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Caltech researchers have discovered a rapid and effective method to total extensive numbers of Feynman diagrams, the basic illustrations physicists employ to depict particle interactions. This innovative technique has already allowed the scientists to resolve a persistent issue in the realms of materials science and physics known as the polaron issue, providing scientists and engineers with a means to forecast electron flow in specific materials, whether traditional or quantum.

In the 1940s, physicist Richard Feynman initially introduced a technique to illustrate the diverse interactions occurring between electrons, photons, and other fundamental particles through 2D illustrations featuring straight and wavy lines converging at vertices. Although they may seem straightforward, these Feynman diagrams enable researchers to calculate the likelihood that a specific collision, or scattering, will occur between particles.

Since particles can interact in numerous ways, a plethora of diagrams is necessary to portray every potential interaction. Each diagram corresponds to a mathematical expression. Thus, by aggregating all possible diagrams, scientists can derive quantitative figures linked to specific interactions and scattering probabilities.

“Accurate summation of all Feynman diagrams is a coveted achievement in theoretical physics,” remarks Marco Bernardi, professor of applied physics, physics, and materials science at Caltech. “We have tackled the polaron issue by summing all diagrams related to the so-called electron-phonon interaction, virtually up to an infinite order.”

In a publication featured in Nature Physics, the Caltech team employs its new method to accurately calculate the strength of electron-phonon interactions and experimentally predict related effects quantitatively. The principal author of the study is graduate student Yao Luo, a member of Bernardi’s team.

For certain materials, such as basic metals, the electrons that move within the crystalline structure will interact only minimally with its atomic vibrations. For these materials, scientists can utilize a technique known as perturbation theory to describe the interactions occurring between electrons and phonons, which can be perceived as “units” of atomic vibration. Perturbation theory serves as a good approximation in these contexts since each subsequent order or interaction becomes less significant. Therefore, computing just one or a few Feynman diagrams—a task that can be accomplished routinely—suffices to obtain precise electron-phonon interactions in these materials.

Introducing Polarons

However, for many other materials, electrons interact much more intensely with the atomic lattice, resulting in entangled electron-phonon states referred to as polarons. Polarons consist of electrons accompanied by the lattice distortion that they create. They are found in various materials, including insulators, semiconductors, electronics, energy devices, and numerous quantum materials. For instance, an electron placed within a material exhibiting ionic bonds will deform the surrounding lattice and generate a localized polaron state, leading to diminished mobility due to the robust electron-phonon interaction. Researchers can analyze these polaron states by measuring the conductivity of electrons or how they alter the atomic lattice surrounding them.

Perturbation theory proves ineffective for these materials because each subsequent order is of greater significance than the previous one. “It’s essentially a nightmare from a scaling perspective,” states Bernardi. “If you can compute the lowest order, it’s very likely that the second order becomes unattainable, and the third order is simply impossible. The computational demand typically scales excessively with interaction order. There are too many diagrams to evaluate, and the higher-order diagrams are prohibitively expensive to compute.”

Summing Feynman Diagrams

Researchers have been on the lookout for a method to aggregate all the Feynman diagrams that outline the myriad ways in which electrons in such materials can engage with atomic vibrations. So far, these calculations have largely relied on methods that allow researchers to fine-tune certain parameters to fit experimental data. “But when you do that, you cannot be sure whether you’ve genuinely comprehended the mechanism,” emphasizes Bernardi. Instead, his team concentrates on tackling problems from “first principles,” which entails starting with nothing more than the spatial arrangements of atoms within a material and utilizing the equations of quantum mechanics.

When contemplating the magnitude of this issue, Luo suggests envisioning the challenge of predicting how the stock market might behave the following day. To undertake this, one would need to consider every interaction between every trader over a specific duration to achieve accurate predictions of the market’s dynamics. Luo aims to grasp all interactions between electrons and phonons within materials where phonons strongly interact with the atoms. However, akin to forecasting the stock market, the number of potential interactions is overwhelmingly vast. “It is virtually impossible to compute directly,” he states. “The only feasible approach is to employ a clever sampling method across all these scattering processes.”

Betting on Monte Carlo

Caltech scientists are tackling this challenge by implementing a technique known as diagrammatic Monte Carlo (DMC), in which an algorithm randomly samples locations within the entirety of Feynman diagrams for a system, guided by some of the most critical areas to sample. “We establish certain rules to traverse effectively with high agility within the realm of Feynman diagrams,” elucidates Bernardi.

The Caltech group overcame the substantial computational load typically associated with employing DMC to investigate real materials using first-principle methodologies by depending on a technique they unveiled last year which compresses the matrices representing electron-phonon interactions. Another significant advancement is the near-elimination of the so-called “sign problem” in electron-phonon DMC through an ingenious technique that interprets diagrams as products of tensors, mathematical constructs expressed as multi-dimensional matrices.

“The innovative diagram sampling, removal of the sign problem, and electron-phonon matrix compression are the three essential elements that have facilitated this transformative shift in tackling the polaron problem,” asserts Bernardi.

In the newly published paper, the researchers have utilized DMC calculations in various systems that contain polarons, including lithium fluoride, titanium dioxide, and strontium titanate. The scientists indicate that their findings pave the way for a broad spectrum of predictions that are pertinent to experiments currently being conducted on both conventional and quantum materials—including electrical transport, spectroscopy, superconductivity, and other characteristics in materials with substantial electron-phonon coupling.

“We have effectively characterized polarons in materials using DMC, but the methodology we’ve developed could also aid in investigating strong interactions between light and matter, or even furnish a framework for efficiently summing Feynman diagrams in entirely distinct physical theories,” concludes Bernardi.

The publication is entitled, “First principles diagrammatic Monte Carlo for electron-phonon interactions and polaron.” In addition to Bernardi and Luo, Jinsoo Park (MS ’20, PhD ’22), currently a visiting associate in applied physics and materials science at Caltech and a postdoctoral research associate at the University of Chicago, is also listed as an author. This research was funded 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 supported by an Eddleman Graduate Fellowship. Calculations concerning transport and polarons in oxides received support from the Air Force Office of Scientific Research and Clarkson Aerospace Corp.

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