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 on Wednesday October 13th, 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.

## Reference:

Luis Elcoro, Benjamin J. Wieder, Zhida Song, Yuanfeng Xu, Barry Bradlyn, B.
Andrei Bernevig. Magnetic topological quantum chemistry. Nature
Communications, 2021; 12 (1)
DOI: 10.1038/s41467-021-26241-8