Adding Up Feynman Diagrams to Make Predictions about Real Materials

Caltech scientists have found a fast and efficient way to add up large numbers of Feynman diagrams, the simple drawings physicists use to represent particle interactions. The new method has already enabled the researchers to solve a longstanding problem in the materials science and physics worlds known as the polaron problem, giving scientists and engineers a way to predict how electrons will flow in certain materials, both conventional and quantum.

In the 1940s, physicist Richard Feynman first proposed a way to represent the various interactions that take place between electrons, photons, and other fundamental particles using 2D drawings that involve straight and wavy lines intersecting at vertices. Though they look simple, these Feynman diagrams allow scientists to calculate the probability that a particular collision, or scattering, will take place between particles.

Since particles can interact in many ways, many different diagrams are needed to depict every possible interaction. And each diagram represents a mathematical expression. Therefore, by summing all the possible diagrams, scientists can arrive at quantitative values related to particular interactions and scattering probabilities.

"Summing all Feynman diagrams with quantitative accuracy is a holy grail in theoretical physics," says Marco Bernardi, professor of applied physics, physics, and materials science at Caltech. "We have attacked the polaron problem by adding up all the diagrams for the so-called electron-phonon interaction, essentially up to an infinite order."

In a paper published in Nature Physics, the Caltech team uses its new method to precisely compute the strength of electron-phonon interactions and to predict associated effects quantitatively. The lead author of the paper is graduate student Yao Luo, a member of Bernardi's group.

For some materials, such as simple metals, the electrons moving inside the crystal structure will interact only weakly with its atomic vibrations. For such materials, scientists can use a method called perturbation theory to describe the interactions that occur between electrons and phonons, which can be thought of as "units" of atomic vibration. Perturbation theory is a good approximation in these systems because each successive order or interaction becomes decreasingly important. That means that computing only one or a few Feynman diagrams – a calculation that can be done routinely – is sufficient to obtain accurate electron-phonon interactions in these materials.

Introducing Polarons

But for many other materials, electrons interact much more strongly with the atomic lattice, forming entangled electron-phonon states known as polarons. Polarons are electrons accompanied by the lattice distortion they induce. They form in a wide range of materials including insulators, semiconductors, materials used in electronics or energy devices, as well as many quantum materials. For example, an electron placed in a material with ionic bonds will distort the surrounding lattice and form a localized polaron state, resulting in decreased mobility due to the strong electron-phonon interaction. Scientists can study these polaron states by measuring how conductive the electrons are or how they distort the atomic lattice around them.

Perturbation theory does not work for these materials because each successive order is more important than the last. "It's basically a nightmare in terms of scaling," says Bernardi. "If you can calculate the lowest order, it's very likely that you cannot do the second order, and the third order will just be impossible. The computational cost typically scales prohibitively with interaction order. There are too many diagrams to compute, and the higher-order diagrams are too computationally expensive."

Summing Feynman Diagrams

Scientists have searched for a way to add up all the Feynman diagrams that describe the many, many ways that the electrons in such a material can interact with atomic vibrations. Thus far such calculations have been dominated by methods where scientists can tune certain parameters to match an experiment. "But when you do that, you don't know whether you've actually understood the mechanism or not," says Bernardi. Instead, his group focuses on solving problems from "first principles," meaning beginning with nothing more than the positions of atoms within a material and using the equations of quantum mechanics.

When thinking about the scope of this problem, Luo says to imagine trying to predict how the stock market might behave tomorrow. To attempt this, one would need to consider every interaction between every trader over some period to get precise predictions of the market's dynamics. Luo wants to understand all the interactions between electrons and phonons in a material where the phonons interact strongly with the atoms in the material. But as with predicting the stock market, the number of possible interactions is prohibitively large. "It is actually impossible to calculate directly," he says. "The only thing we can do is use a smart way of sampling all these scattering processes."

Betting on Monte Carlo

Caltech researchers are addressing this problem by applying a technique called diagrammatic Monte Carlo (DMC), in which an algorithm randomly samples spots within the space of all Feynman diagrams for a system, but with some guidance in terms of the most important places to sample. "We set up some rules to move effectively, with high agility, within the space of Feynman diagrams," explains Bernardi.

The Caltech team overcame the enormous amount of computing that would have normally been required to use DMC to study real materials with first principle methods by relying on a technique they reported last year that compresses the matrices that represent electron-phonon interactions. Another major advance is nearly removing the so-called "sign problem" in electron-phonon DMC using a clever technique that views diagrams as products of tensors, mathematical objects expressed as multi-dimensional matrices.

"The clever diagram sampling, sign-problem removal, and electron-phonon matrix compression are the three key pieces of the puzzle that have enabled this paradigm shift in the polaron problem," says Bernardi.

In the new paper, the researchers have applied DMC calculations in diverse systems that contain polarons, including lithium fluoride, titanium dioxide, and strontium titanate. The scientists say their work opens up a wide range of predictions that are relevant to experiments that people are conducting on both conventional and quantum materials—including electrical transport, spectroscopy, superconductivity, and other properties in materials that have strong electron-phonon coupling.

"We have successfully described polarons in materials using DMC, but the method we developed could also help study strong interactions between light and matter, or even provide the blueprint to efficiently add up Feynman diagrams in entirely different physical theories," says Bernardi.

The paper is titled, "First principles diagrammatic Monte Carlo for electron-phonon interactions and polaron." Along with Bernardi and Luo, Jinsoo Park (MS '20, PhD '22), now a visiting associate in applied physics and materials science at Caltech and a postdoctoral research scholar at the University of Chicago, is also an author. The work was supported by the U.S. Department of Energy's Scientific Discovery through Advanced Computing program, the National Science Foundation, and the National Energy Research Scientific Computing Center, a U.S. Department of Energy Office of Science User Facility. Luo was partially funded by an Eddleman Graduate Fellowship. Calculations of transport and polarons in oxides were supported by the Air Force Office of Scientific Research and Clarkson Aerospace Corp.