Plasmon modes in three-dimensional multifold semimetals. II. Rarita-Schwinger-Weyl and Kramers-Weyl nodes

In the vast landscape of solid-state physics, materials are often categorized by how easily electrons can move through them. Insulators are like locked rooms where electrons are trapped; metals are open halls where they flow freely. Between these two extremes lie semimetals, a unique class of materials where the energy bands of electrons just barely touch, creating points where the density of available states drops to zero. For decades, scientists focused on the simplest of these touchpoints, known as Weyl nodes, which behave like single, isolated particles with a specific "handedness" or chirality. However, nature is rarely so simple. In recent years, researchers have discovered that in certain crystals with strong spin-orbit coupling—a quantum effect linking an electron's spin to its motion—these touchpoints can be more complex, involving multiple bands crossing at once. These are called multifold semimetals. Understanding how electrons behave collectively in these exotic environments is crucial, not just for fundamental physics, but because these collective behaviors dictate how the material responds to light, electricity, and magnetic fields, potentially unlocking new technologies for electronics and sensors.

The paper at hand dives deep into two specific, complex types of these multifold nodes: the Rarita-Schwinger-Weyl node and the Kramers-Weyl node. The researchers set out to map out the "collective charge excitations" within these materials. To understand this, imagine a crowd of people in a room. If you push one person, they bump into their neighbor, who bumps into the next, creating a ripple effect that travels through the entire crowd. In a metal, electrons do the same thing; when disturbed, they oscillate together in waves known as plasmons. The team wanted to know exactly how these waves form, how fast they travel, and how they fade away in these specific, high-degeneracy nodes. They focused on a regime where the material is "doped," meaning extra electrons have been added so that the Fermi level—the energy boundary between occupied and empty states—sits above the crossing point. This creates a situation where multiple concentric shells of electrons exist simultaneously, a feature that makes the physics significantly richer than in simpler materials.

Using a standard theoretical framework called the random-phase approximation, which treats the electrons as a fluid that responds to disturbances, the team calculated how the density of electrons would react to external probes. They looked at two distinct scenarios: one where the disturbance is gentle and spreads over a large area (small momentum transfer), and another where the probe is sharp and localized (arbitrary momentum transfer). For the Rarita-Schwinger-Weyl node, which features four energy bands and two distinct electron shells, the researchers found that the collective waves, or plasmons, are governed by the combined properties of both shells. The speed of these waves is determined by a weighted average of the speeds of electrons in each shell. Crucially, they discovered that these waves can travel without losing energy, but only within a specific range of frequencies and distances. If the wave tries to travel too far or at the wrong frequency, it crashes into the background of individual electron movements and dissipates, a process known as Landau damping.

The situation is even more intriguing for the Kramers-Weyl node. Here, the presence of a quadratic term in the energy equation breaks the symmetry between electrons and holes, creating two electron shells that behave differently. Unlike the first material, this system supports two separate, undamped plasmon modes at long wavelengths. One mode exists at low frequencies, similar to the first material, but a second, distinct mode appears at higher frequencies, sitting in a gap above the main background of electron activity. This second mode is possible because the range of energies where individual electrons can move is finite, creating a "window" where the collective wave can exist without interference. The researchers also calculated how these materials screen a static electric charge, such as a single impurity atom sitting inside the crystal. They found that the electric field around such an impurity does not fade away smoothly; instead, it oscillates, creating ripples in the electron density that extend far into the material. The rate at which these ripples fade is determined by the specific quantum geometry of the electron states, revealing that the Rarita-Schwinger-Weyl node produces ripples that decay more slowly than those in simpler systems, while the Kramers-Weyl node produces a pattern dominated by its inner electron shell.

By comparing these results with previous studies on simpler nodes, the author demonstrated that the collective behavior of electrons is not just a simple sum of the individual electron speeds. Instead, it is deeply encoded in the quantum mechanical "shape" of the electron states themselves. The study shows that even when two materials have similar energy levels, their collective responses can be vastly different if the underlying quantum states differ. This distinction allows scientists to use collective excitations as a tool to identify and characterize these complex nodes, distinguishing them from simpler counterparts. The work provides a complete theoretical map of how these exotic materials respond to charge disturbances, from the gentlest ripples to the sharpest probes, offering a clear picture of the hidden dynamics within these multifold semimetals.