Humans have been aware of the strange phenomenon of magnetism for over 2,000 years. From ancient Greece through modern times, researchers have steadily improved upon humanity’s fundamental understanding of magnets. For over 100 years, magnetism has been known to emerge in solid-state materials when, due to electronic and chemical interactions, the electronic spins (a quantum mechanical property) and their motion around atoms develop a fixed orientation within the material. Ever since this discovery, physicists, chemists, and materials scientists have developed extensive theoretical and experimental machinery to predict and characterize magnetic materials.

Despite an intense effort comprising multiple competing theories (and several Nobel prizes), a unified description of magnetic structures within materials has remained surprisingly elusive. In fact, even the most successful classification system for magnetic materials, developed almost 75 years ago by the Soviet scientist Lev Shubnikov, was incomplete, until now.

An international team of researchers announced this week that it has finally completed the mathematical characterization of Shubnikov’s magnetic and nonmagnetic crystal symmetry groups. The work is the collaborative effort of scientists at the Massachusetts Institute of Technology (MIT); Princeton University; the University of the Basque Country in Bilbao, Spain; Northeastern University; the Max Planck Institute of Microstructure Physics in Halle, Germany; and the University of Illinois Urbana-Champaign.

The team’s results were published Oct. 13, 2021, in *Nature Communications* in the article “Magnetic topological quantum chemistry.”

**A long road from there to here**

One early description of magnetism that gained traction with many researchers was representation theory, which provided a simplified picture in which much of the underlying material structure is ignored, and the magnetism is described through repeating electronic spin waves partially decoupled from the rest of the material. Since the 1950s, the limitations of representation theory have been apparent. In particular, the theory breaks down when even the simplest realistic interactions between electron spins and the underlying atoms are taken into consideration.

In classifying materials by their geometry, Shubnikov, on the other hand, considered all of the complicated crystal symmetries, and then considered the even more complicated ways in which those symmetries can be reduced by magnetic ordering. Shubnikov’s system allows all possible crystals — magnetic or otherwise — to be classified by one of a mere 1,651 collections of symmetries known as the magnetic and nonmagnetic space groups (SGs).

For 230 of Shubnikov’s SGs, the complete mathematical properties — known as the “small corepresentations” (coreps) — have been known for over 50 years. But for the magnetic SGs, the coreps have remained largely unidentified and inaccessible, because of the complicated symmetries of magnetic crystals and the sheer number of magnetic SGs.

In the current study, the team painstakingly derived the over 100,000 small coreps of the MSGs through several redundant calculations to ensure internal consistency.

**Open-access database**

Based on the team’s findings, Luis Elcoro, a professor at the University of the Basque Country and one of the lead authors on the study, wrote computer code to generate an extensive set of publicly available resources on the Bilbao Crystallographic Server, granting researchers around the globe access to the team’s resulting data.

Elcoro comments, “In the crystallography and magnetic structure communities, we have been awaiting an accessible and complete guide to the magnetic coreps since before I was born. We can now robustly characterize all possible magnetic phase transitions in experimental studies of magnetic materials — typically done by neutron diffraction experiments — without falling back on the incomplete representation-theory method.”

**Quantum applications**

Recognizing a mathematical connection between the magnetic coreps and the electronic structure of solid-state materials, the team next performed additional calculations to link the resulting magnetic symmetry data to topological band insulators and semimetals — exotic electronic states having tantalizingly intricate mathematical descriptions. These states hold promise for quantum applications, for example, as platforms for engineering quantum information and quantum spintronic devices.

Benjamin Wieder, a postdoctoral researcher at MIT and Northeastern and a lead author on the study, pored through Elcoro’s symmetry tools to deduce the exhaustive classification of magnetic topological insulators, using a mix of mathematical theory and by-hand, brute-force calculations.

“Over the holidays in 2019, I would email Elcoro my attempted classification for a couple magnetic SGs each day,” remembers Wieder. “I spent most of that holiday break scribbling drafts of the classification between meals and dessert, much to the bewilderment of my friends and family.”

**Magnetic Topological Quantum Chemistry **

In collaboration with Barry Bradlyn, a physics professor at UIUC, the work of Elcoro and Wieder was then combined into a new theory, which they coined Magnetic Topological Quantum Chemistry (MTQC). MTQC is capable of characterizing all possible topological electronic bands in terms of their position-space chemistry and magnetic order. MTQC takes as input the positions and types of atoms in the crystal as well as the magnetic orientation, and outputs the set of allowed topological features. The foundation for MTQC was laid four years ago by members of the same collaboration in a seminal paper entitled *Topological Quantum Chemistry*.

Bradlyn, who was lead author on the original Topological Quantum Chemistry paper, notes, “MTQC answers some of the largest outstanding questions raised by our previous work. If we wanted to consider magnetism in a topological material, we would previously have had to start from scratch each time. By applying the same position-space tools we developed for Topological Quantum Chemistry, we now have a unified understanding of topological insulators in magnetic and nonmagnetic materials.”

**Materials simulation by numerical methods**

Building upon Elcoro and Wieder’s calculations, the team then turned to Zhida Song and Yuanfeng Xu to connect MTQC to numerically efficient symmetry and topological diagnoses of real magnetic materials.

Song, a postdoctoral researcher at Princeton University, is well known for his earlier work on numerical methods for the identification of topological insulators in materials calculations. For this study, Song performed theoretical calculations to link Wieder’s classification to Song’s earlier work on nonmagnetic materials.

Song sums up the outcome of the team’s multilayered efforts, “When the dust settled, we were sitting on the first-ever universal guide to magnetic topological insulators in real materials.”

In the final phase of work for this study, Xu, a postdoctoral researcher at the Max Planck Institute of Microstructure Physics, performed large-scale numerical simulations of theoretical models and real magnetic materials to validate the underlying theory. In addition to his efforts for the present work, Xu was also the lead author on an accompanying study published in *Nature* this past year, in which Xu and the other researchers applied MTQC to perform the first-ever high-throughput search for magnetic topological materials.

Andrei Bernevig, a professor at Princeton University and the principal investigator of both works, emphasized that “MTQC represents over four years of intense study by our collaboration.”

Given that the last two years of collaboration and writing on the two papers — over 400 pages combined — were accomplished remotely during the Covid-19 pandemic, Bernevig concluded: “it is a testament to the otherworldly dedication and focus of our team that we were able to persist and complete this longstanding problem.”

This work was funded by the US Department of Energy, the National Science Foundation, the Simons Foundation, the US Office of Naval Research, the Packard Foundation, the Schmidt Fund for Innovative Research, the US-Israel Binational Science Foundation, the Gordon and Betty Moore Foundation, the John Simon Guggenheim Memorial Foundation, the Government of the Basque Country, the Spanish Ministry of Science and Innovation, the European Research Council, the Max Planck Society, and the Alfred P. Sloan Foundation. The findings are those of the researchers and not necessarily those of the funding agencies.