Electrons whizzing through a grid-like grid don’t behave at all like pretty silver balls in a pinball machine. They blur and bend in collective dances, following the whims of an undulating reality difficult enough to imagine, let alone calculate.

Yet scientists have succeeded in doing just that, capturing the motion of electrons moving around a square lattice in simulations that previously required hundreds of thousands of individual equations to generate.

By using artificial intelligence (AI) to reduce this task to just four equations, physicists have made their task of probing the emergent properties of complex quantum materials much more manageable.

In this way, this computing power could help tackle one of the most difficult problems in quantum physics, the “many-electron” problem, which attempts to describe systems containing large numbers of interacting electrons.

It could also advance a truly legendary tool for predicting electron behavior in solid-state materials, the Hubbard model – while improving our understanding of how practical phases of matter like superconductivity occur.

Superconductivity is a strange phenomenon that occurs when a stream of electrons flows freely through a material, losing almost no energy as they slide from one point to another. Unfortunately, most practical means of creating such a condition rely on insanely low temperatures, if not ridiculously high pressures. Harnessing superconductivity closer to room temperature could lead to far more efficient power grids and devices.

As achieving superconductivity under more reasonable conditions remains a lofty goal, physicists use models to predict how electrons might behave under different circumstances and which materials are therefore suitable conductors or insulators.

These models have their work cut out for them. After all, electrons do not roll through the atomic network like small balls with clearly defined positions and trajectories. Their activity is a jumble of probabilities, influenced not only by their environment but also by their history of interactions with other electrons they encounter along the way.

When electrons interact, their fates can become intimately intertwined, or “tangled.” Simulating the behavior of an electron means tracking the range of possibilities of all electrons in a model system simultaneously, making the computational challenge exponentially more difficult.

The Hubbard model is a decades-old mathematical model that fairly accurately describes the puzzling movement of electrons through an atomic lattice. Over the years, and much to the delight of physicists, the deceptively simple model has been experimentally realized in the behavior of a variety of complex materials.

As computing power has increased, researchers have developed numerical simulations based on Hubbard model physics that allow them to get a handle on the role of the topology of the underlying lattice.

For example, in 2019 researchers proved the Hubble model’s ability to represent superconducting temperatures higher than ultracold and gave researchers the green light to use the model for deeper insights into the area.

This new study could be another big leap, greatly simplifying the number of equations required. Researchers developed a machine learning algorithm to refine a mathematical contraption called a renormalization group, which physicists use to study changes in a material system when properties such as temperature are changed.

“It’s essentially a machine that can discover hidden patterns,” says physicist and lead author Domenico Di Sante of the University of Bologna in Italy of the program the team has developed.

“We start with this giant object made up of all these differential equations coupled together,” each representing pairs of entangled electrons, “then we use machine learning to turn it into something so small you can count on your fingers,” Di Sante says about their approach.

The researchers demonstrated that their data-driven algorithm could efficiently learn and recapitulate the dynamics of the Hubbard model using just a handful of equations – four, to be precise – and without sacrificing accuracy.

“When we saw the result, we said, ‘Wow, that’s more than we expected.’ We were able to really capture the relevant physics,” says Di Sante.

Training the machine learning program on data took weeks, but Di Sante and colleagues say it could now be adapted to work on other, tantalizing condensed-matter problems.

The simulations so far only capture a relatively small number of variables in the lattice network, but the researchers expect their method should be fairly scalable to other systems.

If so, it could be used in the future to study the suitability of conducting materials for applications involving clean power generation, or to help design materials that could one day deliver that elusive room-temperature superconductivity .

The real test, the researchers say, will be how well the approach works on more complex quantum systems, such as materials, where electrons interact over long distances.

For now, the work demonstrates the possibility of using AI to extract compact representations of dynamic electrons, “a goal of paramount importance for the success of state-of-the-art quantum field theoretical methods to deal with the many-electron problem,” the researchers conclude in their abstract.

The study was published in *Physical Verification Letters.*

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