Symmetry-constrained coexistence of charge-three and charge-one Weyl points in nonmagnetic crystals

In the quantum world, electrons do not always behave like simple particles moving through a solid. In certain exotic materials, they can act as if they are massless, racing through the crystal lattice at speeds that mimic light. These materials are known as Weyl semimetals. Inside them, the energy bands of the electrons cross each other at specific points, creating isolated nodes. These nodes act like tiny magnets for a hidden property of the electron's motion called "topological charge." Just as a magnet has a north and south pole, these nodes have a charge that can be positive or negative. In most known materials, these charges are small, usually equal to one. However, physics allows for nodes with much larger charges, such as three. These high-charge nodes are rare and difficult to find because the rules of the crystal structure must be perfectly aligned to protect them. If the symmetry of the crystal is broken, these special nodes can split apart or disappear.

For a long time, scientists have struggled to find a clean, non-magnetic material that hosts these high-charge nodes without interference from other electronic states. The challenge is compounded by a fundamental law of physics: the total charge in a crystal must always sum to zero. This means that if a material contains a node with a charge of three, it must also contain other nodes with negative charges to balance it out. Until now, the specific arrangement of these balancing nodes alongside the high-charge ones remained largely unexplored. Researchers have not had a clear map of which crystal structures could support this delicate coexistence, nor have they found a real-world material that perfectly demonstrates this complex arrangement.

A team of researchers at Yanshan University in China has now filled this gap by combining theoretical symmetry analysis with computer simulations to design and identify such materials. They focused on a specific, complex configuration: a crystal containing two nodes with a charge of three, balanced by six nodes with a charge of negative one. This arrangement forms a stable, charge-neutral system. By analyzing every possible type of crystal symmetry, the team identified twelve specific crystal structures that could host this arrangement without the influence of spin-orbit coupling, a quantum effect that often complicates electronic behavior. When they included this effect, the number of possible structures rose to twenty-three. This systematic search provided a blueprint for finding these rare materials in nature.

Following this theoretical map, the researchers turned their attention to boron, a light element that is naturally weak in the quantum effects that usually obscure these delicate features. They discovered two distinct forms of boron that perfectly match their predictions. The first, called TBR-B48, is a structure made of buckled triangular ribbons. In this material, the high-charge nodes appear in a shape resembling an hourglass, where the energy bands twist and cross in a specific way. The second material, OBD-B48, is a chiral structure, meaning it exists in left-handed and right-handed versions, much like a pair of gloves. In this material, the high-charge nodes do not form an hourglass shape but instead take on a different, non-hourglass configuration. Both materials are predicted to be mechanically and dynamically stable, meaning they can exist as solid crystals without falling apart.

The researchers then simulated the behavior of electrons on the surface of these crystals. Because of the unique arrangement of the charges inside, the surface of the material should display distinct patterns of electron flow, known as Fermi arcs. For the material with the hourglass nodes, the simulation showed that when looking at the surface from one angle, six distinct paths of electrons would emerge from a single point, connecting to six other points on the surface. This is a direct consequence of the high charge of the internal nodes. When viewed from a different angle, these paths split into three separate groups. In the chiral boron material, the researchers found that flipping the crystal from its left-handed form to its right-handed form reversed the sign of every charge inside. The positive nodes became negative, and the negative became positive, yet the overall pattern of the electron paths remained the same, just mirrored. This finding establishes a direct link between the physical twist of the crystal and the twist of the electron's quantum properties.

To prove that these high-charge nodes rely entirely on the crystal's symmetry, the team simulated what would happen if they stretched the TBR-B48 crystal. When they pulled the crystal along a specific direction, they broke the six-fold rotational symmetry that was protecting the high-charge nodes. As predicted, the single node with a charge of three did not vanish; instead, it split apart into three separate nodes, each carrying a charge of one. The complex six-path electron pattern on the surface also reorganized itself into simpler patterns corresponding to these new, smaller charges. This experiment, performed in the computer, demonstrated that the high-charge state is not an accidental feature but a direct result of the crystal's geometric symmetry. If the symmetry is removed, the high-charge state dissolves into conventional ones.

This work provides a clear guide for the search of new quantum materials. It moves beyond simply looking for isolated high-charge nodes and instead focuses on the entire system of charges that must exist together to satisfy the laws of physics. By identifying specific boron allotropes that realize this complex configuration, the researchers have offered tangible targets for experimentalists to synthesize and study. The discovery that the chirality of the crystal can control the sign of the topological charge, and that strain can switch the material between high-charge and low-charge states, opens new avenues for manipulating quantum properties. These findings suggest that by carefully designing the symmetry of a crystal, scientists can engineer materials with exotic electronic behaviors that were previously thought to be too rare to find in nature.