🔬 mesoscale physics

Geometry contribution to sound attenuation in double-Weyl semimetals

This paper demonstrates that while simple Weyl semimetals exhibit sound attenuation via axial coupling to strain, double Weyl semimetals lack this mechanism and instead display a unique geometric contribution to sound attenuation arising solely from the strain-induced deformation of the Fermi surface.

Original authors: Varsha Subramanyan, Shi-Zeng Lin, Avadh Saxena

Published 2026-07-21

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CC BY 4.0

Original authors: Varsha Subramanyan, Shi-Zeng Lin, Avadh Saxena

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the rules of traffic aren't just about speed limits and stop signs, but about the very shape of the road itself. In the strange realm of quantum physics, there are materials called "Weyl semimetals" that act like a super-highway for electrons. Unlike normal metals where electrons bump into each other and get stuck, these electrons zip around in a way that is protected by the material's hidden geometry, almost like they are riding a magic slide. Scientists are obsessed with these materials because they might hold the keys to super-fast computers and new kinds of electronics. But to understand how they work, researchers often poke and prod them, for instance, by sending sound waves through them. Sound isn't just noise; in these materials, it's a rhythmic squeeze and stretch of the atomic lattice. The big question is: how does this squeezing change the flow of electrons, and how much energy does the sound wave lose as it travels? The answer depends on whether the material has "simple" Weyl points or "double" ones, a distinction that changes the physics entirely.

This paper dives into the specific case of "double-Weyl semimetals," a special type of material where the electron highways have a more complex, twisted shape. The authors, Varsha Subramanyan, Shi-Zeng Lin, and Avadh Saxena, set out to figure out exactly how sound waves get slowed down (attenuated) in these materials. They discovered that the usual explanation for sound loss in simpler materials doesn't work here at all. In simple Weyl semimetals, sound waves act like a magnetic force that pushes electrons in opposite directions, draining energy from the sound. However, the authors show that in double-Weyl semimetals, this "magnetic push" mechanism is completely absent. Instead, the sound wave loses energy because it physically reshapes the electron highway.

To visualize this, imagine the electrons in a double-Weyl semimetal are running on a perfectly round, circular track. When a sound wave passes through, it doesn't just push the runners; it squishes the track itself, turning the circle into an oval. The electrons have to adjust to this new, stretched shape. The paper argues that the energy lost by the sound wave comes entirely from this constant reshaping of the track. It's as if the sound wave is trying to run on a trampoline that keeps changing its shape under your feet; the effort to keep up with the changing geometry is what drains the energy. The authors calculated that this "geometric" effect is the only major source of sound loss in these specific materials, provided the sound wave is slow enough and the material is kept cold.

The researchers were very careful to rule out other possibilities. They explicitly showed that the "axial coupling"—the mechanism where sound acts like a magnetic field pushing electrons apart, which is the main culprit in simple Weyl semimetals—simply does not exist in double-Weyl semimetals. They proved that because of the material's symmetry, the sound wave cannot create that kind of magnetic-like effect. Instead, the only thing happening is the deformation of the electron's path. They also noted that if the sound wave were extremely fast or the magnetic field too strong, this geometric effect would vanish, and different rules would apply. But under the conditions they studied, the geometry is king.

Using a set of equations that describe how electrons move and scatter, the team estimated how much sound would be absorbed in a real-world candidate material, HgCr2Se4. They assumed a sound wave with a frequency of about 100 MHz (which is very high-pitched, in the ultrasonic range) and a speed of 1,000 meters per second. With typical values for how long electrons stay on their path before bumping into something, they calculated that the sound attenuation factor would be around 1.5 kHz. This number isn't just a random guess; it's a specific prediction based on their model of how the electron track gets squished and stretched.

The beauty of this finding is that it connects the abstract math of "quantum geometry" to something you can actually measure: the volume of a sound wave. The authors suggest that because the sound wave explicitly breaks the symmetry of the electron track, the amount of energy lost is a direct measure of the track's geometric shape. It's like listening to the sound of a rubber band being stretched; the pitch and volume tell you exactly how the rubber band is deforming. This work suggests that by listening to how sound fades in these materials, scientists might be able to "hear" the geometry of the quantum world, offering a new way to test and understand the exotic properties of these topological materials.

Technical Summary: Geometry Contribution to Sound Attenuation in Double-Weyl Semimetals

Problem Statement
The paper addresses the mechanism of sound attenuation (ultrasonic absorption) in Weyl semimetals (WSMs), specifically contrasting simple WSMs with multi-Weyl semimetals (specifically double-Weyl semimetals with winding number $|n|=2$ ). In simple WSMs, lattice strain couples to the Weyl nodes as an axial (pseudo-)gauge field, shifting the nodes in opposite directions in momentum space. This axial coupling generates an anomalous contribution to sound attenuation. However, in double-Weyl semimetals, the nodes carry a higher topological charge ( $|n|>1$ ) and are protected by crystalline rotational symmetry. The authors investigate how strain couples to these systems, noting that the standard axial coupling mechanism is absent, and seek to identify the alternative relaxation mechanisms governing sound attenuation.

Methodology
The authors employ a semiclassical transport approach based on the Boltzmann equation and chiral kinetic theory to calculate the rate of energy change ( $Q$ ) in the presence of a dynamic strain field (sound waves).

Hamiltonian Formulation: They utilize the low-energy effective Hamiltonian for double-Weyl semimetals, which includes quadratic band crossings in the plane transverse to the symmetry axis. They incorporate strain via the strain tensor $u_{ij}$ , treating it as a symmetry-breaking director field rather than a gauge field.
Energy Dissipation Calculation: The coefficient of energy attenuation, $\Gamma$ , is derived from the rate of energy change $Q = \langle \dot{E} \rangle$ . The authors evaluate the modified electron distribution function $f(E)$ by solving the Boltzmann equation, separating the distribution into an average part and a deviation ( $\delta f$ ).
Geometric Tensor Analysis: A central component of the methodology is the explicit inclusion of the quantum geometric tensor ( $g_{ij}$ ). The authors express the strain-induced energy modification ( $\delta E$ ) as a coupling between the spatial strain tensor and the momentum-space quantum geometric tensor.
Regime Analysis: The calculations are performed in the "slow sound" regime ( $\omega \tau^* \ll 1$ ) and at low temperatures ( $\mu \gg k_B T$ ), assuming weak magnetic fields where Landau quantization is negligible. The authors also consider limiting cases, such as the chemical potential approaching the nodes ( $\mu \to 0$ ) and the presence of donor impurities.

Key Contributions and Results

Absence of Axial Coupling: The authors demonstrate that in double-Weyl semimetals, the deformation potential terms are purely even under inversion due to the quadratic nature of the momentum terms ( $k_x^2 - k_y^2$ , etc.). Consequently, there are no odd (axial) terms analogous to $\lambda^o_{ij}$ that flip sign between nodes. This implies the anomalous relaxation mechanism dominant in simple WSMs vanishes entirely in double-Weyl systems.
Geometric Mechanism of Attenuation: The primary contribution to sound attenuation in double-Weyl semimetals is identified as a geometric contribution. Dynamic strain breaks the $C_4$ rotational symmetry, causing the Fermi surface to oscillate between an isotropic state and a nematic (anisotropic) state. This deformation of the Fermi surface geometry, mediated by the quantum geometric tensor, drives the energy dissipation.
Attenuation Coefficient: The derived expression for the attenuation coefficient $\Gamma$ in the relevant regime is:
$\Gamma \sim \frac{\tau^* \omega^2 \mu}{\rho v_s} \gamma^2 \frac{v_1}{v_2}$
where $\tau^*$ is the effective relaxation time, $\omega$ is the sound frequency, $\mu$ is the chemical potential, $\rho$ is mass density, $v_s$ is sound velocity, $\gamma$ is the Grüneisen parameter, and $v_1, v_2$ are velocity parameters of the Hamiltonian. Crucially, this result is independent of external magnetic fields to second order in strain and depends quadratically on the strain amplitude.
Regime Validity: The geometric contribution dominates when the chemical potential is finite ( $\mu \gg 0$ ) and the system is in the semiclassical regime. If $\mu \to 0$ , the Fermi surface area vanishes, and the geometric contribution disappears. In this limit, or under strong magnetic fields (lowest Landau levels), other mechanisms (such as $u_{zz}$ axial coupling or impurity scattering) become relevant, though the paper notes these contributions are generally smaller or require different treatment (e.g., Fermi's Golden Rule for impurity scattering).

Significance and Claims
The paper claims to identify a distinct physical mechanism for sound attenuation that is unique to multi-Weyl semimetals with even winding numbers. The significance lies in demonstrating that the quantum geometric tensor, which characterizes the geometry of the Fermi surface in momentum space, acts as a metric that couples to lattice deformations.

The authors argue that while simple WSMs rely on axial gauge-field-like couplings, double-Weyl semimetals rely entirely on the deformation of the Fermi surface geometry.
This establishes sound attenuation as a probe for the quantum geometric tensor, offering a way to measure momentum-space geometry through macroscopic transport properties.
The work provides a theoretical estimate for the attenuation factor in candidate materials like HgCr $_2$ Se $_4$ , predicting a magnitude on the order of 1.5 kHz under specific experimental conditions, thereby offering a potential signature for identifying these materials.

The authors maintain a modest scope, focusing strictly on the theoretical derivation of the geometric contribution and noting that the specific relaxation mechanism depends on the inversion symmetry properties of the deformation potential in materials with different winding numbers. They do not propose new experimental setups beyond the analysis of existing candidate materials but emphasize the fundamental difference in relaxation physics between simple and multi-Weyl systems.

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