Landau quantization of $C_3$ high-fold fermions in honeycomb lattices

In the world of solid-state physics, the way electrons move through a material determines everything from how well a wire conducts electricity to whether a device can process information at incredible speeds. For decades, scientists have been fascinated by a specific material called graphene, a single layer of carbon atoms arranged in a honeycomb pattern. In this material, electrons behave in a peculiar way: they act as if they have no mass, racing through the lattice at high speeds and creating unique energy patterns known as Dirac cones. When researchers place such materials into a strong magnetic field, the electrons are forced into specific, quantized energy levels called Landau levels. This phenomenon is a powerful tool for understanding the fundamental nature of electrons, but it usually follows predictable rules based on how the material is structured.

However, nature offers more complex arrangements than simple honeycombs. Scientists have recently begun exploring a new class of materials built from triangular clusters of carbon atoms, known as triangulenes, arranged in two-dimensional crystals. These structures are more intricate than graphene, hosting multiple electronic pathways at each site. The central question for researchers has been whether these complex, multi-orbital structures would simply mimic the behavior of graphene or if they would reveal entirely new physics. Specifically, they wanted to know how the unique geometry of these triangular crystals would alter the way electrons organize themselves under a magnetic field, particularly when the material contains both fast-moving electrons and others that seem to get stuck in flat, non-moving energy states.

A team of researchers at the International Iberian Nanotechnology Laboratory and the University of Aveiro set out to answer this by developing a theoretical framework for these "high-fold" fermions. They focused on a family of models where the number of electron orbitals at different sites in the crystal varies, creating a rich tapestry of energy bands. By combining two powerful computational approaches—one that simulates the entire crystal atom by atom and another that zooms in on the low-energy behavior—they mapped out exactly how these electrons respond to magnetic fields. Their work covers several specific configurations, including crystals where the triangular units are arranged in pairs of two, two and three, three and three, and four and four orbitals.

The researchers found that the answer is a fascinating mix of the familiar and the strange. In the simplest case, where the triangular units are paired in a specific way, the electrons behave almost exactly like those in graphene. They form the expected energy levels that scale in a predictable square-root pattern, confirming that the underlying physics of the massless, fast-moving electrons remains intact even in these more complex structures. However, as the complexity of the crystal increases, new and unexpected behaviors emerge. In crystals with three orbitals per site, the electrons still form the standard Dirac-like levels, but they also begin to interact with a different type of electronic state: a flat band where electrons have zero energy and do not move. This interaction causes the standard energy levels to deviate from the perfect graphene pattern, revealing that the electrons are no longer just racing freely but are being influenced by the crowded, multi-orbital environment.

The most striking discovery occurs in crystals with three and four orbitals per site, where the researchers identified a phenomenon known as a singular flat band. In these materials, a flat energy band touches a curved, parabolic band at a single point. When a magnetic field is applied, this specific geometry creates a dual response. One set of electrons follows the conventional rules, forming energy levels that spread out linearly with the magnetic field strength. But a second set of electrons, trapped in the singular flat band, behaves in a radically different way. Instead of spreading out, their energy levels crowd together extremely tightly, scaling inversely with the energy index. This means that as you look at higher energy states, the gap between them shrinks dramatically, a behavior that is fundamentally different from anything seen in standard materials.

The study also clarified how these electrons organize themselves in space. In the simpler graphene-like cases, the electrons form distinct, flat energy bands that correspond to the bulk of the material, while the edges of the crystal host different, flowing states. In the more complex crystals, the researchers observed that the flat bands associated with the singular points are not just static; they are "singular," meaning their quantum geometry is highly sensitive to direction. This sensitivity causes the electrons to exhibit an anomalous quantization that is rooted in the shape of their wave functions rather than just their speed. The team demonstrated that this behavior is robust across different magnetic field strengths and crystal sizes, suggesting it is an intrinsic property of these triangular lattices.

By comparing their results to known models like the Lieb lattice and the Kagome lattice, the authors confirmed that these triangulene crystals are not just variations of existing materials but represent a distinct class of quantum matter. They showed that the (2,3) configuration is essentially a triangular version of the square-lattice Lieb model, sharing its unique spin-1 Dirac crossing and flat band. Meanwhile, the (3,3) and (4,4) configurations introduce a new layer of complexity where two sets of energy bands are separated by a gap, yet still interact in ways that produce both standard and anomalous Landau levels. The researchers noted that in the (4,4) case, these anomalous levels can even penetrate the insulating gap between the bands, a feature that could lead to unusual magnetic responses even without a traditional flow of electricity.

Ultimately, this work provides a comprehensive map of how electrons behave in these high-fold fermion systems. It establishes that while the familiar Dirac physics of graphene persists, it is now intertwined with a new kind of quantum geometry driven by flat bands. The findings reveal that the nature of these materials is a combination of the relativistic, fast-moving behavior of Dirac fermions and an anomalous quantization rooted in the singular geometry of flat bands. This duality suggests that by tuning the number of orbitals in these triangular crystals, scientists can engineer materials with highly specific electronic properties, opening the door to a new understanding of quantum transport in two-dimensional systems. The study confirms that these materials are not merely theoretical curiosities but realizable systems, with potential applications in artificial lattices made from molecules or atoms on surfaces, where such precise control over electron behavior could be harnessed for future technologies.