Imagine a material one million times thinner than a millimetre, that does not conduct electricity through its interior but allows electricity to flow without resistance along its edges. Cut it into smaller pieces, and the same thing happens again: every new piece has insulating interiors but perfectly conducting edges. This remarkable behaviour is the hallmark of a two-dimensional topological insulator, making these materials particularly interesting for future electronics and quantum technologies.

Topological Quantum Chemistry (TQC) formalism deeply transformed the computational search for topological insulators and other topological materials in 2017, unlocking the characterization of thousands of candidates since then. Despite the theory’s broad success, the distinct symmetries (operations that preserve the arrangement of atoms in a crystal), encoded in two-dimensional (2D) materials have prevented a rigorous application of TQC to them.

Now, a paper from an international collaboration (titled “Two-dimensional topological quantum chemistry and catalog of topological materials” and published in Science on September 24) fills that gap and provides both the adaptation of TQC to explore the topology of any monolayer or multilayer material, plus a comprehensive survey of the electronic structure and topology of nearly 9,000 two-dimensional entries of computational material databases.

“Our new adaptation of TQC to 2D materials adds a new flavour to the framework, opening up a vast new landscape of quantum materials waiting to be explored.,” B. Andrei Bernevig, (Princeton professor of physics and DIPC Visiting Ikerbasque Research Professor) a coauthor of both studies, remarks. “In our work we have increased the catalog of topological or obstructed 2D materials by an order of magnitude approximately. This is fundamental because 2D materials can be combined, we can put many together, and occupy little, so are and will be a fundamental pillar of current and future electronics,” Urko Petralanda, an Assistant Professor at the Department of Applied Mathematics of the EHU, explains.

In a parallel work, titled “2D theoretically twistable material database” and published in Science on the same day, the authors identify materials from the first study whose electronic band structures make them promising building blocks for twisted bilayers with exotic properties, akin to those found in “magic-angle” graphene.

Reading the quantum properties of a layer

The study brings topological quantum chemistry into the realm of nonmagnetic two-dimensional materials. This framework links the chemical composition and crystal symmetries of a material to the topology of its electronic states.

Certain electronic patterns possess a remarkable robustness: their topology cannot be altered without fundamentally changing the underlying electronic structure. One example is a quantum spin Hall insulator, which can remain insulating in its interior while supporting current-carrying channels along its edges that are protected from particular types of perturbations. These robust states provide a platform for investigating quantum phenomena and have consequently inspired proposals for novel electronic devices.

The researchers analysed 8,872 entries from two computational materials databases and identified 4,073 with nontrivial topology or an obstructed atomic limit. The latter describes a different kind of unusual electronic organization: in one such class, symmetry constrains the centres of electronic charge to lie away from the atoms. Cutting the crystal at particular boundaries can then expose distinctive electronic states.

“The symmetry of a crystal gives us rules for how its electronic states can be arranged. We developed those rules for layers and turned them into tools that researchers can apply to thousands of materials,” Luis Elcoro, Professor at the Department of Physics of the EHU, explains.

Urko Petralanda (EHU) and Yi Jiang (DIPC) share first authorship of the study. Alongside the calculations, the team developed crystallographic tables, analysis programs and the Topological 2D Materials Database, distinguishing experimentally reported structures from computational candidates.

Crucially, the catalog provides electronic band structures—the energies available to electrons—for both topologically nontrivial and ordinary layers. With nearly 9,000 entries, it serves as a vast library of Lego-like building blocks: pairing two copies of the same layer produces a homobilayer, while combining different layers gives a heterobilayer.

“The catalog brings together a diverse array of calculated structural, electronic, crystallographic, and topological properties for every material,” Urko Petralanda says.

The researchers have selected several topological insulator and obstructed material examples and carried out large-scale simulations to probe edge states in experimentally realistic ribbons.

“Robust edge states persist with or without hydrogen passivation, underscoring the experimental feasibility of these topological states,” Yi Jiang adds.

Turning building blocks into quantum matter

The shared database, built by Nicolas Regnault (Princeton University), lets researchers compare the individual layers and select candidates for the physics they want to investigate. “In order to make the results accessible to the community, we have built a 2D materials database containing the calculated all properties for monolayers. The value of a catalog is what people can do with it. Researchers can compare the electronic structures, select candidates for the physics they want to study, and bring their own questions to the search,” Regnault explains.

“Like Lego building blocks, monolayers can be stacked and twisted to create entirely new platforms for exploring quantum phenomena. Our catalog lays the foundation for this expanding search," Bernevig concludes.

Additional information

The two studies are the work of a joint theoretical and experimental team that spans more than a dozen institutions. At Princeton are B. Andrei Bernevig (also DIPC and IKERBASQUE), Dumitru Călugăru (also the University of Oxford), Haoyu Hu (also the University of Science and Technology of China), Nicolas Regnault (also the Flatiron Institute and, in Paris, the École normale supérieure and the French National Centre for Scientific Research), Grigorii Skorupskii, Jiaze Xie and Leslie M. Schoop; at the Donostia International Physics Center (DIPC), Yi Jiang, Hanqi Pi, Garen Avedissian, Yongsong Wang, Miguel M. Ugeda (also IKERBASQUE) and Maia G. Vergniory (also the Université de Sherbrooke); at the University of the Basque Country, Urko Petralanda and Luis Elcoro; at the Max Planck Institute for the Structure and Dynamics of Matter in Hamburg, Angel Rubio (also the Flatiron Institute and the University of the Basque Country) and Lede Xian (also the Tsientang Institute for Advanced Study and the Songshan-Lake Materials Laboratory); at RWTH Aachen University, Dante M. Kennes (also the Max Planck Institute in Hamburg); at Sichuan Normal University, Qiaoling Xu (also the Tsientang Institute for Advanced Study); and at the University of Pennsylvania, Martin Claassen. The experimental work involves Peter Höhn, Vicky Haase and Claudia Felser at the Max Planck Institute for Chemical Physics of Solids in Dresden; Rose Albu Mustaf and Emilia Morosan at Rice University; Jiacheng Zhu, Dongyang Yang, Zuhan Geng, Jie Shan and Kin Fai Mak at Cornell University, with Shan and Mak also at the Kavli Institute at Cornell for Nanoscale Science and Yang, Geng, Shan and Mak at the Max Planck Institute in Hamburg; and Abdelmajid Ouahchi, Soumyajit Samal and Dmitri K. Efetov at Ludwig Maximilian University of Munich, with Ouahchi also at the Technical University of Munich and Samal and Efetov at the Munich Center for Quantum Science and Technology.