these studies are guiding us toward how we should develop further the available methods and what the next generation of methods might be,” Georges says. “The ultimate goal is to put these methods to good work in applications like design of interesting electronic materials.” The work also has connections beyond materials: Some of the models share mathematical DNA with theories of quantum gravity.

“Our vision for CCQ has been to build a place where we have several senior and junior faculty who are experts at certain types of methods,” says Georges. “Put all these people under the same roof and have them interact and collaborate and invent the methods of tomorrow through this sort of synergetic effort.”