Scientists at Caltech and Yale University have developed a way to precisely calculate the Kondo effect in specific real materials, something that had not previously been possible. For decades, researchers have largely depended on simplified models that capture the phenomenon only approximately. The new approach instead works directly from a material’s true atomic and electronic structure.
The advance could help pave the way for realistic computer simulations of more complicated quantum materials, including high-temperature superconductors. In these materials, the behavior of any one electron is strongly influenced by what nearby electrons are doing, making the system difficult to describe with conventional approximations.
The researchers report their method and findings in a paper published in Science. The lead authors are Linqing Peng (PhD ’25) and Tianyu Zhu of Yale University. Both Peng and Zhu began working on the project in the laboratory of Garnet Chan, Bren Professor of Chemistry and director of the Rudolph A. Marcus Center for Theoretical Chemistry at Caltech.
“It is now possible to predict the properties of some complicated materials purely through computation without referring to experiment,” says Chan, who is the senior author of the paper and a Simons Investigator in Physics. “These first materials that we have studied are like a baby step, or a prototype problem, along the way to more complex phenomena such as high-temperature superconductors and quantum magnets.”
The Kondo Effect — A Classic Many-Body Challenge
In many materials used in modern electronics, including semiconductors such as silicon, interactions among electrons are weak enough that they can often be ignored when describing the material’s overall behavior. That is not true for strongly correlated materials, which are important for many proposed quantum technologies. In those systems, understanding how electrons influence and scatter from one another is essential.
The Kondo effect appears in one of the simplest examples of a strongly correlated system. It occurs when a single magnetic atom, such as iron or manganese, is placed as an impurity inside a metal such as copper. When the material is cooled below a particular temperature (known as the Kondo temperature), its electrical behavior changes in an unusual way.
Ordinarily, cooling a metal causes its electrical resistance to fall steadily, allowing current to flow more easily. But in a metal containing a magnetic impurity, resistance stops falling when the Kondo temperature is reached. It reaches a minimum and then begins to rise again as the temperature continues to drop.
“That is the signature of the Kondo effect, and it’s a property of the electrons in the impurity interacting with the electrons traveling through the bulk metal,” Chan explains.
Physicists developed the general theoretical picture of the Kondo effect in the 1970s (including, importantly, alumnus Kenneth Wilson, PhD ’61). The phenomenon became a classic example of a many-body problem because it requires describing a huge number of interacting particles — in this case, electrons in the metal — whose behavior cannot be understood simply by treating them one at a time.
The magnetic impurity contains unpaired electrons whose spin gives the atom its magnetism. At higher temperatures, the direction of that magnetism, known as the atom’s magnetic moment, can fluctuate freely. As the material cools, however, electrons moving through the surrounding metal begin interacting strongly with the impurity’s spin.
Those electrons can flip their own spins in ways that partially cancel the magnetic moment of the embedded atom. The added scattering produced by these interactions is responsible for the characteristic flattening and eventual increase in electrical resistance. As more and more electrons participate, they collectively surround the impurity and ultimately conceal its magnetism.
In effect, the surrounding electrons form a cloud that “screens,” or cancels out, the magnetism of the atom.
Moving Beyond Simplified Quantum Models
The Kondo effect has been studied extensively because, although its underlying physics is complex, the basic problem is relatively straightforward to describe. It has therefore become an important benchmark for testing new theoretical and computational techniques.
Until now, however, researchers have not been able to reliably calculate exactly how the resistance of a real material falls and then rises again, or determine the precise temperature at which that change occurs for a specific impurity.
Traditional approaches simplify the electronic structure of a material by reducing it to a small set of orbitals — spaces around an atom’s nucleus where electrons are most likely to be found — and then using an approximate mathematical model to describe the smaller system.
Chan and his colleagues chose a different strategy. They adapted highly accurate computational tools originally developed in quantum chemistry for describing molecules and applied them to quantum materials. This allowed the researchers to represent the magnetic impurities almost as if they were molecules, while retaining the full complexity of their electronic interactions.
Much More Accurate Predictions for Real Materials
The researchers tested their method on seven different transition-metal atoms embedded in copper. For most of the elements they examined, the calculations were as much as two orders of magnitude more accurate than predictions obtained from conventional model-based techniques.
“We are in an exciting era in which faithful predictive quantum descriptions of the full chemical complexity of real materials are coming within reach,” says Peng. “It is becoming realistic to predict material-specific behavior of correlated electrons from first principles, even in some of the most challenging classes of quantum materials.
“This is an important step toward computationally designing materials whose functions emerge from intricate correlated physics, such as high-temperature superconductivity, where the large chemical space and competition among many phases call for predictive theory to help focus the experimental search for new materials,” she adds. “I am excited to see what new materials breakthroughs this theory will enable in the future.”
Additional Caltech authors of the paper, “Toward an exact quantum many-body treatment of Kondo correlation in magnetic impurities,” are Huanchen Zhai, a former postdoctoral scholar; Runze Chi, a current postdoctoral scholar; and Zhi-Hao Cui (PhD ’23), who completed the work as a graduate student.
The research received support from the Air Force Office of Scientific Research through the Multidisciplinary University Research Initiative program, the US Department of Energy and its Center for Molecular Magnetic Quantum Materials, and the US National Science Foundation.
