Anomaly-Induced Hybrid Bulk Electromagnetic Mode in Weyl Semimetals

Technical Summary: Anomaly-Induced Hybrid Bulk Electromagnetic Mode in Weyl Semimetals

Problem Statement
Weyl semimetals (WSMs) exhibit unconventional topological electromagnetic responses, including the chiral anomaly—the nonconservation of chiral charge in parallel electric and magnetic fields ($E \cdot B \neq 0$). While previous studies have explored collective modes in WSMs, such as conventional plasmons, helicons, and chiral zero sound, a gap remains in understanding the dynamical interplay between the chiral anomaly and collective charge oscillations. Specifically, earlier works often treated the chiral anomaly as a steady-state effect or neglected the explicit orientation of the anomaly-activating magnetic field relative to the wave-vector. Consequently, the existence of a bulk electromagnetic mode arising from the dynamic coupling between plasmonic charge oscillations and oscillating valley (chiral) imbalance has not been fully identified or characterized.

Methodology
The authors develop a semiclassical electromagnetic framework to treat the chiral charge density as a dynamical variable that oscillates in phase with the collective electron charge density. The study focuses on a doped WSM with broken time-reversal symmetry, characterized by a Weyl-node separation vector $\mathbf{b}$.

• Setup: The system is subjected to a weak static magnetic field $\mathbf{B}_0$ (confined to the $xy$-plane) and a time-dependent electromagnetic field with wave-vector $\mathbf{q}$. Two configurations are analyzed: parallel ($\mathbf{b} \parallel \mathbf{q}$) and perpendicular ($\mathbf{b} \perp \mathbf{q}$).
• Formalism: The authors solve coupled kinetic-Maxwell equations. The current density includes contributions from AC Drude conductivity, anomalous Hall-like currents (Berry curvature), diffusion, and the chiral magnetic current driven by $\mathbf{B}_0$.
• Dynamics: The continuity equation is modified to include the chiral anomaly source term ($E \cdot B$), allowing the valley-resolved electron densities to oscillate. The analysis focuses on the long-wavelength regime ($q \ll q_c$), where diffusion effects are negligible compared to internode scattering and anomaly-induced drift.
• Calculations: The dielectric response tensor is constructed to include anomaly-induced terms ($\lambda_{ij}$). The dispersion relations are determined by solving $\det[\tilde{M}(\omega, q)] = 0$, and the loss function $L(\omega, q) = -\Im[1/\epsilon(\omega, q)]$ is used to identify observable collective modes and their damping.

Key Contributions and Results
The paper identifies a previously unobserved anomaly-induced hybrid bulk electromagnetic mode. The key findings are:

1. Hybridization Mechanism: When the anomaly-induced chiral magnetic current has a component along the propagation direction ($\mathbf{B}_0 \cdot \mathbf{q} \neq 0$), oscillations of valley imbalance hybridize with plasmonic charge oscillations. This coupling generates a linearly dispersing mode ($\omega_{anom} = uq$) in the long-wavelength limit.
2. Orientation Dependence: The existence and character of this mode are strictly controlled by the angle $\theta$ between the magnetic field and the wave-vector.
   • Non-Perpendicular Configurations ($0 \le \theta < \pi/2$): A linearly dispersing gapless mode appears. In the perpendicular configuration, this mode hybridizes with a gapped bulk plasmon via an avoided crossing, creating a hybrid excitation. In the parallel configuration ($\theta=0$), the anomaly modifies the longitudinal plasmon sector, yielding a well-defined, weakly damped hybrid plasmon.
   • Perpendicular Configuration ($\theta = \pi/2$): When the magnetic field is transverse to the wave-vector, the anomaly-induced coupling vanishes. Consequently, the linear mode and the associated hybridization are absent.
3. Damping Characteristics: The damping of the hybrid mode is highly sensitive to the field orientation.
   • In the parallel configuration ($\theta=0$), the hybrid mode is weakly damped and appears as a sharp peak in the loss spectrum.
   • For oblique angles in the parallel configuration ($0 < \theta < \pi/2$) and in the perpendicular configuration (where the mode exists but does not hybridize in the same manner), the resulting excitations are strongly damped due to internode relaxation, often failing to appear as distinct peaks in the loss spectrum.
4. Distinction from Other Modes: The authors distinguish this mode from:
   • Chiral Magnetic Waves (CMW) and Chiral Zero Sound (CZS): Unlike these modes, which may remain decoupled from charge dynamics or rely on specific node pairings, the identified mode is an intrinsically electromagnetic hybrid arising from the coupling of charge and chiral imbalance oscillations.
   • Chiral Magnetic Plasmons: Previous works described gapped plasmon oscillations; this work identifies a gapless, linearly dispersing branch that hybridizes with bulk plasmons.
   • Surface Modes: This is a three-dimensional bulk excitation, distinct from Fermi-arc surface plasmons.

Significance and Claims
The paper claims to provide a direct theoretical signature for the chiral anomaly and its associated chiral magnetic effect in the collective excitation spectrum of Weyl semimetals. The primary significance lies in:

• Identification of a New Mode: The discovery of a hybrid bulk excitation that is absent in ordinary metals and distinct from previously known anomaly-related modes.
• Experimental Probe: The authors suggest that the hybrid mode, particularly its linear dispersion and specific damping behavior, offers observable features in electron energy-loss spectroscopy (EELS). The presence or absence of the mode depending on the relative orientation of $\mathbf{B}_0$ and $\mathbf{q}$ provides a clear experimental test for the dynamical nature of the chiral anomaly.
• Theoretical Framework: The work establishes a framework for treating the chiral anomaly dynamically, incorporating the orientation of the chiral magnetic response, which fundamentally alters the predicted electromagnetic response of WSMs.

The authors maintain a modest scope, noting that their semiclassical description is valid for magnetic fields up to a few Tesla and specific doping levels, and they do not propose specific experimental setups beyond the general applicability to EELS. They conclude by suggesting that similar hybrid modes might emerge in systems with band tilting where the curl of the tilt velocity acts as an effective magnetic field, though this remains a subject for future investigation.