Longstanding magnetic materials classification problem solved
New classifications have implications for quantum applications

Date:
October 13, 2021
Source:
University of the Basque Country
Summary:
For over 100 years, physicists, chemists, and materials scientists
have developed extensive theoretical and experimental machinery
to predict and characterize the electronic properties of magnetic
materials, but even the most successful classification system,
developed almost 75 years ago by Lev Shubnikov, was incomplete. An
international team of researchers announced this week that it has
finally been completed.



FULL STORY ========================================================================== 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.

========================================================================== Story Source: Materials provided by
University_of_the_Basque_Country. Note: Content may be edited for style
and length.


========================================================================== Journal Reference:
1. 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 ==========================================================================

Link to news story: https://www.sciencedaily.com/releases/2021/10/211013114052.htm

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